# The steady states of strong-KPP reactions
in general domains

Henri Berestycki  HB: Centre d’analyse et de mathématique sociales, EHESS-
CNRS, 54 Boulevard Raspail, 75006, Paris, France  Senior Visiting Fellow,
Institute for Advanced Study, Hong Kong University of Science and Technology
<EMAIL_ADDRESS>and  Cole Graham  CG: Division of Applied Mathematics, Brown
University, 182 George St, Providence, RI 02906, USA<EMAIL_ADDRESS>
###### Abstract.

We study the uniqueness of steady states of strong-KPP reaction–diffusion
equations in general domains under various boundary conditions. We show that
positive bounded steady states are unique provided the domain satisfies a
certain spectral nondegeneracy condition. We also formulate a number of open
problems and conjectures.

##  1\. Overview and main results

We study the steady states of KPP-type reaction–diffusion equations in general
domains in $\mathbb{R}^{d}$. In particular, we consider the existence and
uniqueness of positive bounded solutions to semilinear elliptic equations of
the form

$\begin{cases}-\Delta u=f(u)&\text{in }\Omega,\\\ u=0&\text{on
}\partial\Omega\end{cases}$ (1.1)

in nonempty connected open domains $\Omega\subset\mathbb{R}^{d}$, which need
not be bounded. We also consider several other types of boundary conditions.
We state our assumptions on $f$ and $\Omega$ below.

### Motivation.

Reaction–diffusion equations describe a host of phenomena in diverse fields
including chemistry, ecology, and sociology. For instance, a solution $u$ of
(1.1) could model the population density of a species in a geographic region
$\Omega$. The reaction $f$ represents the growth rate of the species and the
Dirichlet boundary models a hostile border that kills individuals. A solutions
of (1.1) represents an equilibrium or steady state between these two competing
factors. The study of this system naturally begins with a fundamental
question: are such steady states unique? This question has proved to be
exceedingly rich.

The steady states of many homogeneous reaction–diffusion equations posed on
the whole space $\mathbb{R}^{d}$ are well understood; for more details, see
the discussion of related works below. Much less is known on other domains.
Here, we examine the influence of geometry on the solutions of (1.1). We study
homogeneous equations in subsets of $\mathbb{R}^{d}$ with Dirichlet, Robin,
and Neumann boundary conditions. As we shall see, absorbing boundary
conditions of Dirichlet or Robin type exhibit the most complex behavior.

We are further motivated by parabolic reaction–diffusion equations, which
represent population _dynamics_. The parabolic form of (1.1) describes the
propagation of the species throughout its environment, and an elliptic
solution represents a fixed point of these dynamics. Typically, stable steady
states are attractors of the parabolic equation at long times, and unstable
steady states separate domains of attraction. The classification of solutions
to the elliptic problem (1.1) is thus an essential element of a broader
understanding of the time-dependent dynamics.

In our context, several prior works have treated bounded domains; these serve
as important precursors for this paper. However, we expect the parabolic
dynamics to be of greater interest in _unbounded_ domains. On unbounded
domains, solutions have sufficient space to develop coherent long-time
patterns like asymptotic spreading speeds and traveling waves. We are
therefore keenly interested in the classification of steady states in the
unbounded case. A handful of prior works have considered _specific_ unbounded
domains including epigraphs, half-spaces, and cones. However, to our
knowledge, the nature of steady states on general unbounded domains is largely
open.

We aim to fill this gap. In this paper, we show that nontrivial steady states
of so-called strong-KPP reactions are unique under quite general conditions on
the domain and boundary data. Moreover, we demonstrate that the strong-KPP
hypothesis cannot be easily weakened: a slightly more general class of
reactions can support multiple nontrivial steady states. In a forthcoming
paper, we take up the study of more general reactions and obtain various
conditions on the domain that yield uniqueness. In view of our previous work
on the half-space [7], the present paper is the second in a sequence focused
on the steady states and long-time dynamics of reaction–diffusion equations in
the presence of boundary.

### Setup.

We now describe our objects of study in greater detail. To begin, we state our
hypotheses on the reaction $f$ and the domain $\Omega$.

We assume that the nonlinearity ${f\colon[0,\infty)\to\mathbb{R}}$ is
$\mathcal{C}^{1,\gamma}$ for some $\gamma\in(0,1]$ and satisfies
$f(0)=f(1)=0$. As a consequence, $0$ solves (1.1) and $1$ is a supersolution,
in the sense that it satisfies (1.1) with $\geq$ in place of equalities. We
also assume that $f|_{(1,\infty)}<0$, so that the reaction drives large values
down towards $1$. The maximum principle then implies that all positive bounded
solutions of (1.1) take values between $0$ and $1$. We are thus primarily
interested in the behavior of $f$ on $(0,1)$. We consider three nested classes
of reactions: _positive_ , _weak-KPP_ , and _strong-KPP_.

All our reactions will be _positive_ in the sense that:

  1. (P)

$f|_{(0,1)}>0$ and $f^{\prime}(0^{+})>0$.

Positive reactions are sometimes known as “monostable.” We use distinct
terminology to emphasize the positive derivative at $u=0$, which plays a
significant role in our analysis. In many sources, “monostability” only
denotes the first condition in (P).

Our _weak-KPP_ reactions are positive and additionally satisfy:

  1. (W)

$f(s)\leq f^{\prime}(0)s$ for all $s\in[0,1]$.

Finally, _strong-KPP_ reactions are weak-KPP and satisfy a form of strict
convexity:

  1. (S)

The map $s\mapsto\frac{f(s)}{s}$ is strictly decreasing on $(0,1]$.

Both (W) and (S) have been termed the “KPP condition” in recognition of the
pioneering work of Kolmogorov, Piskunov, and Petrovsky [20] on a similar class
of reactions. In many settings, the distinction is unimportant. For example,
weak and strong-KPP reactions exhibit identical propagation phenomena in the
whole space. The same holds in half-spaces [7] and cones [23].

However, the steady states of weak and strong-KPP reactions behave quite
differently in general domains. As we shall see, the strong KPP hypothesis (S)
ensures broad uniqueness for steady states. In marked contrast, both positive
and weak-KPP reactions can easily support multiple steady states. These
divergent behaviors lead us to carefully delineate between the hypotheses (W)
and (S). We devote most of our attention to strong-KPP reactions.

Throughout, we assume that the domain $\Omega$ is open, nonempty, connected,
and uniformly $\mathcal{C}^{2,\gamma}$ smooth. We make this precise in
Definition A.1 in the appendix. For now, we merely note that this hypothesis
includes a uniform interior ball condition.

In the remainder of the paper, we consider more general boundary conditions.
Let $\nu$ denote the outward normal vector field on $\partial\Omega$. Given
$\varrho\in[0,1]$, let
$\mathcal{N}_{\varrho}\colon\mathcal{C}^{1}\big{(}\overline{\Omega}\big{)}\to\mathcal{C}^{0}(\partial\Omega)$
denote the boundary operator

$\mathcal{N}_{\varrho}u\coloneqq\varrho\partial_{\nu}u+(1-\varrho)u|_{\partial\Omega}.$
(1.2)

Then we study a natural generalization of (1.3):

$\begin{cases}-\Delta u=f(u)&\text{in }\Omega,\\\
\mathcal{N}_{\varrho}u=0&\text{on }\partial\Omega.\end{cases}$ (1.3)

The values $\varrho=0$, $\varrho\in(0,1)$, and $\varrho=1$ correspond to
Dirichlet, Robin, and Neumann boundary conditions, respectively.

Even in its weak form, the KPP condition (W) ensures that the “rate of growth”
$f(s)/s$ is maximized at $s=0$, where $f$ resembles its linearization
$f^{\prime}(0)s$. KPP phenomena are thus often “linearly determined:” they can
be understood via the linearization of (1.3) about $0$. And indeed, much of
our analysis revolves around spectral properties of the linearization. We
express our results in terms of the generalized principal eigenvalue
introduced in [15] and applied to unbounded domains in [16].

Let $\mathcal{L}$ be an elliptic operator on $\Omega$. Then the _generalized
principal eigenvalue_ of $-\mathcal{L}$ on $\Omega$ with boundary parameter
$\varrho$ is given by

$\lambda(-\mathcal{L},\Omega,\varrho)\coloneqq\sup\big{\\{}\mu\in\mathbb{R}\mid\exists\,\phi\in
W_{\mathrm{loc}}^{2,d}(\Omega)_{+}\hskip 4.30554pt\text{s.t.}\hskip
4.30554pt\mathcal{N}_{\varrho}\phi\geq 0,\hskip
4.30554pt-\mathcal{L}\phi\geq\mu\phi\big{\\}}.$ (1.4)

Here $W_{\mathrm{loc}}^{2,d}(\Omega)_{+}$ denotes the set of positive
functions in $W_{\mathrm{loc}}^{2,d}(\Omega)$. This formulation with boundary
conditions first appeared in [25].

The definition (1.4) is closely related to the principal eigenvalue familiar
from bounded domains; see Proposition 2.1 below for details. We will work
almost exclusively with the Laplacian, so we make liberal use of the shorthand

$\lambda(\Omega,\varrho)\coloneqq\lambda(-\Delta,\Omega,\varrho).$

As we shall see, this eigenvalue is intimately linked with existence in (1.3).

First, however, we take up the question of uniqueness. In the strong-KPP case,
Rabinowitz [27] showed that (1.3) admits at most one positive solution on
bounded domains; the first author subsequently established the same in a more
general framework [3]. Thus any obstruction to uniqueness can only occur “at
infinity.” We therefore take some care with the structure of $\Omega$ at
infinity. Given a sequence of points $(x_{n})_{n\in\mathbb{N}}$ in $\Omega$,
we say that $\Omega^{*}$ is the _connected limit_ of $\Omega$ along
$(x_{n})_{n}$ if the sequence of domains $(\Omega-x_{n})_{n\in\mathbb{N}}$
converges in a suitable sense and $\Omega^{*}$ is the connected component of
the limit that contains the origin. For a precise definition, see Definitions
A.2 and 2.2. This terminology accounts for the fact that connected domains may
have disconnected limits; see Figure 1.

Figure 1. A horseshoe domain $\Omega$ such that $\lim_{n\to\infty}(\Omega-
x_{n})$ is disconnected. The sequence illustrated here selects the upper strip
as the connected limit $\Omega^{*}$. There are exactly two other
possibilities, up to translation: the lower strip and $\Omega$ itself, which
is the connected limit of all bounded sequences.

We next introduce the collection of eigenvalues on all connected limits.

######  Definition 1.1.

The _principal limit spectrum_ is

$\Sigma(\Omega,\varrho)\coloneqq\big{\\{}\lambda(\Omega^{*},\varrho)\mid\Omega^{*}\text{
is a connected limit of }\Omega\big{\\}}.$

We refer to the elements of $\Sigma$ as _(principal) limit eigenvalues_. We
denote the closure by $\overline{\Sigma}(\Omega,\rho)$.

### Results.

With these notions, we can state our main results for strong-KPP reactions. We
show that strong-KPP steady states are unique provided the principal limit
spectrum avoids the critical eigenvalue $f^{\prime}(0)$.

######  Theorem 1.1.

Suppose $f$ satisfies (S) and
$f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,\varrho)$. Then (1.3) has at
most one positive bounded solution.

In bounded domains, this follows from Corollary 4.1 in [27]. The argument in
[27] extends to unbounded domains provided $u$ does not vanish locally
uniformly along a subsequence in $\Omega$. However, if $\Omega$ has a limit
eigenvalue above $f^{\prime}(0)$, then every solution _does_ vanish along the
corresponding subsequence. On the other hand, we show that (1.3) satisfies the
maximum principle when $\lambda(\Omega,\varrho)>f^{\prime}(0)$. We can combine
these tools to recover uniqueness whenever the limit eigenvalues above
$f^{\prime}(0)$ can be cleanly separated from those below. To this end, we
prove the following geometric result that may be of independent interest. We
state the simpler Dirichlet case here; for the full form, see Theorem 4.3
below.

######  Theorem 1.2.

A domain $\Omega$ satisfies $f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,0)$
if and only if $\Omega$ can be written as the union of two uniformly
$\mathcal{C}^{2,\gamma}$ open sets $\Omega_{-}$ and $\Omega_{+}$ such that
$\Sigma(\Omega_{-},0)\subset[0,f^{\prime}(0))$ and
$\Sigma(\Omega_{+},0)\subset(f^{\prime}(0),\infty)$.

For a visualization of the decomposition in Theorem 1.2, see Figure 2. The
sets $\Omega_{-}$ and $\Omega_{+}$ typically overlap; thus the decomposition
is by no means unique. Moreover, the sets need not be connected and may be
empty.

Our construction of $\Omega_{+}$ makes essential use of a deep result of Lieb
concerning principal eigenvalues on intersections of open sets [21]. The
decomposition in Theorem 1.2 is also similar in spirit to ideas at the heart
of the moving plane method. Indeed, in that method, as presented in [14], one
decomposes $\Omega$ into two parts, one with certain compactness properties
and the other with a maximum principle. It seems possible that Theorem 1.2
could lead to further developments in this direction. We likewise anticipate
that this decomposition could bear fruit in the study of inhomogeneous
equations in the whole space.

Figure 2.  Illustrations of the decomposition in Theorem 1.2 with $\Omega_{-}$
in blue and $\Omega_{+}$ in red. Strictly speaking, the sets are smooth and
overlap; we omit these features for clarity. (a) An infinite sequence of large
balls joined by narrow tubes of varying lengths. (b) An infinite “comb” with
varying tooth width. As these examples show, both $\Omega_{+}$ and
$\Omega_{-}$ may be disconnected with infinitely many components.

At present, our methods cannot treat critical eigenvalues approaching or equal
to $f^{\prime}(0)$. Nonetheless, we conjecture that strong-KPP uniqueness
holds in general.

######  Conjecture 1.3.

In a general domain, if $f$ is strong-KPP, then (1.3) admits at most one
positive bounded solution.

To support this conjecture, we prove it in the special case of a bounded
domain augmented by a half-cylinder with a critical limit eigenvalue; see
Section 5 for details.

The strong KPP property (S) is essential to the uniqueness in Theorem 1.1. In
fact, under Dirichlet or Robin conditions, weak-KPP reactions can have
multiple positive solutions even on bounded domains.

######  Proposition 1.4.

For every bounded domain $\Omega$ and boundary parameter $\varrho\in[0,1)$,
there exists a reaction $f$ satisfying (W) such that (1.3) admits multiple
positive bounded solutions.

On the other hand, under Neumann conditions ($\varrho=1$), there is no
absorption in the system. Then we expect the reaction $f$ to push every
nontrivial steady state to its unique stable root, namely $1.$ The first
author, Hamel, and Nadirashvili confirmed this intuition in domains that
satisfy a certain geometric condition [10]. Using the same approach, Rossi
removed this constraint and thereby showed uniqueness for positive reactions
in general domains with Neumann boundary—see Corollary 2.8 in [28].

In the special case that $f$ is strong-KPP, Neumann uniqueness also follows
from our Theorem 1.1. Indeed, Proposition 2.5 below states that
$\lambda(\Omega,1)=0$ for any uniformly $\mathcal{C}^{2,\gamma}$ domain
$\Omega$. Hence $(\Omega,1)$ always satisfies the hypothesis of Theorem 1.1,
and the Neumann solution $1$ of (1.3) is always unique.

We also study existence in (1.3). In the following, we use the term _periodic_
to indicate that $\Omega$ is invariant under all translations in some full-
rank lattice.

######  Theorem 1.5.

If $f$ is positive in the sense of (P) and
$\lambda(\Omega,\varrho)<f^{\prime}(0),$ then (1.3) admits a positive bounded
solution. Conversely, if $f$ satisfies the weak KPP hypothesis (W) and
$\lambda(\Omega,\varrho)>f^{\prime}(0),$ then (1.3) has no positive bounded
solution. If, in addition, $f$ is strong-KPP in the sense of (S) and $\Omega$
is bounded or periodic, then (1.3) has no positive bounded solution when
$\lambda(\Omega,\varrho)=f^{\prime}(0).$

Together with Theorem 1.1, this completes the classification of bounded steady
states in periodic domains.

######  Corollary 1.6.

If $f$ is strong-KPP and $\Omega$ is periodic, then (1.3) has no positive
bounded solution when $\lambda(\Omega,\varrho)\geq f^{\prime}(0)$ and exactly
one such solution when $\lambda(\Omega,\varrho)<f^{\prime}(0)$.

Theorem 1.5 is analogous to existing results for variable-coefficient
operators in $\mathbb{R}^{d}$; see Theorem 1.3(1) and Theorem 6.1 in [12]. In
all these results, the critical case of equality is subtle and largely open.
If $\lambda(\Omega,\varrho)=f^{\prime}(0)$ and $f(s)\equiv f^{\prime}(0)s$ in
a nontrivial interval containing $0$, then (1.3) may admit positive bounded
solutions in the form of principal eigenfunctions. In this case, any small
multiple of the eigenfunction solves (1.3), so uniqueness fails. However, the
strong KPP hypothesis precludes this structure, and in bounded domains with
$\lambda(\Omega,\varrho)=f^{\prime}(0)$, there are no positive bounded
solutions. We expect unbounded critical domains to exhibit the same behavior:

######  Conjecture 1.7.

If $f$ satisfies (S) and $\lambda(\Omega,\varrho)=f^{\prime}(0),$ then (1.3)
has no positive bounded solution.

When (1.3) admits a unique positive bounded solution $u$, we expect nontrivial
solutions of the corresponding parabolic problem to converge to $u$ at large
times. Let $v\in\mathcal{C}^{2}([0,\infty)\times\Omega;\mathbb{R})$ solve

$\begin{cases}\partial_{t}v=\Delta v+f(v)&\text{in }\Omega,\\\
\mathcal{N}_{\varrho}v=0&\text{on }\partial\Omega\\\ \end{cases}$ (1.5)

with strong-KPP $f$. Then $v$ satisfies the so-called “hair-trigger effect.”

######  Proposition 1.8.

Suppose $f$ is positive in the sense of (P) and
$\lambda(\Omega,\varrho)<f^{\prime}(0)$. Then (1.3) admits a minimal positive
bounded solution $u$ such that every positive bounded solution $\tilde{u}$
satisfies $\tilde{u}\geq u$. Moreover, if $v$ solves (1.5) with nonnegative
bounded initial data $v(0,\,\cdot\,)\not\equiv 0$, then

$\liminf_{t\to\infty}v(t,\,\cdot\,)\geq u.$

Suppose, in addition, that $f$ is strong-KPP in the sense of (S) and
$f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,\varrho)$, so $u$ is the unique
positive bounded solution. Then

$v(t,\,\cdot\,)\to u$

locally uniformly in $\overline{\Omega}$ as $t\to\infty$.

This proposition raises the question of the asymptotic speed of propagation:
how rapidly does the disturbance spread through the domain? This matter was
addressed in [10] for KPP reactions with Neumann boundary conditions. The
nature of spreading in general domains with Dirichlet or Robin conditions
remains open.

### Related works.

As indicated earlier, our investigation is part of a very broad study of
reaction–diffusion dynamics. Here, we review a selection of works that pertain
to our own. Naturally, our discussion is by no means comprehensive.

#### Homogeneous problems

In this paper, we consider steady states in subsets of $\mathbb{R}^{d}$. In
$\mathbb{R}^{d}$ itself, $1$ is a solution due to the absence of boundary. For
positive reactions, the “hair-trigger effect” first observed by Aronson and
Weinberger [2] ensures that $1$ is the unique positive bounded steady state.
Other reaction classes can exhibit far more elaborate behavior. For instance,
steady states of bistable reactions on the whole space are the subject of the
celebrated de Giorgi conjecture [19]. See [29, 26] for landmark results in
this direction.

#### Bounded domains

Rabinowitz [27] showed uniqueness for equations of the form
$-\mathcal{L}u=f(u)$ for self-adjoint $\mathcal{L}$ and strong-KPP $f$ on
bounded domains. His argument pervades this paper. The first author later
extended uniqueness to non-self-adjoint operators [3]. We emphasize that in
such generality, the result is false in unbounded domains. To see this, recall
that KPP equations on the line admit traveling wave solutions that move with
positive speed. If we shift into a co-moving frame, such a wave becomes a
positive bounded steady-state in addition to $1$. Crucially, the shifted
operator includes a first-order term and is not self-adjoint. It seems likely
that some form of Theorem 1.1 remains true for more general self-adjoint
operators. For further discussion along this line, see [12].

#### Unbounded domains

The study of steady states on unbounded domains has largely focused on domains
with particular structure. In extensive collaboration with Caffarelli and
Nirenberg, the first author has examined qualitative properties of elliptic
solutions on half-spaces [4], cylinders [5], and epigraphs [6]. Although
uniqueness was not the principal focus of these works, [5] establishes
uniqueness for strong-KPP reactions on cylinders and [6] shows uniqueness for
positive reactions on uniformly Lipschitz epigraphs. We also highlight a
collaboration [16] of Rossi and the first author, which analyzes the linear
problem in general unbounded domains. We draw heavily on characterizations of
the generalized principal eigenvalue from [16].

Under Neumann boundary conditions, the classification of steady states is
somewhat simpler, as $1$ is a bounded solution of (1.3). Several prior works
have touched on Neumann uniqueness with weak-KPP [10] or positive reactions
[28]. As we discussed earlier, the former treats domains satisfying a certain
geometric constraint and the latter removes the constraint, thereby proving
uniqueness on general (possibly unbounded) domains.

#### Inhomogeneous equations

Boundary conditions are one way to model the effects of inhomogeneous
environments. Of course, one can also study inhomogeneous _equations_ on the
whole space. In collaboration with Hamel and Rossi, the first author has
studied the uniqueness of steady states in such systems [12]. This work is a
major inspiration for our own; it establishes uniqueness under certain
conditions on the generalized principal eigenvalues of limiting problems.
Their conditions are more restrictive than ours, however. In our language, the
main Theorem 1.3(2) of [12] is analogous to uniqueness when
$\Sigma(\Omega,\varrho)\subset[0,f^{\prime}(0)).$ Thus [12] does not discuss
systems with limit eigenvalues both above and below $f^{\prime}(0)$. It would
be interesting to see whether a more general condition like
$f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,\varrho)$ could ensure
uniqueness in the variable-coefficient setting.

#### Homogeneous evolution

As noted earlier, one can view solutions of (1.3) as steady states of time-
dependent reaction–diffusion equations. Transitions from one steady state to
another have featured prominently in the theory of such parabolic equations.
In the homogeneous KPP case, $0$ is unstable and $1$ is stable. Any initial
disturbance will trigger a transition from $0$ to $1$ that spreads in space at
the asymptotic speed of propagation introduced by Aronson and Weinberger [2].

#### Evolution in periodic media

The parabolic theory of inhomogeneous systems is dominated by the study of
periodic equations in periodic domains. In such systems, the asymptotic speed
of propagation may depend on the direction. These speeds were characterized in
increasing generality by Freidlin and Gärtner [18] and Weinberger [30]. Hamel,
Nadirashvili, and the first author have examined detailed properties of the
speeds for weak-KPP reactions [9]. In collaboration with Hamel and Roques
[11], the first author has studied steady states and spreading properties in
periodic media. In this case, the strong-KPP condition implies the uniqueness
of steady states.

#### General inhomogeneous evolution

The literature on aperiodic systems is comparatively sparse. In [10], Hamel,
Nadirashvili, and the first author consider the asymptotic speed of
propagation of KPP reactions in general domains with Neumann boundary data.
Under geometric conditions on the domain, they show that $1$ is the unique
steady state and that solutions propagate at the usual linearly-determined
speed in “free” directions that avoid the boundary. Because it treats Neumann
data, this work does not incorporate the boundary absorption that we emphasize
in the present paper.

A handful of results are known for specific domains with Dirichlet data. Luo
and Lu [23] have established the uniqueness of steady states and the
asymptotic speed of propagation for KPP reactions on cones. For analogous
results on half-spaces with a variety of boundary conditions and reactions,
see our prior work [7].

Hamel, Nadin, and the first author have studied general time-dependent
inhomogeneous reactions in the whole space [8]. When the reaction is strong-
KPP, they prove the uniqueness of a uniformly-positive entire solution that is
analogous to our steady states. In a further collaboration, Nadin and the
first author have extensively studied weak-KPP spreading in quite general
inhomogeneous media [13].

There is also a growing body of work on spreading in random media, which falls
within the broader theory of stochastic homogenization. For results in this
direction, we direct the reader to work of Lin and Zlatoš [22] and the
references therein.

### Organization.

The present paper is organized as follows. In Section 2, we establish
fundamental properties of the generalized principal eigenvalue
$\lambda(\Omega,\varrho)$. Drawing on these preliminaries, we prove a maximum
principle and Theorem 1.5 in Section 3. We devote Section 4 to the
decomposition in Theorem 1.2 and our uniqueness results Theorem 1.1 and
Proposition 1.4. Finally, we treat a special example with critical limit
eigenvalue in Section 5. An appendix puts our notions of smoothness and
convergence for domains on rigorous footing.

##  2\. Properties of the principal eigenvalue

In this section, we discuss the essential properties of the generalized
principal eigenvalue. We consider elliptic operators of the form

$\mathcal{L}\coloneqq\Delta+q(x)$ (2.1)

with a uniformly $\mathcal{C}^{\gamma}$ potential $q$. We let
$\mathcal{C}_{\mathrm{b}}^{\gamma}(\Omega)$ denotes the space of uniformly
$\mathcal{C}^{\gamma}$ functions on $\Omega$. This naturally generalizes to
$\mathcal{C}_{\mathrm{b}}^{k,\alpha}(\Omega)$.

In the introduction, we confined our attention to constant boundary
conditions. However, as we cut our domains into pieces, we may employ Robin
conditions on some parts of the boundary and Dirichlet conditions on others.
We encompass such mixed boundary conditions through variable boundary
parameters ${\sigma\colon\partial\Omega\to[0,1]}$. Unless otherwise stated, we
will assume that $\sigma$ is uniformly $\mathcal{C}^{2,\gamma}$. We often
abbreviate and refer to the pair $(\Omega,\sigma)$ as a uniformly
$\mathcal{C}^{2,\gamma}$ domain. We let $\mathcal{N}_{\sigma}$ denote the
corresponding mixed boundary operator as in (1.2). Then the definition (1.4)
of the generalized principal eigenvalue extends to mixed boundary conditions.
Throughout the paper, we use $\varrho$ and $\sigma$ to denote constant and
mixed parameters, respectively.

We often wish to compare $\lambda(\Omega,\sigma)$ with the principal
eigenvalue on a subset ${\Omega^{\prime}\subset\Omega}$. Under Dirichlet
conditions, this is immediate:
$\lambda(\Omega^{\prime},0)\geq\lambda(\Omega,0)$. Unfortunately, Robin and
Neumann conditions do not yield such monotonicity. Nonetheless, domain
monotonicity does hold if we employ suitable boundary conditions on
$\Omega^{\prime}$. To capture this phenomenon, we define a partial order on
pairs $(\Omega,\sigma)$ of domains and boundary parameters.

######  Definition 2.1.

We write $(\Omega^{\prime},\sigma^{\prime})\preceq(\Omega,\sigma)$ if
$\Omega^{\prime}\subset\Omega$, $\sigma^{\prime}\leq\sigma$ on the common
boundary $\partial\Omega^{\prime}\cap\partial\Omega$, and $\sigma^{\prime}=0$
on $\partial\Omega^{\prime}\cap\Omega$.

We shall see below that
$\lambda(\Omega^{\prime},\sigma^{\prime})\geq\lambda(\Omega,\sigma)$ when
$(\Omega^{\prime},\sigma^{\prime})\preceq\lambda(\Omega,\sigma)$. Note the
reversed order.

As a special case, we frequently consider the intersection between $\Omega$
and a ball $B$. Motivated by Definition 2.1, we impose $\sigma$-conditions on
the “original” boundary $\partial\Omega\cap B$ and Dirichlet conditions on the
“artificial” boundary $\Omega\cap\partial B$. We use the notation
$\lambda(-\mathcal{L},\Omega\mid B,\sigma)$ to denote the corresponding
generalized principal eigenvalue. Thus

$\lambda(-\mathcal{L},\Omega\mid
B,\sigma)\coloneqq\lambda(-\mathcal{L},\Omega\cap
B,\sigma\textup{{1}}_{\partial\Omega\cap B}).$ (2.2)

Clearly, $(\Omega\cap B,\sigma\textup{{1}}_{\partial\Omega\cap
B})\preceq(\Omega,\sigma)$ under Definition 2.1.

We observe that the domain $\Omega\cap B$ typically has corners and the mixed
boundary condition $\sigma\textup{{1}}_{\partial\Omega\cap B}$ is typically
discontinuous. This is our one routine exception to the assumption that
$\Omega$ and $\sigma$ are uniformly $\mathcal{C}^{2,\gamma}$. Because
$\Omega\cap B$ is bounded, the corresponding eigenvalue problem largely falls
within established work on low-regularity domains. We also note that
$\Omega\cap B$ may be disconnected. This is not a major hindrance—our
definition of $\lambda$ extends to disconnected domains. It is straightforward
to check that the generalized principal eigenvalue of a disconnected domain is
the infimum of the eigenvalues of the connected components. Also, we adopt the
convention that $\lambda=+\infty$ if the domain is empty.

We now turn to characterizations of the generalized principal eigenvalue.
Given $\sigma\colon\partial\Omega\to[0,1]$, let $\mathcal{H}_{\sigma}$ denote
the closure of
$\mathcal{C}_{0}^{\infty}\big{(}\overline{\Omega}\setminus\\{\sigma=0\\}\big{)}$
in $H^{1}(\Omega)$. Informally, $\mathcal{H}_{\sigma}$ is the set of $H^{1}$
functions that vanish on the “Dirichlet part” of $\partial\Omega$ where
$\sigma=0.$ This is the natural space of test functions for a variational
formulation of $\lambda$. In the following proposition, we collect various
fundamental properties of $\lambda$.

######  Proposition 2.1.

Let $(\Omega,\sigma)$ be a uniformly $\mathcal{C}^{2,\gamma}$ domain. Let
$\mathcal{L}$ be an elliptic operator of the form (2.1) with potential
$q\in\mathcal{C}_{\mathrm{b}}^{\gamma}(\Omega)$. Then the following hold.

  1. (i)

If $(\Omega^{\prime},\sigma^{\prime})\preceq(\Omega,\sigma)$, then
$\lambda(\Omega^{\prime},\sigma^{\prime})\geq\lambda(\Omega,\sigma)$.

  2. (ii)

Let $R_{\textnormal{in}}$ denote the (possibly infinite) inradius of $\Omega$.
Then

$-\sup q\leq\lambda(-\mathcal{L},\Omega,\sigma)\leq-\inf
q+\lambda(B_{1},0)R_{\textnormal{in}}^{-2}.$

  3. (iii)

For every $x\in\mathbb{R}^{d}$,

$\lambda(-\mathcal{L},\Omega,\sigma)=\lim_{R\to\infty}\lambda(-\mathcal{L},\Omega\mid
B_{R}(x),\sigma).$ (2.3)

  4. (iv)

There exists
$\varphi\in\mathcal{C}_{\mathrm{loc}}^{2,\gamma}\big{(}\overline{\Omega}\big{)}$
such that $\varphi>0$ and

$-\mathcal{L}\varphi=\lambda(-\mathcal{L},\Omega,\sigma)\varphi\quad\textrm{and}\quad\mathcal{N}_{\sigma}\varphi=0.$
(2.4)

  5. (v)

Recalling the space $\mathcal{H}_{\sigma}$ from above, we have

$\lambda(-\mathcal{L},\Omega,\sigma)=\inf_{\phi\in\mathcal{H}_{\sigma}\setminus\\{0\\}}\frac{\int_{\Omega}\left[\left|\nabla\phi\right|^{2}-q\phi^{2}\right]+\int_{\partial\Omega}\sigma^{-1}(1-\sigma)\phi^{2}}{\int_{\Omega}\phi^{2}}.$
(2.5)

  6. (vi)

If $\Omega$ is bounded, then $\lambda(-\mathcal{L},\Omega,\sigma)$ is the
classical principal eigenvalue of $-\mathcal{L}$ on $\Omega$ with boundary
parameter $\sigma$.

These properties are already known in the Dirichlet setting; see Proposition
2.3 of [16] and the references therein.

###### Proof.

(i). By Definition 2.1, a supersolution for $\lambda(\Omega,\varrho)$ in the
sense of (1.4) is also a supersolution for
$\lambda(\Omega^{\prime},\varrho^{\prime})$.

(ii). The constant supersolution $\phi=1$ in (1.4) yields the lower bound. For
the upper bound, let $B\subset\Omega$ be a ball of radius $R>0$. Using (1.4),
(i), and scaling, we find

$\lambda(-\mathcal{L},\Omega,\sigma)\leq\lambda(-\mathcal{L},B,0)\leq-\inf
q+\lambda(B,0)=-\inf q+\lambda(B_{1},0)R^{-2}.$

Varying over all inscribed balls, we obtain (ii).

(iii) and (iv). By (i), the sequence $\lambda(-\mathcal{L},\Omega\mid
B_{R}(x),\sigma)$ is decreasing in $R$ and

$\lambda(-\mathcal{L},\Omega,\sigma)\leq\lim_{R\to\infty}\lambda(-\mathcal{L},\Omega\mid
B_{R}(x),\sigma).$ (2.6)

In the Dirichlet case $\sigma\equiv 0$, [15] shows the existence of a
principal eigenfunction $\varphi_{R}>0$ corresponding to
$\lambda(-\mathcal{L},\Omega\mid B_{R}(x),\sigma)$, despite the possible
corners in $\Omega\cap B_{R}$. Similar arguments apply in the Robin case; we
omit the proof. Precisely, there exists $\varphi_{R}$ on $\Omega\cap B_{R}(x)$
such that

$-\mathcal{L}\varphi_{R}=\lambda(-\mathcal{L},\Omega\mid
B_{R}(x),\sigma)\varphi_{R}\quad\text{in }\Omega\cap B$

and

$\mathcal{N}_{\sigma\textup{{1}}_{\partial\Omega\cap
B_{R}(x)}}\varphi_{R}=0\quad\text{ almost everywhere on }\partial(\Omega\cap
B_{R}(x)).$

Assume without loss of generality that $0\in\Omega$ and
$R\geq\left|x\right|+1$ so that $0\in\Omega\cap B_{R}(x)$. We normalize
$\varphi_{R}$ by $\varphi_{R}(0)=1$. Fix $K\Subset\overline{\Omega}$ and
$R_{K}>0$ such that $K\subset B_{R_{K}-1}(x)$. Then Harnack and Schauder
estimates up to the boundary imply that the family $(\varphi_{R})_{R\geq
R_{K}}$ is uniformly $\mathcal{C}^{2,\gamma}$ smooth on $K$. Fix
$\alpha\in(0,\gamma)$. By Arzelà–Ascoli and diagonalization, we can extract a
subsequence of radii tending to infinity along which $\varphi_{R}$ converges
locally uniformly in $\mathcal{C}^{2,\alpha}$ to a limit
$\varphi\in\mathcal{C}^{2,\gamma}(\Omega)$. We can easily check that $\varphi$
is an eigenfunction in the sense of (2.4) with eigenvalue

$\lim_{R\to\infty}\lambda(-\mathcal{L},\Omega\mid B_{R}(x),\sigma).$

Since $\varphi(0)=1$ and $\varphi\geq 0$, the strong maximum principle implies
that $\varphi>0$ in $\Omega$. Using $\varphi$ as a supersolution in (1.4), we
see that

$\lambda(-\mathcal{L},\Omega,\sigma)\geq\lim_{R\to\infty}\lambda(-\mathcal{L},\Omega\mid
B_{R}(x),\sigma).$

In combination with (2.6), this proves (2.3) and (iv).

(v). The variational formulation (2.5) is classical when $\Omega$ is bounded.
The general form follows from (iii) and a standard approximation argument.

(vi). It is well known that the classical principal eigenvalue is given by the
Rayleigh quotient (2.5), so (vi) follows from (v). ∎

Many of our key arguments rely on various manipulations of the domain
$\Omega$. We often take limits or intersect domains with balls as in (2.2).
The next two results control
$\lambda(\Omega,\sigma)=\lambda(-\Delta,\Omega,\sigma)$ under these
operations.

The precise sense in which a sequence of domains converges to another is
technical and has a distinct flavor from the rest of the paper. We therefore
collect such details in Appendix A. Definition A.2 establishes our notion of
locally uniform $\mathcal{C}^{2,\alpha}$ convergence
$(\Omega_{n},\sigma_{n})\to(\Omega_{\infty},\sigma_{\infty})$. We supplement
this definition with a corresponding compactness result. Corollary A.2 ensures
that any uniformly $\mathcal{C}^{2,\gamma}$ sequence $(\Omega_{n},\sigma_{n})$
has a subsequence that converges in this sense (provided $\alpha<\gamma$).

We now turn to the notion of _connected_ limits. Given $x\in\mathbb{R}^{d}$,
define the translation operator

$\tau_{x}\Omega\coloneqq\Omega-x\quad\textrm{and}\quad\tau_{x}\sigma\coloneqq\sigma(\,\cdot\,+x).$

Then we define connected limits as follows.

######  Definition 2.2.

Let $(\Omega_{n},\sigma_{n})_{n\in\mathbb{N}}$ be a uniformly
$\mathcal{C}^{2,\gamma}$ sequence of domains such that
$0\in\overline{\Omega}_{n}$. We say that $(\Omega^{*},\sigma^{*})$ is the
_connected limit_ of $(\Omega_{n},\sigma_{n})_{n\in\mathbb{N}}$ if the
sequence converges locally uniformly in $\mathcal{C}^{2,\alpha}$ for some
$\alpha\in(0,\gamma)$ to a domain $(\Omega_{\infty},\sigma_{\infty})$,
$\Omega^{*}$ is the connected component of $\Omega_{\infty}$ such that
$0\in\overline{\Omega}^{*}$, and
$\sigma^{*}=\sigma_{\infty}|_{\partial\Omega^{*}}$. We also say that
$(\Omega^{*},\sigma^{*})$ is the connected limit of a single domain
$(\Omega,\sigma)$ _along_ $(x_{n})_{n\in\mathbb{N}}\subset\Omega$ if
$(\Omega^{*},\sigma^{*})$ is the connected limit of
$(\tau_{x_{n}}\Omega,\tau_{x_{n}}\sigma)_{n\in\mathbb{N}}$.

If $(x_{n})_{n\in\mathbb{N}}$ is _any_ sequence in $\Omega$, then the uniform
$\mathcal{C}^{2,\gamma}$-smoothness of $\Omega$ and Corollary A.2 ensure that
the domains $(\tau_{x_{n}}\Omega,\tau_{x_{n}}\sigma)_{n\in\mathbb{N}}$ have a
nonempty uniformly $\mathcal{C}^{2,\gamma}$ subsequential limit
$(\Omega^{*},\sigma^{*})$. We note that the uniform interior ball condition on
$\Omega$ prevents the limit from degenerating to $\emptyset$.

We extend the definition of the principal limit spectrum to
$\Sigma(\Omega,\sigma)$ in the natural manner. Note that every translate of
$(\Omega,\sigma)$ to the origin is a connected limit along a constant
sequence. Therefore

$\lambda(\Omega,\sigma)\in\Sigma(\Omega,\sigma).$ (2.7)

We now constrain the behavior of $\lambda$ under connected limits.

######  Lemma 2.2.

Suppose $(\Omega^{*},\sigma^{*})$ is the connected limit of a uniformly
$\mathcal{C}^{2,\gamma}$ sequence $(\Omega_{n},\sigma_{n})_{n\in\mathbb{N}}$.
Then

$\lambda(\Omega^{*},\sigma^{*})\geq\limsup_{n\to\infty}\lambda(\Omega_{n},\sigma_{n}).$
(2.8)

That is, the generalized principal eigenvalue is upper semicontinuous in
$(\Omega,\sigma)$. Lemma 2.2 is analogous to Proposition 5.3 in [12], and the
proof is similar. The proof also recalls that of Proposition 2.1(iii) above.

Figure 3. A narrowing strip.

######  Remark 2.1.

In general, this result cannot be improved—volume can “escape to infinity” and
vanish in the limit, leading to strict inequality in (2.8). For example,
consider the narrowing strip $\Omega$ in Figure 3. Using Proposition 2.1, one
can check that $\lambda(\Omega,0)=(\pi/L_{+})^{2}$. This principal eigenvalue
is determined by the half-cylinder on the left of width $L_{+}$.

Let us translate the domain steadily to the left via
${\Omega_{n}\coloneqq\Omega-(n,0)}$. The principal eigenvalue is translation-
invariant, so $\lambda(\Omega_{n},0)=(\pi/L_{+})^{2}$ for all
$n\in\mathbb{N}$. However, in the $n\to\infty$ limit we are left with a
narrower cylinder $\Omega^{*}$ of width $L_{-}<L_{+}$. This cylinder satisfies
$\lambda(\Omega_{*},0)=(\pi/L_{-})^{2}$, so

$\lambda(\Omega^{*},0)>\limsup_{n\to\infty}\lambda(\Omega_{n},0).$

Conversely, if the sequence $(\Omega_{n})_{n}$ of domains is uniformly
bounded, then volume cannot escape to infinity. In this case, equality holds
in (2.8). That is, $\lambda_{1}$ is continuous in $(\Omega,\sigma)$ when
$\Omega$ is uniformly bounded. This continuity holds in significantly weaker
topologies; see, for instance, [17].

###### Proof of Lemma 2.2.

By Definition A.2, the connected limit $\Omega^{*}$ must share a point in
common with all $\Omega_{n}$ for sufficiently large $n$. Dropping smaller
$n\in\mathbb{N}$ and shifting, we are free to assume that
$0\in\bigcap_{n}\Omega_{n}\cap\Omega^{*}$. By Proposition 2.1(iv), each
$\Omega_{n}$ admits a principal eigenfunction $\varphi_{n}>0$ such that
$\varphi_{n}(0)=1$. By Corollary A.2, we can extract a subsequence converging
to an eigenfunction $\varphi^{*}$ on $\Omega^{*}$ with eigenvalue

$\limsup_{n\to\infty}\lambda(\Omega_{n},\sigma_{n}).$

Since $\varphi^{*}(0)=1$ and $\varphi^{*}\geq 0$, the strong maximum principle
implies that $\varphi^{*}>0$. Using $\varphi^{*}$ as a supersolution in (1.4),
we see that

$\lambda(\Omega^{*},\sigma^{*})\geq\limsup_{n\to\infty}\lambda(\Omega_{n},\sigma_{n}).\qed$

Next, we establish a Robin version of a beautiful result of Lieb. The
following is an elementary extension of Theorem 1.1 in [21] to non-Dirichlet
boundary conditions.

######  Theorem 2.3.

Let $(\Omega,\sigma)$ be uniformly $\mathcal{C}^{2,\gamma}$. Then for all
$R>0$,

$\inf_{x\in\mathbb{R}^{d}}\lambda(\Omega\mid
B_{R}(x),\sigma)\leq\lambda(\Omega,\sigma)+\lambda(B_{1},0)R^{-2}.$

This is an essential tool in our analysis of the principal eigenvalue.
Informally, it states that $\lambda$ is local. To determine $\lambda$ to
accuracy $\varepsilon$, it suffices to examine $\Omega$ at scale
$\propto\varepsilon^{-2}$.

###### Proof.

Fix $\varepsilon>0$. By Proposition 2.1(v) and the definition of
$\mathcal{H}_{\sigma}$, there exist
$f\in\mathcal{C}_{0}^{\infty}\big{(}\overline{\Omega}\setminus\\{\sigma=0\\}\big{)}$
and $g\in\mathcal{C}_{0}^{\infty}(B_{R})$ with unit $L^{2}$ norms such that

$\int_{\Omega}\left|\nabla
f\right|^{2}+\int_{\partial\Omega}\sigma^{-1}(1-\sigma)f^{2}<\lambda(\Omega,\sigma)+\varepsilon/2$
(2.9)

and

$\int_{B_{R}}\left|\nabla g\right|^{2}<\lambda(B_{R},0)+\varepsilon/2.$ (2.10)

We extend $g$ by $0$ to $\mathbb{R}^{d}$ and define

$h_{x}(y)\coloneqq f(y)g(y-x)$

for $x\in\mathbb{R}^{d}$ and $y\in\Omega$. Then
$h_{x}\in\mathcal{H}_{\sigma}$. Next, we define

$T(x)\coloneqq\int_{\Omega}\left|\nabla
h_{x}\right|^{2}+\int_{\partial\Omega}\sigma^{-1}(1-\sigma)h_{x}^{2}\quad\textrm{and}\quad
D(x)\coloneqq\int_{\Omega}h_{x}^{2}.$

Fubini and our $L^{2}$-normalization of $f$ and $g$ imply that
$\int_{\mathbb{R}^{d}}D(x)=1$. For $T$, we can compute

$\left|\nabla h_{x}(y)\right|^{2}=\left|\nabla
f(y)\right|^{2}g(y-x)^{2}+f(y)^{2}\left|\nabla
g(y-x)\right|^{2}+\frac{1}{2}\nabla(f^{2})(y)\cdot\nabla(g^{2})(y-x).$

Writing $\nabla(g^{2})(y-x)$ as $-\nabla_{x}[g(y-x)^{2}]$, we see that the
final term vanishes when we integrate in $x$. Thus by Fubini,

$\displaystyle\int_{\mathbb{R}^{d}\times\Omega}\left|\nabla h_{x}\right|^{2}$
$\displaystyle=\int_{\mathbb{R}^{d}\times\Omega}\left[\left|\nabla
f(y)\right|^{2}g(y-x)^{2}+f(y)^{2}\left|\nabla g(y-x)\right|^{2}\right]$
(2.11) $\displaystyle=\int_{\Omega}\left|\nabla
f\right|^{2}+\int_{B_{R}}\left|\nabla g\right|^{2}.$

Also,

$\int_{\mathbb{R}^{d}\times\partial\Omega}\sigma^{-1}(1-\sigma)h_{x}^{2}=\int_{\partial\Omega}\sigma^{-1}(1-\sigma)f(y)^{2}.$
(2.12)

Combining (2.11) and (2.12) with (2.9) and (2.10), we see that

$\int_{\mathbb{R}^{d}}T(x)<\lambda(\Omega,\sigma)+\lambda(B_{R},0)+\varepsilon.$

Since $\int D=1$, we have

$\int_{\mathbb{R}^{d}}\left[T(x)-(\lambda(\Omega,\sigma)+\lambda(B_{R},0)+\varepsilon)D(x)\right]<0.$

Therefore

$(\lambda(\Omega,\sigma)+\lambda(B_{R},0)+\varepsilon)D(x)>T(x)$ (2.13)

on a set of positive measure. In particular, there exists $x\in\mathbb{R}^{d}$
satisfying (2.13). If we substitute $h_{x}$ into the Rayleigh quotient in
Proposition 2.1(v), we obtain $T(x)/D(x)$. Thus by our choice of $x$, we see
that

$\lambda(\Omega\mid
B_{R}(x),\sigma)\leq\lambda(\Omega,\sigma)+\lambda(B_{R},0)+\varepsilon=\lambda(\Omega,\sigma)+\lambda(B_{1},0)R^{-2}+\varepsilon.$

Since $\varepsilon>0$ was arbitrary, the theorem follows. ∎

Next, we establish the continuity of $\lambda$ in the boundary parameter. We
do not make use of this result, but it may be of independent interest. The
proof is a typical application of Lieb’s theorem.

######  Proposition 2.4.

Let $\Omega$ be uniformly $\mathcal{C}^{2,\gamma}$. Suppose
$(\sigma_{n})_{n\in\mathbb{N}}$ is a uniformly $\mathcal{C}^{2,\gamma}$
sequence of boundary parameters converging in $L^{\infty}(\partial\Omega)$ to
a $\mathcal{C}_{\mathrm{b}}^{2,\gamma}$ function $\sigma$. Then

$\lim_{n\to\infty}\lambda(\Omega,\sigma_{n})\to\lambda(\Omega,\sigma).$

###### Proof.

By Proposition 2.1(i), $\lambda(\Omega,\sigma)$ is a decreasing function of
$\sigma$. We are therefore free to assume that the sequence
$(\sigma_{n})_{n\in\mathbb{N}}$ is monotone. First suppose
$\sigma_{n}\nearrow\sigma$. Without loss of generality, we may assume that
$0\in\Omega$. By Proposition 2.1(iv), for each $n$, there exists a
$\sigma_{n}$-principal eigenfunction $\varphi_{n}$ with $\varphi_{n}(0)=1$.
Elliptic estimates allow us to extract a locally convergent subsequence with
limit $\varphi$, which is a positive eigenfunction with boundary parameter
$\sigma$ and eigenvalue

$\lim_{n\to\infty}\lambda(\Omega,\sigma_{n}).$

Using $\varphi$ as a supersolution in (1.4), we see that

$\lambda(\Omega,\sigma)\geq\lim_{n\to\infty}\lambda(\Omega,\sigma_{n}).$

The reverse inequality follows from $\sigma_{n}\nearrow\sigma$ and Proposition
2.1(i), so in fact

$\lambda(\Omega,\sigma)=\lim_{n\to\infty}\lambda(\Omega,\sigma_{n}).$

Now suppose $\sigma_{n}\searrow\sigma$. This case is more subtle. Fix
$\varepsilon>0$ and choose $R>0$ such that
$\lambda(B_{1},0)R^{-2}<\varepsilon$. For each $n\in\mathbb{N}$, Theorem 2.3
provides $x_{n}\in\mathbb{R}^{d}$ such that

$\lambda(\Omega\mid
B_{R}(x_{n}),\sigma_{n})\leq\lambda(\Omega,\sigma_{n})+\varepsilon.$ (2.14)

The point $x_{n}$ need not lie in $\Omega$, but it is no more than distance
$R$ away from $\Omega$, since $B_{R}(x_{n})\cap\Omega$ must be nonempty. Let
$y_{n}$ be a point in the component of $\Omega\cap B_{R}(x_{n})$ with minimal
eigenvalue. Taking $n\to\infty$, we can extract a nonempty connected limit
$(\Omega^{*},\sigma^{*})$ of $(\Omega,\sigma_{n})$ along
$(y_{n})_{n\in\mathbb{N}}$. The sequence $(x_{n}-y_{n})_{n}$ lies in $B_{R}$,
so we can extract a further limit point $z$ of $(x_{n}-y_{n})_{n}$. We note
that $B_{R}(z)\cap\Omega^{*}$ is nonempty, for otherwise $\lambda(\Omega\mid
B_{R}(x_{n}),\sigma_{n})$ would tend to infinity, contradicting (2.14).
Restricting to a subsequence, there exist positive eigenfunctions $\psi_{n}$
corresponding to $\lambda(\Omega\mid B_{R}(x_{n}),\sigma_{n})$ such that
$\tau_{y_{n}}\psi_{n}$ converges as $n\to\infty$ to a positive eigenfunction
$\psi^{*}$ for $(\Omega^{*}\mid B_{R}(z),\sigma^{*})$ with eigenvalue

$\lim_{n\to\infty}\lambda(\Omega\mid B_{R}(x_{n}),\sigma_{n}).$

Because $\Omega^{*}\cap B_{R}$ is bounded, every positive eigenfunction is
principal, so in fact

$\lambda(\Omega^{*}\mid B_{R}(z),\sigma)=\lim_{n\to\infty}\lambda(\Omega\mid
B_{R}(x_{n}),\sigma_{n}).$ (2.15)

Therefore Lemma 2.2, (2.14), and (2.15) yield

$\displaystyle\lambda(\Omega,\sigma)\leq\lambda(\Omega^{*},\sigma^{*})\leq\lambda(\Omega^{*}\mid
B_{R}(z),\sigma^{*})$ $\displaystyle=\lim_{n\to\infty}\lambda(\Omega\mid
B_{R}(x_{n}),\sigma_{n})$
$\displaystyle\leq\lim_{n\to\infty}\lambda(\Omega,\sigma_{n})+\varepsilon.$

Since $\varepsilon>0$ was arbitrary,

$\lambda(\Omega,\sigma)\leq\lim_{n\to\infty}\lambda(\Omega,\sigma_{n}).$

The reverse inequality follows from $\sigma_{n}\searrow\sigma$ and Proposition
2.1(i). ∎

To close the section, we show that the Neumann generalized principal
eigenvalue is zero. Our argument is similar to the proof of Theorem 2.3 in
[10].

######  Proposition 2.5.

If $\Omega$ is uniformly $\mathcal{C}^{2,\gamma}$, then $\lambda(\Omega,1)=0$.

###### Proof.

Without loss of generality, assume $0\in\Omega$. Given $R>1$, we abbreviate
$\Omega\cap B_{R}$ by $\Omega_{R}$. We control $\lambda(\Omega,1)$ via the
variational formula (2.5). Let $\operatorname{dist}_{\Omega}$ denote the
distance induced on the metric space $\Omega$ through its inclusion in
$\mathbb{R}^{d}$, and define the test function

$\psi_{R}(y)\coloneqq\min\big{\\{}\operatorname{dist}_{\Omega}(y,\partial\Omega_{R}\cap\Omega),1\big{\\}}.$

Then

$\displaystyle\left|\nabla\psi\right|$
$\displaystyle\leq\textup{{1}}_{\Omega_{R}\setminus\Omega_{R-1}}\quad$
$\displaystyle\text{in }\Omega_{R},$ (2.16) $\displaystyle\psi$
$\displaystyle=1$ $\displaystyle\text{in }\Omega_{R-1},$ (2.17)
$\displaystyle\psi$ $\displaystyle=0$ $\displaystyle\text{on
}\partial\Omega_{R}\cap\Omega.$

Recall the space of test functions $\mathcal{H}_{1}=H^{1}(\Omega)$ defined
before Proposition 2.1. Clearly, $\psi_{R}\in\mathcal{H}_{1}$. Using (2.16)
and (2.17), we find

$\int_{\Omega_{R}}\left|\nabla\psi_{R}\right|^{2}\leq\left|\Omega_{R}\setminus\Omega_{R-1}\right|\quad\textrm{and}\quad\int_{\Omega_{R}}\psi^{2}\geq\left|\Omega_{R-1}\right|.$

Therefore

$\lambda(\Omega,1)\leq\frac{\left|\Omega_{R}\setminus\Omega_{R-1}\right|}{\left|\Omega_{R-1}\right|}\quad\quad\textrm{for
all }R>1.$ (2.18)

Suppose for the sake of contradiction that there exists $\delta>0$ such that

$\frac{\left|\Omega_{R}\setminus\Omega_{R-1}\right|}{\left|\Omega_{R-1}\right|}>\delta\quad\textrm{for
all }R>1.$

Rearranging, we find

$\left|\Omega_{R}\right|\geq(1+\delta)\left|\Omega_{R-1}\right|\quad\textrm{for
all }R>1.$

That is, the volume of $\Omega_{R}$ grows exponentially as $R\to\infty$, which
contradicts $\Omega_{R}\subset B_{R}$. Therefore by Proposition 2.1(ii) and
(2.18),

$0\leq\lambda(\Omega,1)\leq\inf_{R>1}\frac{\left|\Omega_{R}\setminus\Omega_{R-1}\right|}{\left|\Omega_{R-1}\right|}=0.\qed$

##  3\. Existence

We now consider the existence of positive bounded solutions to (1.3). As noted
in the introduction, this problem is linearly determined: existence can be
characterized in terms of the generalized principal eigenvalue. Throughout the
section, we assume that $\Omega$ is a uniformly $\mathcal{C}^{2,\gamma}$
domain.

We begin by proving a maximum principle for the linear problem with variable
potential. This is a consequence of recent work of Nordmann [24, Theorem 1],
but for the reader’s convenience we provide a self-contained proof in our
setting. As in Section 2, let $\mathcal{L}\coloneqq\Delta+q(x)$ with
$q\in\mathcal{C}_{\mathrm{b}}^{\gamma}(\Omega)$.

######  Proposition 3.1.

Let $\sigma\colon\partial\Omega\to[0,1]$ be a uniformly
$\mathcal{C}^{2,\gamma}$ boundary parameter. Suppose
$\lambda(-\mathcal{L},\Omega,\sigma)>0.$ If
$w\in\mathcal{C}_{\mathrm{b}}^{2}(\Omega)\cap\mathcal{C}\big{(}\,\overline{\Omega}\,\big{)}$
satisfies

$-\mathcal{L}w\geq 0\quad\textrm{and}\quad\mathcal{N}_{\sigma}w\geq 0,$ (3.1)

then $w\geq 0.$

The proof presented here could be easily extended to a wider class of self-
adjoint operators. However, other methods are necessary when $\mathcal{L}$ is
not self-adjoint; see [16, Theorem 1.6(i)] for a pointwise approach.

###### Proof.

Define $\zeta(x)\coloneqq\operatorname{sech}\left|x\right|$ and
$\zeta_{\delta}(x)\coloneqq\zeta(\delta x)$ for $\delta>0$. Then

$\left|\nabla\zeta_{\delta}(x)\right|=\delta\zeta_{\delta}(x)\left|\tanh\left|\delta
x\right|\right|\leq\delta\zeta_{\delta}(x)\quad\textrm{for all
}x\in\mathbb{R}^{d}.$ (3.2)

Let $w_{-}\coloneqq\max\\{-w,0\\}$ denote the negative part of $w.$ Recall the
Hilbert space $\mathcal{H}_{\sigma}$ from Section 2. Because $\zeta_{\delta}$
decays exponentially, $w$ is bounded, and $\mathcal{N}_{\sigma}w\geq 0$, the
product $\zeta_{\delta}w_{-}$ lies in $\mathcal{H}_{\sigma}$. We make use of
the following identity, which is easily verified through repeated integrations
by parts:

$\int_{\Omega}\left|\nabla(fg)\right|^{2}=\int_{\partial\Omega}f^{2}g\partial_{\nu}g+\int_{\Omega}\left(g^{2}\left|\nabla
f\right|^{2}-f^{2}g\Delta g\right).$

We take $f=\zeta_{\delta}$ and $g=w_{-}$. Using (3.1) and (3.2), we find

$\displaystyle\int_{\Omega}\left|\nabla(\zeta_{\delta}w_{-})\right|^{2}$
$\displaystyle=\int_{\partial\Omega}\zeta_{\delta}^{2}w_{-}\partial_{\nu}w_{-}+\int_{\Omega}\left(-\zeta_{\delta}^{2}w_{-}\Delta
w_{-}+w_{-}^{2}\left|\nabla\zeta_{\delta}\right|^{2}\right)$ (3.3)
$\displaystyle\leq-\int_{\partial\Omega}\sigma^{-1}(1-\sigma)(\zeta_{\delta}w_{-})^{2}+\int_{\Omega}(q+\delta^{2})\left|\zeta_{\delta}w_{-}\right|^{2}.$

On the other hand, Proposition 2.1(v) states that

$\int_{\Omega}\left|\nabla\phi\right|^{2}\geq-\int_{\partial\Omega}\sigma^{-1}(1-\sigma)\phi^{2}+\int_{\Omega}\left[q+\lambda(-\mathcal{L},\Omega,\sigma)\right]\phi^{2}$
(3.4)

for all $\phi\in\mathcal{H}_{\sigma}$. Setting $\phi=\zeta_{\delta}w_{-}$,
(3.3) and (3.4) yield

$\int_{\Omega}\left[\delta^{2}-\lambda(-\mathcal{L},\Omega,\sigma)\right]\left|\zeta_{\delta}w_{-}\right|^{2}\geq
0.$

Taking $\delta^{2}\in\big{(}0,\lambda(-\mathcal{L},\Omega,\sigma)\big{)}$, we
conclude that $w_{-}\equiv 0$. That is, $w\geq 0$. ∎

With the maximum principle in hand, we can prove our main existence result.

###### Proof of Theorem 1.5.

Throughout, we fix $\varrho\in[0,1]$. Let $f$ be positive in the sense of (P)
and suppose $\lambda(\Omega,\varrho)<f^{\prime}(0)$. By Proposition 2.1(iii),
there exists a ball $B$ such that

$\mu\coloneqq\lambda(\Omega\mid B,\varrho)<f^{\prime}(0).$

Let $\Omega_{B}$ denote the connected component of $\Omega\cap B$ whose
eigenvalue is $\mu$. Let $\varphi$ denote the corresponding positive principal
eigenfunction normalized by ${\left\|\varphi\right\|_{L^{\infty}}=1}$. Since
$f^{\prime}$ is continuous, there exists $\varepsilon\in(0,1)$ such that

$\inf_{(0,\varepsilon]}\frac{f(s)}{s}\geq\mu.$

It follows that $\varepsilon\varphi$ is a subsolution of (1.3). We recall that
$1$ is a supersolution.

Let $\mathcal{P}_{t}$ denote the parabolic flow given by
$\mathcal{P}_{t}u=v(t,\,\cdot\,)$ where $v$ solves (1.5) with
$v(0,\,\cdot\,)=u$. By the comparison principle,
$\mathcal{P}_{t}(\varepsilon\varphi)$ is increasing in $t$ and
$\varepsilon\varphi\leq\mathcal{P}_{t}(\varepsilon\varphi)\leq 1$. Thus the
limit $u\coloneqq\mathcal{P}_{\infty}(\varepsilon\varphi)$ exists and
satisfies $\varepsilon\varphi\leq u\leq 1$. By standard parabolic estimates,
$u$ solves (1.3). Moreover, the strong maximum principle implies that $u>0$ in
$\Omega$. So $u$ is a positive bounded solution of (1.3).

Next, let $f$ satisfy the weak KPP hypothesis (W). Suppose
$\lambda(\Omega,\varrho)>f^{\prime}(0)$ and $u$ is a nonnegative bounded
solution of (1.3). Define $q\coloneqq f(u)/u$ under the convention that
$q=f^{\prime}(0)$ where $u=0$. Let $\mathcal{L}\coloneqq\Delta+q$. By the weak
KPP hypothesis (W), $q\leq f^{\prime}(0)$. Thus the definition (1.4) of
$\lambda$ implies that

$\lambda(-\mathcal{L},\Omega,\varrho)\geq\lambda(\Omega,\varrho)-f^{\prime}(0)>0.$

Because $u$ solves (1.3), $\mathcal{L}u=0$ and $\mathcal{N}_{\varrho}u=0$. By
standard elliptic estimates,
${u\in\mathcal{C}_{\mathrm{b}}^{2}(\Omega)\cap\mathcal{C}\big{(}\overline{\Omega}\big{)}}$.
Since $f$ is $\mathcal{C}^{1,\gamma}$ and $f(0)=0$,
$q\in\mathcal{C}_{\mathrm{b}}^{\gamma}(\Omega)$. Thus $u$ satisfies the
hypotheses of the maximum principle. By Proposition 3.1, $u=0$. Thus there is
no positive bounded solution of (1.3).

We now tackle the critical case $\lambda(\Omega,\varrho)=f^{\prime}(0)$ under
additional hypotheses. For the remainder of the proof, we assume that $f$ is
strong-KPP in the sense of (S). Suppose $\Omega$ is bounded and there exists a
positive bounded solution $u$ of (1.3). Then $u$ lies in the Hilbert space
$\mathcal{H}_{\varrho}$ from Section 2. We use it as a test function in the
Rayleigh quotient (2.5). By (S), $f(u)<f^{\prime}(0)u$. Integrating by parts,
we find:

$\int_{\Omega}\left|\nabla
u\right|^{2}+\int_{\partial\Omega}\varrho^{-1}(1-\varrho)u^{2}=\int_{\Omega}uf(u)<f^{\prime}(0)\int_{\Omega}u^{2}.$
(3.5)

Thus by Proposition 2.1(v), $\lambda(\Omega,\varrho)<f^{\prime}(0)$. In
particular, there is no positive bounded solution when
$\lambda(\Omega,\varrho)=f^{\prime}(0)$.

Now suppose $\Omega$ is periodic with lattice $\Lambda$ and there exists a
positive bounded solution $v$ of (1.3). As noted in the introduction, $0<v\leq
1$. Hence $u\coloneqq\mathcal{P}_{\infty}1$ solves (1.3) and satisfies
$0<v\leq u\leq 1$. Moreover, since $\Omega$ and $1$ are periodic, so is $u$.
That is, there exists a positive bounded _periodic_ solution $u$ of (1.3).

Let $\Gamma\coloneqq\Omega/\Lambda$, so that $\overline{\Gamma}$ is a compact
manifold with boundary $\partial\Gamma\coloneqq\partial\Omega/\Lambda$. By
Theorem 1.7 and Proposition 2.5 of [16], $\lambda(\Omega,\varrho)$ is the
principal eigenvalue on $\Gamma$ with boundary parameter $\varrho$. (Strictly
speaking, [16] only treats Dirichlet boundaries, but the Robin and Neumann
proofs are identical.) Let $\mathcal{H}_{\varrho}$ be $H_{0}^{1}(\Gamma)$ if
$\varrho=0$ and $H^{1}(\Gamma)$ otherwise. Then it is well known that the
principal eigenvalue on $\Gamma$ satisfies (2.5) with $\Gamma$ in place of
$\Omega$. Since $u$ is periodic, it can also be viewed as an element of
$\mathcal{H}_{\varrho}$. Arguing as in (3.5), we again find
$\lambda(\Omega,\varrho)<f^{\prime}(0)$. ∎

######  Remark 3.1.

We have not used the full strength of the strong KPP hypothesis in this
proof—we only required $f(u)<f^{\prime}(0)u$. We will use the full form of (S)
in the following section.

######  Remark 3.2.

Periodic domains are akin to periodic reactions $f(x,u)$ and diffusions
$\nabla\cdot[A(x)\nabla u]$ in the whole space. For an analogous existence
result for periodic equations, see Theorem 2.1 of [11].

As stated in Conjecture 1.7, we do not expect any critical domain with a
strong-KPP reaction to support a positive bounded solution. To motivate this
conjecture, we observe that we can rule out two extremes: solutions in
$L^{2}(\Omega)$ and solutions that do not vanish locally uniformly along any
subsequence. To see the first case, we note that (3.5) forbids all solutions
in $H^{1}$, and elliptic estimates ensure that $L^{2}$ solutions are
automatically $H^{1}$. For the second, one can employ a cutoff argument as in
the proof of Proposition 2.5 to derive a similar contradiction.

Thus the only alternative is a positive bounded solution of (1.3) that is not
square integrable but does decay along some subsequence. We are not presently
able to rule out this possibility, but it does tax the imagination.

##  4\. Uniqueness

We now come to the heart of the paper: the uniqueness of solutions to (1.3).
Throughout this section, we assume that $f$ is a strong-KPP reaction
satisfying (S) and that $\Omega$ is a uniformly $\mathcal{C}^{2,\gamma}$
domain with constant boundary parameter $\varrho\in[0,1]$.

###  4.1. Bounded domains

Rabinowitz [27] has previously proven uniqueness on bounded domains. We recall
the proof for the reader’s convenience. It serves as a blueprint for our
arguments in the unbounded case.

######  Theorem 4.1 ([27]).

If $\Omega$ is bounded and $\varrho=0$, then (1.3) admits at most one
solution.

As noted in the introduction, Rabinowitz in fact treats more general self-
adjoint elliptic operators, and the first author has shown that self-
adjointness is not necessary on bounded domains [3].

###### Proof.

We are free to assume there exists a positive bounded solution. Let $u_{1}$
and $u_{2}$ be two such solutions (not necessarily distinct). By the strong
maximum principle and the Hopf lemma, $u_{1}$ and $u_{2}$ are comparable in
the sense that

$C^{-1}u_{2}\leq u_{1}\leq Cu_{2}\quad\text{for some }C\in[1,\infty).$ (4.1)

Define

$\underline{\mu}\coloneqq\inf\left\\{\mu>0\mid u_{1}\leq\mu
u_{2}\enspace\text{in }\Omega\right\\},$

which is finite by (4.1).

We claim that $\underline{\mu}\leq 1$. Suppose for the sake of contradiction
that $\underline{\mu}>1$. Let $w\coloneqq\underline{\mu}u_{2}-u_{1}\geq 0$. By
the strong KPP property and $u_{2}>0$,

$-\Delta
w=\underline{\mu}f(u_{2})-f(u_{1})>f(\underline{\mu}u_{2})-f(u_{1})=qw$

for some $q\in L^{\infty}(\Omega)$. By the strong maximum principle, $w>0$.
Again, the strong maximum principle and the Hopf lemma imply that $w$ is
comparable to $u_{2}$. Thus there exists $\varepsilon>0$ such that
$w\geq\varepsilon u_{2}$. That is,
$u_{1}=\underline{\mu}u_{2}-w\leq(\underline{\mu}-\varepsilon)u_{2}$,
contradicting the definition of $\underline{\mu}$. So in fact $u_{1}\leq
u_{2}$, and $u_{1}=u_{2}$ by symmetry. ∎

This proof relies on the Hopf lemma, and thus on the interior ball property.
To our knowledge, uniqueness in less regular domains is open. Even if $\Omega$
is somewhat irregular, it seems plausible that all positive bounded solutions
of fairly general elliptic problems on $\Omega$ are comparable to one another.
Indeed, any such solution is likely comparable to the generalized principle
eigenfunction constructed in [15]. If this is case, the above proof of
uniqueness can be extended to irregular domains. We expect this to hold,
conservatively, in the case of Lipschitz boundary.

######  Conjecture 4.2.

If $f$ is strong-KPP, $\Omega$ is a bounded Lipschitz domain, and $\varrho=0$,
then (1.3) admits at most one solution.

###  4.2. Domain decomposition

The proof of Theorem 4.1 points to certain difficulties in unbounded domains.
When $\Omega$ is unbounded, positive solutions may vanish at infinity. If
distinct solutions vanish at different rates, they need not be comparable in
the sense of (4.1). It is reasonable to expect all solutions to vanish at
comparable rates, but it is not clear how to prove such a fact.

We sidestep this issue by decomposing $\Omega$ into two parts, $\Omega_{-}$
and $\Omega_{+}$. Roughly speaking, $\Omega_{-}$ is the region on which
solutions cannot vanish at infinity. Solutions do vanish at infinity in the
remainder $\Omega_{+}$, but it is sufficiently “narrow” that we can apply our
maximum principle. This additional tool substitutes for precise knowledge
about rates of decay and suffices to show uniqueness.

To capture these properties, we rely on the principal limit spectrum $\Sigma$
defined in the introduction. In the following, $\Omega_{\pm}$ is a uniformly
$\mathcal{C}^{2,\gamma}$ open set that need not be connected (and may be
empty) and $\sigma_{\pm}$ is a uniformly $\mathcal{C}^{2,\gamma}$ boundary
parameter on $\Omega_{\pm}$. These notions of regularity are made precise in
Definition A.1.

######  Definition 4.1.

Fix $\mu>0$. A pair $(\Omega_{-},\sigma_{-})$ is $\mu$-_ample_ if
$\Sigma(\Omega_{-},\sigma_{-})\subset[0,\mu)$. A pair
$(\Omega_{+},\sigma_{+})$ is $\mu$-_narrow_ if
$\Sigma(\Omega_{+},\sigma_{+})\subset(\mu,\infty)$ or, equivalently,
$\lambda(\Omega_{+},\sigma_{+})>\mu$. We say
$(\Omega_{-},\Omega_{+},\sigma_{-},\sigma_{+})$ is a $\mu$-_ample–narrow
decomposition_ of $(\Omega,\varrho)$ if the following hold:
$(\Omega_{\pm},\sigma_{\pm})\preceq(\Omega,\varrho)$,
$\Omega=\Omega_{-}\cup\Omega_{+}$, $(\Omega_{-},\sigma_{-})$ is $\mu$-ample,
$(\Omega_{+},\sigma_{+})$ is $\mu$-narrow, and
$\max\\{\sigma_{+},\sigma_{-}\\}=\varrho$ on $\partial\Omega$, where we extend
$\sigma_{\pm}$ by $0$ to $\partial\Omega$.

Conceptually, $\lambda(\Omega,\sigma)>\mu$ if the absorption from the
$\sigma$-boundary is stronger than rate-$\mu$ growth in the interior. This
means that every point in $\Omega$ is relatively close to $\partial\Omega$;
that is, the domain is “narrow.” Conversely, if
$\lambda(\Omega^{*},\sigma^{*})<\mu$ for every connected limit
$(\Omega^{*},\sigma^{*})$, then $(\Omega,\sigma)$ must be sufficiently
capacious even at infinity. We express this notion through the word “ample.”

We wish to prove uniqueness on domains $\Omega$ that are _noncritical_ in the
sense that $f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,\varrho)$. However,
this condition on limit eigenvalues is difficult to use directly. Instead, we
prove a geometric result relating this condition to ample and narrow sets.

######  Theorem 4.3.

Fix $\mu>0.$ Then $(\Omega,\varrho)$ admits a $\mu$-ample–narrow decomposition
if and only if $\mu\not\in\overline{\Sigma}(\Omega,\varrho)$.

The ample–narrow decomposition is far from unique. Roughly speaking, we can
augment $\Omega_{-}$ and curtail $\Omega_{+}$ by any bounded region and still
preserve the decomposition.

We will make use of one technical operation. Given open $U\subset\Omega$, we
define the nested sets

$U_{r}\coloneqq\big{\\{}x\in
U\mid\operatorname{dist}_{\Omega}(x,\Omega\setminus
U)>r\big{\\}}\quad\text{for }r>0.$ (4.2)

We wish to “cut off” $\Omega$ so that it lies within $U$ but contains $U_{1}$.

######  Definition 4.2.

Let $(\Omega,\sigma)$ be a uniformly $\mathcal{C}^{2,\gamma}$ domain and let
$U\subset\Omega$ be open. Then $(\Omega_{U},\sigma_{U})$ is a _truncation_ of
$(\Omega,\sigma)$ in $U$ if it is a uniformly $\mathcal{C}^{2,\gamma}$ open
set such that $U_{1}\subset\Omega_{U}\subset U$,
$(\Omega_{U},\sigma_{U})\preceq(\Omega,\sigma)$, and $\sigma_{U}=\sigma$ on
$\partial\Omega\cap\overline{U}_{1}$.

######  Lemma 4.4.

If $(\Omega,\sigma)$ is uniformly $\mathcal{C}^{2,\gamma}$ and
$U\subset\Omega$ is open, there exists a truncation of $(\Omega,\sigma)$ in
$U$ in the sense of Definition 4.2.

This lemma comes as no surprise, but for lack of a clean reference, we provide
a proof.

###### Proof.

We first construct a uniformly smooth set $\Omega_{U}\subset\Omega$ such that

$U_{3/4}\subset\Omega_{U}\subset U_{1/4}.$ (4.3)

Define $f(x)\coloneqq\operatorname{dist}_{\Omega}(x,\Omega\setminus U)$ and
$g(x)\coloneqq\min\\{\max\\{f(x),\,1/3\\},\,2/3\\}$. Since $\Omega$ is
uniformly smooth,

$\operatorname{dist}_{\mathbb{R}^{d}}\big{(}f^{-1}(1/3,2/3),f^{-1}(1/4,3/4)^{c}\big{)}>0.$

It follows that there is a mollification $h$ of $g$ such that

$\operatorname{dist}_{\Omega}\big{(}h^{-1}(1/3,2/3),f^{-1}(1/4,3/4)^{c}\big{)}>0.$
(4.4)

Sard’s theorem provides $c\in(1/3,2/3)$ such that $h^{-1}(c)$ is a (locally)
smooth manifold, perhaps with corners where the level set meets
$\partial\Omega$. Let ${V\coloneqq h^{-1}(c,1]}$, which satisfies
$U_{3/4}\subset V\subset U_{1/4}$. In fact, (4.4) implies that $\partial V$ is
uniformly separated from both $U_{1/4}$ and $U_{3/4}^{c}$. We therefore have
space to uniformly smooth $\partial V$ and make $\partial V$ smoothly join
$\partial\Omega$ while still remaining in the strip
$U_{1/4}\setminus\overline{U}_{3/4}$. Let $\Omega_{U}$ be this uniformly
smooth modification, which satisfies (4.3).

Next, let $Z\coloneqq\overline{U}_{3/4}\setminus(\overline{\Omega\setminus
U_{3/4}})$ and $W\coloneqq(\Omega\setminus U_{1})\setminus\overline{U}_{1}$.
Then $\\{Z,W\\}$ is an open cover of $\overline{\Omega}$ viewed as a manifold
with boundary. Let $\\{\psi,\phi\\}$ be a smooth partition of unity on
$\overline{\Omega}$ adapted to this cover. Because
$\operatorname{dist}_{\Omega}(Z,\overline{\Omega}\setminus W)>0$, we can
arrange $\psi$ (and $\phi$) to be uniformly $\mathcal{C}^{2,\gamma}$.

Finally, let $\sigma_{U}\coloneqq\psi\sigma\textup{{1}}_{\partial\Omega}$ on
$\partial\Omega_{U}$. Because $\psi$ is uniformly smooth and supported in
$U_{3/4}$, $\sigma_{U}$ is uniformly smooth and
$(\Omega_{U},\sigma_{U})\preceq(\Omega,\sigma)$. Moreover, $\psi\equiv 1$ on
$U_{1}$, so $\sigma_{U}=\sigma$ on $\partial\Omega\cap\overline{U}_{1}$. Hence
$(\Omega_{U},\sigma_{U})$ is a truncation of $(\Omega,\sigma)$ in $U$. ∎

We now prove our main geometric result.

###### Proof of Proposition 4.3.

One direction is straightforward. Suppose $(\Omega,\varrho)$ admits a
$\mu$-ample–narrow decomposition
$(\Omega_{-},\Omega_{+},\sigma_{-},\sigma_{+})$. Let $\Omega^{*}$ be a
connected limit of $\Omega$ along a sequence
$(x_{n})_{n\in\mathbb{N}}\subset\Omega$, which necessarily inherits the
boundary parameter $\varrho$. Then $\Omega^{*}$ is a connected component of
the limit of the translates $\Omega-x_{n}$. Recall that
$\operatorname{dist}_{\Omega}$ denotes the distance induced on the metric
space $\Omega$ through its inclusion in $\mathbb{R}^{d}$. If
$\operatorname{dist}_{\Omega}(x_{n},\Omega_{-})\to\infty$ as $n\to\infty$,
then $(\Omega^{*},\varrho)$ is a connected limit of $(\Omega_{+},\sigma_{+})$
alone. Recall that $(\Omega_{+},\sigma_{+})$ is $\mu$-narrow. By (2.7), Lemma
2.2, and Definition 4.1,

$\lambda(\Omega^{*},\varrho)\geq\lambda(\Omega_{+},\sigma_{+})>\mu.$

Next, suppose a subsequence of
$\operatorname{dist}_{\Omega}(x_{n},\Omega_{-})$ remains bounded as
$n\to\infty$. Restricting to a further subsequence, we can extract a connected
limit $(\Omega_{-}^{*},\sigma_{-}^{*})$ of $(\Omega_{-},\sigma_{-})$. It is
straightforward to check that the partial order $\preceq$ is preserved under
connected limits, so by Definition 4.1,
$(\Omega_{-}^{*},\sigma_{-}^{*})\preceq(\Omega^{*},\varrho)$. By Proposition
2.1(i),

$\lambda(\Omega^{*},\varrho)\leq\lambda(\Omega_{-}^{*},\sigma_{-}^{*})\leq\sup\Sigma(\Omega_{-},\sigma_{-}).$

Because $(\Omega_{-},\sigma_{-})$ is $\mu$-ample,
$\sup\Sigma(\Omega_{-},\sigma_{-})\leq\mu$. We claim that the inequality is
strict. To see this, suppose otherwise. Then there exists a sequence of
connected limits $(\Omega_{-}^{n},\sigma_{-}^{n})$ of
$(\Omega_{-},\sigma_{-})$ such that

$\lambda(\Omega_{-}^{n},\sigma_{-}^{n})\nearrow\mu\quad\text{as }n\to\infty.$
(4.5)

For each $n\in\mathbb{N}$, $(\Omega_{-}^{n},\sigma_{-}^{n})$ is constructed
from a sequence of translates

$(\tau_{x_{n,m}}\Omega_{-},\tau_{x_{n,m}}\sigma_{-})_{m\in\mathbb{N}}.$

Restricting to a subsequence of $n$, we can assume that the sequence
$(\Omega_{-}^{n})_{n\in\mathbb{N}}$ has its own connected limit
$(\Omega_{-}^{\infty},\sigma_{-}^{\infty})$. By Lemma 2.2 and (4.5),

$\lambda(\Omega_{-}^{\infty},\sigma_{-}^{\infty})\geq\mu.$ (4.6)

However, we can diagonalize along the double sequence
$(\tau_{x_{n,m}}\Omega_{-},\tau_{x_{n,m}}\sigma_{-})_{n,m\in\mathbb{N}}$ to
see that $(\Omega_{-}^{\infty},\sigma_{-}^{\infty})$ is itself a connected
limit of $(\Omega_{-},\sigma_{-})$. Thus (4.6) contradicts the $\mu$-ampleness
of $(\Omega_{-},\sigma_{-})$. It follows that
$\sup\Sigma(\Omega_{-},\sigma_{-})<\mu$, as claimed.

If $\Omega^{*}$ is a connected limit of $\Omega$, we have now shown that one
of two alternatives holds:

$\lambda(\Omega^{*},\varrho)\geq\lambda(\Omega_{+},\sigma_{+})>\mu$

or

$\lambda(\Omega^{*},\varrho)\leq\sup\Sigma(\Omega_{-},\sigma_{-})<\mu.$

It follows that $\mu\not\in\overline{\Sigma}(\Omega,\sigma)$.

We now tackle the reverse direction. Assume
$\mu\not\in\overline{\Sigma}(\Omega,\varrho)$, so there exists $\delta>0$ such
that

$\Sigma(\Omega,\varrho)\cap(\mu-2\delta,\mu+2\delta)=\emptyset.$ (4.7)

We wish to show that $(\Omega,\varrho)$ admits a $\mu$-ample–narrow
decomposition. We first construct the narrow part $(\Omega_{+},\sigma_{+})$.
Fix $R>0$ such that

$\lambda(B_{1},0)R^{-2}<\delta.$ (4.8)

In the following, let $\operatorname{comp}_{x}U$ denote the connected
component of a domain $U$ containing $x$. Let
$\sigma\coloneqq\varrho\textup{{1}}_{\partial\Omega}$. We use the boundary
parameter $\sigma$ on various subsets of $\Omega$ via restriction. Define

$\Omega_{+}^{0}\coloneqq\big{\\{}x\in\Omega\mid\lambda\big{(}\operatorname{comp}_{x}(\Omega\cap
B_{2R}(x)),\sigma\big{)}>\mu+\delta\big{\\}}.$

This is our “first pass” at the narrow part. It is not difficult to verify
that the map

$x\mapsto\lambda\big{(}\operatorname{comp}_{x}(\Omega\cap
B_{2R}(x)),\sigma\big{)}$

is continuous, so $\Omega_{+}^{0}$ is open.

Fix $y\in\mathbb{R}^{d}$ and suppose $B_{R}(y)$ intersects $\Omega_{+}^{0}$.
Take $x\in\Omega_{+}^{0}\cap B_{R}(y)$. Because $\left|x-y\right|<R$, we have
$B_{R}(y)\subset B_{2R}(x)$. It follows that

$\operatorname{comp}_{x}(\Omega_{+}^{0}\cap
B_{R}(y))\subset\operatorname{comp}_{x}(\Omega\cap B_{2R}(x)).$

By Proposition 2.1(i) and the definition of $\Omega_{+}^{0}$,

$\lambda\big{(}\operatorname{comp}_{x}(\Omega_{+}^{0}\cap
B_{R}(y)),\sigma\big{)}\geq\lambda\big{(}\operatorname{comp}_{x}(\Omega\cap
B_{2R}(x)),\sigma\big{)}>\mu+\delta.$

Now, $x\in\Omega_{+}^{0}\cap B_{R}(y)$ was arbitrary, so every connected
component of $\Omega_{+}^{0}\cap B_{R}(y)$ has principal eigenvalue at least
$\mu+\delta$. Therefore

$\lambda\big{(}\Omega_{+}^{0}\cap
B_{R}(y),\sigma\big{)}=\lambda\big{(}\Omega_{+}^{0}\mid
B_{R}(y),\sigma\big{)}\geq\mu+\delta.$ (4.9)

Since this holds for all $y\in\mathbb{R}^{d}$, Theorem 2.3 and our choice
(4.8) morally imply that

$\lambda(\Omega_{+}^{0},\sigma)>\mu.$

This is the desired inequality for $\mu$-narrowness.

There is a catch, however. Generally, $\Omega_{+}^{0}$ has corners and
$\sigma$ is discontinuous. To circumvent this, we let
$(\Omega_{+},\sigma_{+})$ be a truncation of $(\Omega,\varrho)$ in
$\Omega_{+}^{0}$, as provided by Lemma 4.4. Using Proposition 2.1(i) and
(4.9), we see that

$\lambda\big{(}\Omega_{+}\mid B_{R}(y),\sigma_{+}\big{)}\geq\mu+\delta$

for all $y\in\mathbb{R}^{d}$. Thus by Theorem 2.3 and (4.8),

$\lambda(\Omega_{+},\sigma_{+})>\mu.$

That is, $(\Omega_{+},\sigma_{+})$ is $\mu$-narrow.

Next, we construct the ample part. In the following, let
$G\coloneqq\Omega\setminus\Omega_{+}^{0}$. We claim that there exists $K\geq
2$ such that

$\lambda\big{(}\operatorname{comp}_{x}(\Omega\cap
B_{K}(x)),\sigma\big{)}\leq\mu-\delta\quad\textrm{for all }x\in G.$ (4.10)

To see this, suppose otherwise. Then there exists a sequence
$(x_{n})_{n\in\mathbb{N}}\subset G$ such that

$\lambda\big{(}\operatorname{comp}_{x_{n}}(\Omega\cap
B_{n}(x_{n})),\sigma\big{)}>\mu-\delta$ (4.11)

for all $n\in\mathbb{N}$. On the other hand, because
$x_{n}\not\in\Omega_{+}^{0}$, we have

$\lambda\big{(}\operatorname{comp}_{x_{n}}(\Omega\cap
B_{2R}(x_{n})),\sigma\big{)}\leq\mu+\delta.$ (4.12)

Extracting a subsequence, we can assume that
$(\tau_{x_{n}}\Omega)_{n\in\mathbb{N}}$ has a connected limit $\Omega^{*}$.
Let $\sigma^{*}\coloneqq\varrho\textup{{1}}_{\partial\Omega^{*}}$. Morally,
Lemma 2.2 and (4.11) imply that

$\lambda(\Omega^{*},\varrho)\geq\mu-\delta.$

Strictly speaking, we cannot apply Lemma 2.2 because
$\operatorname{comp}_{x_{n}}(\Omega\cap B_{n}(x_{n}))$ is generally not smooth
and $\sigma$ is generally discontinuous. Again, we can bypass these
difficulties by dealing with a truncation
$(\tilde{\Omega}_{n},\tilde{\sigma}_{n})$ of $(\Omega,\varrho)$ in
$\operatorname{comp}_{x_{n}}(\Omega\cap B_{n}(x_{n}))$. By Proposition 2.1(i)
and (4.11),

$\lambda(\tilde{\Omega}_{n},\tilde{\sigma}_{n})\geq\mu-\delta.$

Moreover, $(\tilde{\Omega}_{n},\tilde{\sigma}_{n})$ coincides with
$(\operatorname{comp}_{x_{n}}(\Omega\cap B_{n}(x_{n})),\sigma)$ within
$B_{n-1}(x_{n})$, so the sequence
$(\tau_{x_{n}}\tilde{\Omega}_{n},\tau_{x_{n}}\tilde{\sigma}_{n})_{n}$ also has
connected limit $(\Omega^{*},\varrho)$. Therefore Lemma 2.2 _does_ imply that

$\lambda(\Omega^{*},\varrho)\geq\mu-\delta.$ (4.13)

On the other hand, locally uniform $\mathcal{C}^{2,\alpha}$ convergence
implies that

$(\operatorname{comp}_{0}(\tau_{x_{n}}\Omega\cap
B_{2R}),\tau_{x_{n}}\sigma)\to(\operatorname{comp}_{0}(\Omega^{*}\cap
B_{2R}),\sigma^{*}).$

As noted in Remark 2.1, the principal eigenvalue is continuous on uniformly
bounded domains. Thus Proposition 2.1(i) and (4.12) yield

$\displaystyle\lambda(\Omega^{*},\varrho)$
$\displaystyle\leq\lambda\big{(}\operatorname{comp}_{0}(\Omega^{*}\cap
B_{2R}),\sigma^{*}\big{)}$ (4.14) $\displaystyle\hskip
42.67912pt=\lim_{n\to\infty}\lambda\big{(}\operatorname{comp}_{0}(\tau_{x_{n}}\Omega\cap
B_{2R}),\tau_{x_{n}}\sigma\big{)}\leq\mu+\delta.$

In combination with (4.13), we see that $\Omega_{*}$ is a connected limit of
$\Omega$ with

$\lambda(\Omega^{*},\varrho)\in[\mu-\delta,\mu+\delta].$

This contradicts (4.7). Therefore (4.10) does hold for some $K\geq 2$.

Next, define

$\Omega_{-}^{0}\coloneqq\bigcup_{x\in G}\operatorname{comp}_{x}(\Omega\cap
B_{K}(x)).$

We note that $G\subset\Omega_{-}^{0}$. Moreover, because $K\geq 2$,

$\Omega_{-}^{0}\cup\Omega_{+}=\Omega.$ (4.15)

Suppose $x\in\Omega_{-}^{0}$, so that
${x\in\operatorname{comp}_{x_{0}}(\Omega\cap B_{K}(x_{0}))}$ for some
$x_{0}\in G$. It follows that $B_{K}(x_{0})\subset B_{2K}(x)$ and

$\operatorname{comp}_{x_{0}}(\Omega\cap
B_{K}(x_{0}))\subset\operatorname{comp}_{x}(\Omega_{-}^{0}\cap B_{2K}(x)).$

By Proposition 2.1(i) and (4.10), we have

$\lambda\big{(}\operatorname{comp}_{x}(\Omega_{-}^{0}\cap
B_{2K}(x)),\sigma\big{)}\leq\lambda\big{(}\operatorname{comp}_{x_{0}}(\Omega\cap
B_{K}(x_{0})),\sigma\big{)}\leq\mu-\delta.$ (4.16)

We now smooth $\Omega_{-}^{0}$ into $\Omega_{-}$. Define
$U\coloneqq\\{x\in\Omega\mid\operatorname{dist}_{\Omega}(x,\Omega_{-}^{0})<1\\}$
and let $(\Omega_{-},\sigma_{-})$ be a truncation of $(\Omega,\varrho)$ in
$U$. Because $\Omega_{-}^{0}\subset U_{1}$ in the notation of (4.2), we have
$(\Omega_{-}^{0},\sigma)\preceq(\Omega_{-},\sigma_{-})$. Now take
$x\in\Omega_{-}$. Since $x\in U$,
$\operatorname{dist}_{\Omega}(x,\Omega_{-}^{0})<1$. Hence there exists
$x_{0}\in\Omega_{-}^{0}$ such that

$\operatorname{comp}_{x_{0}}(\Omega_{-}^{0}\cap
B_{2K}(x_{0}))\subset\operatorname{comp}_{x}(\Omega_{-}\cap B_{2K+1}(x)).$

By Proposition 2.1(i) and (4.16),

$\operatorname{comp}_{x}(\Omega_{-}\cap
B_{2K+1}(x))\leq\mu-\delta\quad\textrm{for all }x\in\Omega_{-}.$

Mimicking the reasoning leading to (4.14), we see that every connected limit
$(\Omega_{-}^{*},\sigma_{-}^{*})$ of $(\Omega_{-},\sigma_{-})$ satisfies

$\lambda(\Omega_{-}^{*},\sigma_{-}^{*})\leq\mu-\delta.$

Therefore $(\Omega_{-},\sigma_{-})$ is $\mu$-ample.

Since $\Omega_{-}^{0}\subset\Omega_{-}$, (4.15) implies that
$\Omega=\Omega_{-}\cup\Omega_{+}$. Similarly, $K\geq 2$ and our truncation
constructions of $\sigma_{\pm}$ ensure that
$\max\\{\sigma_{+},\sigma_{-}\\}=\varrho$. Therefore
$(\Omega_{-},\Omega_{+},\sigma_{-},\sigma_{+})$ is a $\mu$-ample–narrow
decomposition of $(\Omega,\varrho)$. ∎

###  4.3. Uniqueness

We are nearly in a position to prove Theorem 1.1. Our final auxiliary result
constrains the vanishing of solutions at infinity.

######  Lemma 4.5.

Let $\Omega^{*}$ be a connected limit of $\Omega$ along
$(x_{n})_{n}\subset\Omega$. Let $u$ (resp. $v$) be a positive
$\mathcal{C}_{\mathrm{b}}^{2,\gamma}$ supersolution (resp. subsolution) of
(1.3) on $(\Omega,\varrho)$. If $\lambda(\Omega^{*},\varrho)<f^{\prime}(0),$
then $(\tau_{x_{n}}u)_{n}$ does not vanish locally uniformly along any
subsequence. Conversely, if $\lambda(\Omega^{*},\varrho)>f^{\prime}(0),$ then
$\tau_{x_{n}}v\to 0$ locally uniformly as $n\to\infty$.

###### Proof.

We prove the second part first. By elliptic regularity, every subsequential
limit $v^{*}$ of $\tau_{x_{n}}v$ is a nonnegative subsolution of (1.3) on
$(\Omega^{*},\varrho)$. Let $\mathcal{P}_{t}^{*}$ denote the parabolic flow on
$(\Omega^{*},\varrho)$ by analogy with (1.5). Because $v^{*}$ is a
subsolution, $\mathcal{P}_{\infty}^{*}v^{*}\geq v^{*}$ is a bounded
nonnegative solution of (1.3) on $(\Omega^{*},\varrho)$. By Theorem 1.5,
$\mathcal{P}_{\infty}^{*}v^{*}=0$ if
$\lambda(\Omega^{*},\varrho)>f^{\prime}(0)$. It follows that $v^{*}=0$.
Because the subsequential limit is unique, $\tau_{x_{n}}v\to 0$ along the
entire sequence.

On the other hand, suppose a supersolution $u$ vanishes along $(x_{n})_{n}$.
Define

$w_{n}\coloneqq\frac{\tau_{x_{n}}u}{u(x_{n})}>0,$

so that $w_{n}(0)=1$. Then $(w_{n})_{n}$ has a locally uniform subsequential
limit $w^{*}$ that satisfies

$\begin{cases}-\Delta w^{*}\geq f^{\prime}(0)w^{*}&\text{in }\Omega^{*},\\\
\mathcal{N}_{\varrho}w^{*}\geq 0&\text{on }\partial\Omega^{*},\\\
w^{*}(0)=1.\end{cases}$

By the strong maximum principle, $w^{*}>0$ in $\Omega^{*}$. Using $w^{*}$ in
(1.4), we see that $\lambda(\Omega^{*},\varrho)\geq f^{\prime}(0)$. Taking the
contrapositive, we obtain Lemma 4.5. ∎

We can now prove our main uniqueness result.

###### Proof of Theorem 1.1.

Suppose $f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,\varrho)$. Then by
Proposition 4.3, $(\Omega,\varrho)$ admits an $f^{\prime}(0)$-ample–narrow
decomposition $(\Omega_{-},\Omega_{+},\sigma_{-},\sigma_{+})$ in the sense of
Definition 4.1.

If $\lambda(\Omega,\varrho)>f^{\prime}(0)$, then Theorem 1.5 states that (1.3)
admits no positive bounded solution. We cannot have
$\lambda(\Omega,\varrho)=f^{\prime}(0)$ because
$\lambda(\Omega,\varrho)\in\Sigma(\Omega,\varrho)$. Thus it suffices to treat
the case $\lambda(\Omega,\varrho)<f^{\prime}(0)$, which implies that
$\Omega_{-}$ is nonempty. Moreover, Theorem 1.5 ensures the existence of a
positive bounded solution of (1.3). Let $u_{1},u_{2}>0$ be two such solutions
(not necessarily distinct).

First, we claim there exists $\mu<\infty$ such that $u_{1}\leq\mu u_{2}$ in
$\Omega_{-}$. For the sake of contradiction, suppose instead that
$\inf_{\Omega_{-}}(u_{2}-u_{1}/n)<0$ for all $n\in\mathbb{N}$. Then for each
$n\in\mathbb{N}$, there exists $x_{n}\in\Omega_{-}$ such that

$u_{2}(x_{n})-u_{1}(x_{n})/n<0.$ (4.17)

Because $\sigma_{-}\leq\varrho$ and $u_{2}\geq 0$, we have

$\mathcal{N}_{\sigma_{-}}u_{2}=\frac{\sigma_{-}}{\varrho}\mathcal{N}_{\varrho}u_{2}+\left(1-\frac{\sigma_{-}}{\varrho}\right)u_{2}\geq
0.$

So $u_{2}$ is a supersolution of (1.3) on $(\Omega_{-},\sigma_{-})$.

We take $n\to\infty$; for simplicity, we suppress subsequential notation.
There exist subsequential connected limits $\Omega_{-}^{*}\subset\Omega^{*}$
of $\Omega_{-}\subset\Omega$ along the sequence $(x_{n})_{n}$. Also, there
exists a limit $u_{2}^{*}$ of $(\tau_{x_{n}}u_{2})_{n}$ solving (1.3) on
$(\Omega^{*},\varrho)$. By ampleness, Lemma 4.5, and the strong maximum
principle, $u_{2}^{*}>0$ in $\Omega_{-}^{*}$. Since $\Omega^{*}$ is connected,
$u_{2}^{*}>0$ in $\Omega^{*}$. On the other hand, (4.17) and the boundedness
of $u_{1}$ imply that $u_{2}^{*}(0)=0$. Therefore $0\in\partial\Omega^{*}$.
This contradicts the boundary condition $\mathcal{N}_{\varrho}u_{2}^{*}=0$ and
the Hopf lemma unless $\varrho=0$. So, for the time being, assume $\varrho=0.$
Let $\nu^{*}$ denote the outward normal vector field on $\partial\Omega^{*}$.
We claim that $\partial_{\nu^{*}}u_{2}^{*}(0)=0$, which again contradicts the
Hopf lemma.

To see this, suppose for the sake of contradiction that
$m\coloneqq-\partial_{\nu^{*}}u_{2}^{*}(0)>0$. Recall that $\nu$ denotes the
outward normal on the original boundary $\partial\Omega$. By the collar
neighborhood theorem and the uniform regularity of $\Omega$, there exists
$\delta>0$ such that $\nu$ has a natural smooth extension to the
$\delta$-neighborhood $\partial^{\delta}\Omega$ of the boundary. Precisely,
the vector field $\nu$ identifies $\partial^{\delta}\Omega$ with the product
manifold $\partial\Omega\times[0,\delta)$. We denote product coordinates by
$x=(\overline{x},h)$, so that the “height” $h$ denotes distance from
$\partial\Omega$ and $\frac{\partial}{\partial h}=\nu$.

Since $0\in\partial\Omega^{*}$, $d(x_{n},\partial\Omega)\to 0$ as
$n\to\infty$. In particular, there exists $N_{0}\in\mathbb{N}$ such that
$x_{n}\in\partial^{\delta}\Omega$ for all $n\geq N_{0}$. When $n\geq N_{0}$,
the derivative $\partial_{\nu}u_{2}(x_{n})$ is well-defined. Elliptic
regularity implies that

$-\partial_{\nu}u_{2}(x_{n})\to-\partial_{\nu^{*}}u_{2}^{*}(0)=m>0\quad\text{as
}n\to\infty.$

Moreover, $\mathcal{C}^{1,\alpha}$ elliptic estimates provide $\varepsilon>0$
and $N_{1}\geq N_{0}$ such that for all $n\geq N_{1}$,

$-\partial_{\nu}u_{2}(x_{n})\geq m/2\quad\text{on
}\overline{B}_{\varepsilon}(\overline{x}_{n})\times(0,\varepsilon).$ (4.18)

Here $\overline{x}_{n}$ denotes the projection of $x_{n}$ to $\partial\Omega$
and $\overline{B}_{\varepsilon}(\overline{x}_{n})$ denotes its
$\varepsilon$-neighborhood in $\partial\Omega$ with Riemannian metric induced
from $\mathbb{R}^{d}$. The height $h_{n}$ of $x_{n}$ tends to $0$, so we may
assume that $h_{n}<\varepsilon$ for all $n\geq N_{1}$.

On the other hand, uniform $\mathcal{C}^{1}$-regularity implies that
$M\coloneqq\sup_{\Omega}\left|\nabla u_{1}\right|<\infty$. Integrating from
the boundary, (4.18) yields

$u_{2}(\overline{x},h)\geq\frac{m}{2}h\quad\textrm{and}\quad
u_{1}(\overline{x},h)\leq Mh\quad\text{on
}\overline{B}_{\varepsilon}(\overline{x}_{n})\times(0,\varepsilon)$ (4.19)

for all $n\geq N_{1}$. Since $x_{n}$ lies in the neighborhood, (4.19)
contradicts (4.17) when $n>2M/m$.

We now return to the general setting $\varrho\in[0,1]$. Together, these
contradictions imply the existence of $\mu<\infty$ such that $u_{1}\leq\mu
u_{2}$ in $\Omega_{-}$. Therefore

$\underline{\mu}\coloneqq\inf\left\\{\mu>0\mid u_{1}\leq\mu
u_{2}\enspace\text{in }\Omega_{-}\right\\}$

is finite. By continuity, $w\coloneqq\underline{\mu}u_{2}-u_{1}\geq 0$ in
$\Omega_{-}$. We claim that $\underline{\mu}\leq 1$. Suppose otherwise for the
sake of contradiction.

Consider $w$ on $\Omega_{+}$. The strong KPP property (S) implies that

$-\Delta
w=\underline{\mu}f(u_{2})-f(u_{1})>f(\underline{\mu}u_{2})-f(u_{1})\eqqcolon
qw.$

Setting $\mathcal{L}\coloneqq\Delta+q$, this reads $-\mathcal{L}w>0$. Now
$u_{i}\in\mathcal{C}_{\mathrm{b}}^{2,\gamma}$ and
$f\in\mathcal{C}^{1,\gamma}$, so the difference quotient $q$ is uniformly
$\mathcal{C}^{\gamma}$. Moreover, (S) yields $q\leq f^{\prime}(0)$.

Both $u_{1}$ and $u_{2}$ satisfy the $\varrho$-boundary condition on
$\partial\Omega$, so $\mathcal{N}_{\varrho}w=0$ there. Now
$(\Omega_{+},\sigma_{+})\preceq(\Omega,\varrho)$, so $\sigma_{+}=0$ on
$\partial\Omega_{+}\cap\Omega$ and $\sigma_{+}\leq\varrho$ on
$\partial\Omega_{+}\cap\partial\Omega$. Because
$\Omega=\Omega_{-}\cup\Omega_{+}$, we have
$\partial\Omega_{+}\cap\Omega\subset\Omega_{-}$. Hence $w\geq 0$ there, and it
follows that $\mathcal{N}_{\sigma_{+}}w=w\geq 0$ on
$\partial\Omega_{+}\cap\Omega$. Next, Definition 4.1 ensures that
$\sigma_{+}=\varrho$ on $\partial\Omega_{+}\setminus\partial\Omega_{-}$.
There, we have $\mathcal{N}_{\sigma_{+}}w=\mathcal{N}_{\varrho}w=0$. Finally,
on $\partial\Omega_{+}\cap\partial\Omega_{-}$, we have $w\geq 0$ and
$\sigma_{+}\leq\varrho$, so

$\mathcal{N}_{\sigma_{+}}w=\frac{\sigma_{+}}{\varrho}\mathcal{N}_{\varrho}w+\left(1-\frac{\sigma_{+}}{\varrho}\right)w\geq
0.$

We have thus shown that

$\begin{cases}-\mathcal{L}w>0&\text{in }\Omega_{+},\\\
\mathcal{N}_{\sigma_{+}}w\geq 0&\text{on }\partial\Omega_{+}.\end{cases}$

Because $(\Omega_{+},\sigma_{+})$ is $f^{\prime}(0)$-narrow, we have
$\lambda(\Omega_{+},\sigma_{+})>f^{\prime}(0)$. Since $q\leq f^{\prime}(0)$,
this implies that $\lambda(-\mathcal{L},\Omega_{+},\sigma_{+})>0$. By
Proposition 3.1, $w\geq 0$ in $\Omega_{+}$, and hence in the whole domain
$\Omega$.

For each $n\in\mathbb{N}$, there exists $y_{n}\in\Omega_{-}$ such that

$\left(\underline{\mu}-\frac{1}{n}\right)u_{2}(y_{n})-u_{1}(y_{n})<0.$ (4.20)

We take limits along $(y_{n})_{n}$ and suppress subsequential notation. There
exist connected limits $\Omega_{-}^{\star}\subset\Omega^{\star}$ and limiting
solutions $u_{i}^{\star}$ for $i\in\\{1,2\\}$ of (1.3) on
$(\Omega^{\star},\varrho)$. Then
$w^{\star}\coloneqq\underline{\mu}u_{2}^{\star}-u_{1}^{\star}$ is a limit of
$w$. It follows that $w^{\star}\geq 0$. As for $u_{2}^{\star}$ above,
$f^{\prime}(0)$-ampleness and Lemma 4.5 imply that $u_{i}^{\star}>0$ in
$\Omega_{-}^{\star}$ and hence in $\Omega^{\star}.$ By the strong KPP property
(S) and the assumption $\underline{\mu}>1$, $w^{\star}$ satisfies

$-\Delta w^{\star}+q^{\star}w^{\star}>0$

for a bounded difference quotient $q^{\star}$. Thus by the strong maximum
principle, $w^{\star}>0$ in $\Omega^{\star}$. However, (4.20) implies that
$w^{\star}(0)=0$, so $0\in\partial\Omega^{\star}$. This contradicts the Hopf
lemma unless $\varrho=0$, so temporarily assume this.

Again, we claim that $\partial_{\nu^{\star}}w^{\star}(0)=0$, which contradicts
the Hopf lemma. Supposing otherwise, we have
${m^{\star}\coloneqq-\partial_{\nu^{\star}}w^{\star}(0)>0}$. We follow the
reasoning from above. Uniform $\mathcal{C}^{1,\alpha}$-regularity implies that
$-\partial_{\nu}w\geq m^{\star}/2$ in a uniform neighborhood of $y_{n}$.
However, $\partial_{\nu}u_{2}$ is bounded, so ${-\partial_{\nu}(w-u_{2}/n)\geq
0}$ near $y_{n}$ once $n$ is sufficiently large. Integrating from the
boundary, we see that $w(y_{n})-u_{2}(y_{n})/n>0$ for such $n$, contradicting
(4.20).

Returning to the general case $\varrho\in[0,1]$, these contradictions imply
that $\underline{\mu}\leq 1$, so $u_{1}\leq u_{2}$ in $\Omega_{-}$. Mimicking
the maximum principle argument on $\Omega_{+}$, we can extend this relation to
$\Omega_{+},$ and thus to the entire domain $\Omega$. By symmetry,
$u_{1}=u_{2}$. ∎

Figure 4. A comb domain in $\mathbb{R}^{2}$ whose teeth have width $L_{i}$ for
$i\in\mathbb{Z}$.

To understand when Theorem 1.1 applies, it may help to consider the comb
domain $\Omega$ in Figure 4. Because $\Omega$ contains the lower half plane
(and thus arbitrarily large balls), we have $\lambda(\Omega,0)=0$. If we take
a sequence of points ascending “tooth $i$”, the corresponding connected limit
is a strip of width $L_{i}$. Moreover, if $L$ is a subsequential limit of
$(L_{i})_{i\in\mathbb{Z}}$, then we can take an ascending sequence of points
from that subsequence of teeth to obtain a strip of width $L$ as a connected
limit. The principal eigenvalue of a strip of width $L$ is $\pi^{2}L^{-2}$.
Thus, these observations imply that

$\Sigma(\Omega,0)=\\{0\\}\cup\overline{\\{\pi^{2}L_{i}^{-2}\\}}_{i\in\mathbb{Z}}.$

We note that, in this case, $\Sigma(\Omega,0)$ is closed. Therefore Theorem
1.1 applies to $\Omega$ if and only if $f^{\prime}(0)\not\in\Sigma(\Omega,0)$.
Equivalently, if and only if

$\inf_{i\in\mathbb{Z}}\left|f^{\prime}(0)-\pi^{2}L_{i}^{-2}\right|>0.$

In other words, to apply Theorem 1.1, $\Omega$ must be _noncritical_ in the
sense that the widths $L_{i}$ uniformly avoid the critical value $\pi
f^{\prime}(0)^{-1/2}$.

######  Remark 4.1.

It seems plausible that the limit spectrum $\Sigma(\Omega,\varrho)$ is
_always_ closed, but we are not presently able to show this.

We close this section with the complete classification of positive bounded
solutions in periodic domains.

###### Proof of Corollary 1.6.

By Theorem 1.5, (1.3) admits a positive bounded solution if and only if
${\lambda(\Omega,\varrho)<f^{\prime}(0)}$. We assume this condition.

Since $\Omega$ is periodic, every connected limit of $\Omega$ is a translate
of $\Omega$ itself. It follows that $\lambda(\Omega,\varrho)$ is the unique
limit eigenvalue. Hence the critical value $f^{\prime}(0)$ is not in
$\overline{\Sigma}(\Omega,\varrho)=\\{\lambda(\Omega,\varrho)\\}$. By Theorem
1.1, the positive bounded solution is unique. ∎

###  4.4. Weak-KPP nonuniqueness

We next demonstrate the importance of the strong KPP hypothesis (S) by
constructing examples of nonuniqueness for weak-KPP reactions.

###### Proof of Proposition 1.4.

Let $\Omega$ be a bounded domain with boundary parameter ${\varrho\in[0,1)}$.
For a graphical representation of the various reactions used in this proof,
see Figure 5.

Figure 5.  Reactions in the proof of Proposition 1.4. (a) An ignition
reaction. (b) A weak-KPP reaction on a restricted interval. (c) A nonnegative
reaction with multiple solutions satisfying the weak-KPP inequality (W). (d) A
true weak-KPP reaction with multiple solutions.

Let $h\colon\mathbb{R}\to[0,\infty)$ satisfy $h|_{(1/2,1)^{c}}\equiv 0$,
$h|_{(1/2,1)}>0$, and $h\in\mathcal{C}^{1,\gamma}([1/2,1])$. Then $h$ is a so-
called “ignition reaction” with ignition temperature $1/2$. We define its
antiderivative

$H(s)\coloneqq\int_{0}^{s}h(r)\;\mathrm{d}r.$

We claim that (1.3) admits a positive bounded solution with reaction $h$
provided $h$ is sufficiently large. To see this, note that solutions of (1.3)
with reaction $h$ are critical points of the energy functional

$E[v]\coloneqq\int_{\Omega}\left[\frac{1}{2}\left|\nabla
v\right|^{2}-H(v)\right]+\varrho^{-1}(1-\varrho)\int_{\partial\Omega}v^{2},$

where we interpret the boundary term as $0$ if $\varrho=0$ and $v=0$ on
$\partial\Omega$. By making $h$ large near $s=3/4$, say, we can evidently
arrange that $\inf E<0$. Then the direct method from the calculus of
variations yields a minimizer $u_{+}^{h}$ such that $0<u_{+}^{h}<1$. This is a
positive bounded solution of (1.3). We claim that $\sup u_{+}^{h}>1/2$.
Otherwise, $h(u_{+}^{h})=0$ and $u_{+}^{h}$ is harmonic. It thus achieves its
maximum on $\partial\Omega,$ which contradicts
$\mathcal{N}_{\varrho}u_{+}^{h}=0$ because $\varrho<1$. So indeed $\sup
u_{+}^{h}>1/2$.

Next, let $g$ be a weak-KPP reaction on the restricted interval $(0,1/2)$ such
that

$g^{\prime}(0)>\max\left\\{\lambda(\Omega,\varrho),\,\sup_{s\in[1/2,1]}\frac{h(s)}{s}\right\\}.$
(4.21)

By Theorem 1.5, (1.3) has a positive bounded solution $u_{-}$ with reaction
$g$. Because $g$ vanishes on $[1/2,\infty)$, we have $0<u_{-}\leq 1/2$.
Moreover, the boundary condition $\mathcal{N}_{\varrho}u_{-}=0$ ensures that
$u_{-}$ is not constant, so by the strong maximum principle and the
boundedness of $\Omega$, $0<u_{-}\leq\sup u_{-}<1/2$. We define

$\varepsilon\coloneqq 1/2-\sup u_{-}>0.$ (4.22)

Now let $f_{0}\coloneqq g+h$. By (4.21), $f_{0}$ satisfies the weak-KPP
inequality (W). Because $h\equiv 0$ on $[0,1/2]$, $f_{0}(u_{-})=g(u_{-})$ and
$u_{-}$ solves (1.3) with reaction $f_{0}$. While the same cannot be said of
$u_{+}^{h}$, we do have $f_{0}(u_{+}^{h})\geq h(u_{+}^{h})$. It follows that
$u_{+}^{h}$ is a _sub_ solution for the reaction $f_{0}$. Using the parabolic
semigroup $\mathcal{P}_{t}^{f_{0}}$ defined by (1.5) with reaction $f_{0}$, we
can construct a solution
$u_{+}^{f_{0}}\coloneqq\mathcal{P}_{\infty}^{f_{0}}u_{+}^{h}\geq u_{+}^{h}$ of
(1.3) with reaction $f_{0}$. It follows that $\sup u_{+}^{f_{0}}>1/2$. Since
$\sup u_{-}<1/2$, we have in particular that $u_{-}\neq u_{+}^{f_{0}}$. That
is, $f_{0}$ supports two distinct positive bounded solutions.

The reaction $f_{0}$ has one deficiency: it vanishes at the intermediate point
$s=1/2$ and is merely Lipschitz there. Recall the “gap” $\varepsilon$ from
(4.22). We can increase $f_{0}$ on the interval
$(1/2-\varepsilon,1/2+\varepsilon)$ so that it becomes
$\mathcal{C}^{1,\gamma}$ and strictly positive on $(0,1)$. By making a
sufficiently small change, we can also ensure that the modified reaction lies
under the tangent line $f_{0}^{\prime}(0)s$. Let $f$ denote this modification
of $f_{0}$. By construction, $f$ is a true weak-KPP reaction.

Because $f=f_{0}=g$ on $[0,1/2-\varepsilon]$, $u_{-}$ is a solution of (1.3)
with reaction $f$. Moreover, because $f\geq f_{0}$, $u_{+}^{f_{0}}$ is a
subsolution for the reaction $f$. Again, there exists a solution $u_{+}$ of
(1.3) with reaction $f$ such that $u_{+}\geq u_{+}^{f_{0}}$. Then $\sup
u_{+}>1/2$, so $u_{+}\neq u_{-}$. Therefore $f$ is a weak-KPP reaction such
that (1.3) admits two distinct positive bounded solutions on
$(\Omega,\varrho)$. ∎

In our definition of $f_{0}=g+h$, $g$ only needs to be a sufficiently large
KPP reaction. In particular, if we fix $g$ and define $f_{\mu}\coloneqq
f_{0}+\mu g$ for $\mu\geq 0$, then $f_{\mu}$ would also support multiple
positive bounded solutions on $\Omega$. This corresponds to a taller first
“hump” in Figure 5(c).

It is interesting to consider the behavior of solutions under the limit
$\mu\to\infty$. Let $u_{-}^{\mu}$ and $u_{+}^{\mu}$ denote the minimal and
maximal solutions of (1.3) on $\Omega$ with reaction $f_{\mu}$. As $f_{\mu}$
tends to $+\infty$ on $(0,1/2)$, it will push solutions into the range
$[1/2,1]$ in the interior of $\Omega$. Indeed, it is straightforward to check
that $u_{-}^{\mu}\to 1/2$ locally uniformly in $\Omega$ as $\mu\to\infty$.
Thus $u_{-}^{\mu}$ develops a boundary layer around $\partial\Omega$ wherein
it rapidly transition from $0$ to $1/2$.

It is reasonable to expect $u_{+}^{\mu}$ to develop a similar boundary layer.
This suggests that $u_{+}^{\mu}$ converges locally uniformly to a solution of
the problem

$\begin{cases}-\Delta u=h(u)&\text{in }\Omega,\\\ u=1/2&\text{on
}\partial\Omega,\\\ u>1/2&\text{in }\Omega.\end{cases}$

If $h$ is strong-KPP on the interval $[1/2,1]$, then this solution is unique.
Finally, we observe that Morse theory suggests the existence of at least one
unstable solution between $u_{-}^{\mu}$ and $u_{+}^{\mu}$. We conjecture that
all such solutions converge to $1/2$ locally uniformly as $\mu\to\infty$ if
$h$ is strong-KPP on $[1/2,1]$.

###  4.5. The hair-trigger effect

To close the section, we prove the hair-trigger effect for strong-KPP
reactions.

###### Proof of Proposition 1.8.

First assume $f$ is positive and $\lambda(\Omega,\varrho)<f^{\prime}(0)$. By
Theorem 1.5, (1.3) admits a positive bounded solution. Because
$\lambda(\Omega,\varrho)<f^{\prime}(0)$, Proposition 2.1(iii) implies that
${\lambda(\Omega\mid B_{R},\varrho)<f^{\prime}(0)}$ for some large $R>0.$
Because $f$ is $\mathcal{C}^{1,\gamma}$, there exists $\eta\in(0,1)$ such that

$\inf_{s\in(0,\eta]}\frac{f(s)}{s}\geq\lambda(\Omega\mid B_{R},\varrho).$

Let $\varphi$ denote the positive principal eigenfunction associated to
$\lambda(\Omega\mid B_{R},\varrho)$ normalized by
$\left\|\varphi\right\|_{\infty}=1$, which we extend by $0$ to all of
$\Omega$. Then $\varepsilon\varphi$ is a subsolution of (1.3) for all
$\varepsilon\in[0,\eta]$.

Let $u$ be a positive bounded solution of (1.3), so $u\geq\varepsilon\varphi$
for some $\varepsilon\in(0,\eta]$. Let $\overline{\varepsilon}$ be the largest
such $\varepsilon$, and suppose for the sake of contradiction that
$\overline{\varepsilon}\in(0,\eta)$. Then
$u\geq\overline{\varepsilon}\varphi$, and by the strong maximum principle
$u>\overline{\varepsilon}\varphi$ in $\Omega$. Then the compactness of
$\overline{B}_{R}$ and the Hopf lemma imply that
$u\geq(\overline{\varepsilon}+\delta)\varphi$ for some $\delta>0$,
contradicting the definition of $\overline{\varepsilon}$. From this, we see
that $u\geq\eta\varphi$.

Recall the parabolic semigroup $(\mathcal{P}_{t})_{t\geq 0}$ associated with
(1.5). Because $\eta\varphi$ is a subsolution of (1.3),
$\mathcal{P}_{t}(\eta\varphi)$ is increasing in $t$ and thus has a limit
$u_{-}\coloneqq\mathcal{P}_{\infty}(\eta\varphi)$, which is a positive bounded
solution of (1.3). By the parabolic comparison principle, $u\geq u_{-}$. That
is, $u_{-}$ is minimal. Similarly, every positive bounded solution is bounded
by the supersolution $1$, so $u_{+}\coloneqq\mathcal{P}_{\infty}1$ is the
maximal solution of (1.3). Every positive bounded solution $u$ of (1.3)
satisfies $u_{-}\leq u\leq u_{+}$.

Now let $v$ solve (1.5) with $v(0,\,\cdot\,)$ bounded, nonnegative, and not
identically $0$. By the parabolic strong maximum principle and the Hopf lemma,
$v(1,\,\cdot\,)\geq\varepsilon\varphi$ for some $\varepsilon\in(0,\eta]$.
Because $\varepsilon\varphi$ is a (nonzero) subsolution of (1.3),
$\mathcal{P}_{\infty}(\varepsilon\varphi)$ is a positive bounded solution of
(1.3). Therefore $\mathcal{P}_{\infty}(\varepsilon\varphi)\geq u_{-}$. In
fact, the comparison principle implies that
$\mathcal{P}_{\infty}(\varepsilon\varphi)=u_{-}$ for all
$\varepsilon\in(0,\eta]$. Also, comparison yields
$v(t,\,\cdot\,)\geq\mathcal{P}_{t-1}(\varepsilon\varphi)$ for all $t\geq 1$.
Taking $t\to\infty$, we see that

$\liminf_{t\to\infty}v(t,\,\cdot\,)\geq u_{-}.$ (4.23)

Because $M\coloneqq\sup v(0,\,\cdot\,)<\infty$, similar reasoning implies that

$\limsup_{t\to\infty}v(t,\,\cdot\,)\leq\mathcal{P}_{\infty}M=\mathcal{P}_{\infty}1=u_{+}.$
(4.24)

Now suppose in addition that $f$ is strong-KPP and
${f^{\prime}(0)\not\in\overline{\Sigma}(\Omega,\varrho)}$. By Theorem 1.1,
(1.3) admits a unique solution $u$. It follows that $u_{-}=u_{+}=u$. Combining
(4.23) and (4.24), we obtain

$\lim_{t\to\infty}v(t,\,\cdot\,)=u.$

Standard parabolic estimates imply that this limit holds locally uniformly in
$\overline{\Omega}$. ∎

##  5\. A critical example

Recall Conjecture 1.3: we expect uniqueness to hold in general for strong-KPP
reactions. We would like to adapt the proof of Theorem 1.1 to general domains,
but critical limit domains pose difficulties. To probe this issue, consider
the “bulb domain” in Figure 6.

Figure 6. A bulb domain in $\mathbb{R}^{2}$.

We impose Dirichlet conditions ($\varrho=0$) and assume that the round portion
is sufficiently large that $\lambda(\Omega,0)<f^{\prime}(0)$. Then by Theorem
1.5, (1.3) admits at least one positive bounded solution on $\Omega$. We
choose coordinates so that the shaded region $\Gamma$ is
$\mathbb{R}_{+}\times(0,L)$.

We can explicitly compute the limit spectrum of $\Omega$. Let $\Omega^{*}$ be
a connected limit of $\Omega$ along a sequence
$(x_{n},y_{n})_{n\in\mathbb{N}}$. If $(x_{n})_{n}$ is bounded, then
$\Omega^{*}$ is a translate of $\Omega$ and
$\lambda(\Omega^{*},0)=\lambda(\Omega,0)$. On the other hand, if $(x_{n})_{n}$
tends to infinity, then $\Omega^{*}$ is a cylinder of width $L$, and one can
easily use (1.4) and Proposition 2.1(v) to check that

$\lambda(\Omega^{*},0)=\lambda\big{(}(0,L),0\big{)}=\frac{\pi^{2}}{L^{2}}.$

Therefore

$\Sigma(\Omega,0)=\big{\\{}\lambda(\Omega,0),\,\pi^{2}/L^{2}\big{\\}}.$

From this and our assumption that $\lambda(\Omega,0)<f^{\prime}(0)$, we see
that Theorem 1.1 applies if and only if $L\neq\pi f^{\prime}(0)^{-1/2}$.

In this section, we assume that $L=\pi f^{\prime}(0)^{-1/2}$ is critical. In
this case, the nonlinear structure of $f$ begins to play a role. We assume
that $f$ has the form

$f(s)=f^{\prime}(0)s-Bs^{2}+\mathcal{O}(s^{3})$ (5.1)

for some $B>0$.

######  Lemma 5.1.

Let $f$ be a strong-KPP reaction satisfying (5.1). Then every solution $u$ of
(1.3) on $(\Omega,0)$ satisfies

$u(x,y)\sim\frac{9\pi}{4Bx^{2}}\sin\left(\frac{\pi y}{L}\right)$ (5.2)

uniformly in $y$ as $x\to\infty$.

This resolves one of the main questions in our proof of uniqueness—it shows
that any two solutions are comparable. Our proof will make use of the
following characterization of the principal eigenvalue problem on cylinders.

######  Lemma 5.2.

Fix $\varrho\in[0,1]$ and suppose $\Omega=\mathbb{R}\times\omega$ for some
bounded domain $\omega\subset\mathbb{R}^{d-1}$. Then
$\lambda(\Omega,\varrho)=\lambda(\omega,\varrho)$. Moreover, if $\varphi$ is a
positive principal eigenfunction on $(\Omega,\varrho)$ in the sense that

$\begin{cases}-\Delta\varphi=\lambda(\Omega,\varrho)\varphi&\text{in
}\Omega,\\\ \mathcal{N}_{\varrho}\varphi=0,\end{cases}$

then $\varphi$ is a multiple of the principal eigenfunction on
$(\omega,\varrho)$. In particular, $\varphi$ is independent of the first
coordinate.

We defer the proof to the end of the section.

###### Proof of Lemma 5.1.

Let $u$ be a positive bounded solution of (1.3) and consider its translates
$\tau_{(n,0)}u$. As $n\to\infty$, we can extract a subsequence that converges
to a bounded nonnegative limit $u^{*}$ solving (1.3) on the cylinder
$\mathbb{R}\times(0,L)$. We know $u^{*}$ lies under the supersolution $1$.
Recall the parabolic evolution from (1.5). By the parabolic comparison
principle, $0\leq u^{*}\leq\mathcal{P}_{\infty}1$. But $\mathcal{P}_{\infty}1$
is independent of $x$, and thus corresponds to a nonnegative bounded solution
of (1.3) on the cross-section $(0,L)$. This problem is critical, so by Theorem
4.1 for bounded domains, $\mathcal{P}_{\infty}1=0$. Hence $u^{*}=0$. The limit
is unique, so in fact $u$ converges to $0$ uniformly as $x\to\infty$. We
determine the asymptotics of this limit.

Let $\\{\varphi_{k}\\}_{k\in\mathbb{N}}$ denote an orthonormal basis of
eigenfunctions of $-\Delta$ on $(0,L)$, so that

$\varphi_{k}(y)=\sqrt{\frac{2}{L}}\sin\left(\frac{\pi ky}{L}\right)$ (5.3)

for $k\in\mathbb{N}$. These have eigenvalues $(\pi k/L)^{2}.$ Since $L=\pi
f^{\prime}(0)^{-1/2}$, the principal eigenvalue is $f^{\prime}(0)$ and
$\varphi_{1}$ solves the linearization of (1.3) about $0$ on
$\mathbb{R}\times(0,L)$. However, because $f^{\prime}(0)u-f(u)>0$ for $u>0$,
there is nonlinear absorption. We have assumed that it takes the standard form
(5.1).

Because $u\in\mathcal{C}_{\mathrm{b}}^{2,\gamma}$, its Fourier series

$u(x,y)=\sum_{k\geq 1}\alpha_{k}(x)\varphi_{k}(y)$ (5.4)

converges absolutely and
$\alpha_{k}\in\mathcal{C}_{\mathrm{b}}^{2,\gamma}(\mathbb{R}_{+})$. Let
$\braket{\,\cdot\,,\,\cdot\,}$ denote the inner product on
$L^{2}\big{(}(0,L)\big{)}$. Since $u(x,\,\cdot\,)\to 0$ uniformly as
$x\to\infty$, the inner product representation

$\alpha_{k}(x)=\braket{u(x,\,\cdot\,),\varphi_{k}}$ (5.5)

implies that $\alpha_{k}(x)\to 0$ as $x\to\infty$.

We claim that the first term of (5.4) dominates as $x\to\infty$. To see this,
consider the sequence $v_{n}\coloneqq\tau_{(n,0)}u/\alpha_{1}(n)$. By (5.5),
we have

$\braket{v_{n}(0,\,\cdot\,),\varphi_{1}}=1.$ (5.6)

Since $\alpha_{1}(n)\to 0$ as $n\to\infty$, $v_{n}$ converges along a
subsequence to a solution $v^{*}$ of the linearization of (1.3) about $0$ on
$\mathbb{R}\times(0,L)$. By Lemma 5.2, $v^{*}$ is a multiple of the transverse
principal eigenfunction $\varphi_{1}$. Moreover, (5.6) yields
$\braket{v^{*},\varphi_{1}}=1,$ so $v^{*}=\varphi_{1}$. Since the limit is
unique, $v_{n}$ converges locally uniformly along the whole sequence to
$\varphi_{1}$. That is,

$\lim_{x\to\infty}\frac{u(x,y)}{\alpha_{1}(x)}=\varphi_{1}(y).$ (5.7)

Elliptic estimates imply that this limit is uniform in $y$ and valid in
$\mathcal{C}^{2}$. Taking the inner product between (5.7) and $\varphi_{k}$,
we see that $\alpha_{k}\ll\alpha_{1}$ as $x\to\infty$ for all $k\geq 2$.

Write $u=\alpha_{1}\varphi_{1}+u^{\perp}$. Substituting this in (1.3) and
using (5.1), we obtain

$-\alpha_{1}^{\prime\prime}\varphi_{1}-\Delta
u^{\perp}=f^{\prime}(0)u^{\perp}-B\varphi_{1}^{2}\alpha_{1}^{2}-2B\varphi_{1}\alpha_{1}u^{\perp}-B\big{(}u^{\perp}\big{)}^{2}+\mathcal{O}(u^{3}).$

We take the inner product with $\varphi_{1}$ and observe that $\braket{\Delta
u^{\perp},\varphi_{1}}=0=\Braket{u^{\perp},\varphi_{1}}$. Also, (5.7) yields
$u^{\perp}\ll\alpha_{1}\varphi_{1}$ and
$u^{3}\ll\varphi_{1}^{2}\alpha_{1}^{2}$. Therefore

$\alpha_{1}^{\prime\prime}=B\braket{\varphi_{1}^{2},\varphi_{1}}\alpha_{1}^{2}\left[1+\beta(x)\right]$
(5.8)

for some $\beta(x)\to 0$ as $x\to\infty$. Define

$\zeta\coloneqq
B\braket{\varphi_{1}^{2},\varphi_{1}},\quad\underline{\beta}(x)\coloneqq\inf_{[x,\infty)}\beta,\quad\textrm{and}\quad\overline{\beta}(x)\coloneqq\sup_{[x,\infty)}\beta.$

Then $\underline{\beta}$ and $\overline{\beta}$ are monotone increasing and
decreasing towards $0$, respectively. Fix $m$ such that
$\underline{\beta}(m)\geq-1$. Then on $[m,\infty)$, (5.8) implies that
$\alpha_{1}$ is convex and decreasing towards $0$. We assume for the time
being that $x\geq m$. We note that elliptic estimates imply that
$\alpha_{1}^{\prime}\to 0$ as $x\to\infty$ as well. Multiplying (5.8) by
$\alpha_{1}^{\prime}$ and integrating on $(x,\infty)$, we find

$\frac{2\zeta}{3}\alpha_{1}^{3}(1+\underline{\beta})\leq(\alpha_{1}^{\prime})^{2}\leq\frac{2\zeta}{3}\alpha_{1}^{3}(1+\overline{\beta}).$

Rearranging, we have

$\sqrt{\frac{\zeta}{6}(1+\underline{\beta})}\leq\left(\frac{1}{\sqrt{\alpha_{1}}}\right)^{\prime}\leq\sqrt{\frac{\zeta}{6}(1+\overline{\beta})}.$

Integrating on $(x_{0},x)$ and taking $x\to\infty$, we can check that

$\left[\frac{\zeta}{6}\big{(}1+\overline{\beta}(x_{0})\big{)}\right]^{-1}\leq\liminf_{x\to\infty}x^{2}\alpha_{1}(x)\leq\limsup_{x\to\infty}x^{2}\alpha_{1}(x)\leq\left[\frac{\zeta}{6}\big{(}1+\underline{\beta}(x_{0})\big{)}\right]^{-1}.$

Since $\underline{\beta},\overline{\beta}\to 0$ as $x\to\infty$, we can take
$x_{0}\to\infty$ to see that

$\lim_{x\to\infty}x^{2}\alpha_{1}(x)=\frac{6}{\zeta}.$

That is,

$\alpha_{1}(x)\sim\frac{6}{B\braket{\varphi_{1}^{2},\varphi_{1}}x^{2}}\quad\text{as
}x\to\infty.$ (5.9)

We can use (5.3) to compute

$\frac{\varphi_{1}(y)}{\braket{\varphi_{1}^{2},\varphi_{1}}}=\frac{3\pi}{8}\sin\left(\frac{\pi
y}{L}\right).$

Then (5.2) follows from (5.7) and (5.9). Elliptic estimates up to the boundary
imply that (5.2) holds uniformly in $y$. ∎

With explicit knowledge of the behavior of solutions at infinity, we can
easily show uniqueness.

######  Proposition 5.3.

Let $f$ be a strong-KPP reaction satisfying (5.1). Then (1.3) has a unique
positive bounded solution on $(\Omega,0)$.

###### Proof.

Suppose $u_{1}$ and $u_{2}$ are both positive bounded solutions of (1.3).
Following the proof of Theorem 1.1, define

$\underline{\mu}\coloneqq\inf\\{\mu\mid u_{1}\leq\mu u_{2}\\}.$

By the strong maximum principle and the Hopf lemma, $u_{1}$ and $u_{2}$ are
comparable on any compact $K\Subset\overline{\Omega}$. That is, there exists
$C(K)\geq 1$ such that

$C(K)^{-1}u_{2}\leq u_{1}\leq C(K)u_{2}\quad\text{in }K.$

On the other hand, Lemma 5.1 implies that $u_{1}\sim u_{2}$ at infinity. It
follows that $\underline{\mu}<\infty$.

Suppose for the sake of contradiction that $\underline{\mu}>1$. Define

$w\coloneqq\underline{\mu}u_{2}-u_{1}\geq 0.$

As in the proof of Theorem 1.1, the strong KPP property (S) ensures that
$-\Delta w>qw$ for a suitable difference quotient $q\in L^{\infty}(\Omega)$.
By the strong maximum principle, $w>0$.

When $x\to\infty$, Lemma 5.1 implies that $w\sim(\underline{\mu}-1)u_{2}$
uniformly in $y$. Thus if $\delta\in(0,\underline{\mu}-1)$, there exists
compact $K\Subset\Omega$ such that $w\geq\delta u_{2}$ on $\Omega\setminus K$.
Moreover, the strong maximum principle and the Hopf lemma imply that $w$ and
$u_{2}$ are comparable on $K$. In particular, perhaps after reducing
$\delta>0$, we have $w\geq\delta u_{2}$ on $K$. Hence $w\geq\delta u_{2}$ on
the entire domain $\Omega$. This contradicts the definition of
$\underline{\mu}$, so in fact $\underline{\mu}\leq 1$. Uniqueness follows by
symmetry. ∎

We close the section with a proof of our symmetry result.

###### Proof of Lemma 5.2.

We denote coordinates on $\Omega=\mathbb{R}\times\omega$ by $(x,y)$. Let
$\psi$ be a positive principal eigenfunction on $(\omega,\varrho)$. Then we
can view $\psi$ as a function on $\Omega$ as well; using it in (1.4), we see
that $\lambda(\Omega,\varrho)\geq\lambda(\omega,\varrho)$. On the other hand,
we can use the test functions $\cos(\varepsilon
x)\textup{{1}}_{\left|x\right|<\pi/2\varepsilon}\psi(y)$ in (2.5) to verify
the opposite inequality when $\varepsilon\searrow 0$. Thus
$\lambda(\Omega,\varrho)=\lambda(\omega,\varrho).$

Now let $\varphi$ be a positive principal eigenfunction on $(\Omega,\varrho)$
and define

$E(x)\coloneqq\int_{\omega}\varphi(x,y)\psi(y)\;\mathrm{d}y.$

Integrating by parts several times and using
$\mathcal{N}_{\varrho}\varphi=\mathcal{N}_{\varrho}\psi=0$, we can check that
$E^{\prime\prime}=0$. Since $\varphi>0,$ $E$ is a positive affine function,
and hence constant. By Harnack and the Hopf lemma, $\varphi$ is comparable to
$\psi$ in the sense that

$C^{-1}\psi\leq\varphi\leq C\psi$

for some $C\geq 1$. Let

$\underline{\mu}\coloneqq\inf\\{\mu>0\mid\varphi\leq\underline{\mu}\psi\\}.$

Then $\underline{\mu}\in\mathbb{R}_{+}$. Consider
$w\coloneqq\underline{\mu}\psi-\varphi\geq 0$. If $w=0$, we are done.
Otherwise, the strong maximum principle implies that $w>0$. Thus $w$ is a
positive principal eigenfunction on $(\Omega,\varrho)$. Replacing $\varphi$ by
$w$ in our work above, we see that $w$ is itself comparable to $\psi$. In
particular, there exists $\varepsilon>0$ such that $w\geq\varepsilon\psi$.
Therefore

$\varphi\leq(\underline{\mu}-\varepsilon)\psi,$

contradicting the definition of $\underline{\mu}$. So in fact
$\varphi=\underline{\mu}\psi$. ∎

##  Appendix A Smoothness and convergence of domains

In this appendix, we precisely define our notion of smoothness for domains and
prove an essential compactness result. These ideas are widely understood
folklore, but we use them so extensively that we elect to put the theory on
solid ground.

We begin with some preliminary notation. Given $r>0$, we let $B_{r}$ and
$B_{r}^{d-1}$ denote the $r$-balls in $\mathbb{R}^{d}$ and $\mathbb{R}^{d-1}$,
respectively. We define the cylinder

$\Gamma_{r}\coloneqq B_{r}^{d-1}\times(-r,r),$

on which we use coordinates
$(x^{\prime},y)\in\mathbb{R}^{d-1}\times\mathbb{R}$. Given a function
$\phi\in\mathcal{C}^{2,\gamma}(B_{r}^{d-1})$, we define its hypograph

$\Pi_{r}(\phi)\coloneqq\big{\\{}(x^{\prime},y)\in\Gamma_{r}\mid
y<\phi(x^{\prime})\big{\\}}.$

Such hypographs serve as building blocks for our smooth sets. Finally, we let
$\operatorname{Isom}_{0}(\mathbb{R}^{d})$ denote the group of orientation-
preserving isometries of $\mathbb{R}^{d}$. It is generated by translations and
rotations.

######  Definition A.1.

Given $r\in(0,1]$ and $K>0,$ we say that an open set
$\Omega\subset\mathbb{R}^{d}$ is $(\gamma,r,K)$-_smooth_ if one of following
holds for every $x\in\mathbb{R}^{d}$:

  1. (S1)

$B_{r/8}(x)\subset\Omega$;

  2. (S2)

$B_{r/8}(x)\subset\Omega^{c}$;

  3. (S3)

There exist $g\in\operatorname{Isom}_{0}(\mathbb{R}^{d})$ and
$\phi\in\mathcal{C}^{2,\gamma}\big{(}B_{r}^{d-1};[-r/4,r/4]\big{)}$ such that
${g(x)=0}$, $\left\|\phi\right\|_{\mathcal{C}^{2,\gamma}(B_{r}^{d-1})}\leq K$,
and $g\Omega\cap\Gamma_{r}=\Pi_{r}(\phi)$.

It follows that $\partial\Omega$ is a uniformly $\mathcal{C}^{2,\gamma}$
submanifold of $\mathbb{R}^{d}$. We say $\sigma\colon\partial\Omega\to[0,1]$
is uniformly $\mathcal{C}^{2,\gamma}$ smooth when this holds in the standard
sense used in differential geometry. Also, we say a collection $\mathcal{U}$
of open sets is $(\gamma,r,K)$-smooth if every element of $\mathcal{U}$ is
$(\gamma,r,K)$-smooth.

We often suppress parameters and say that $\Omega$ or $\mathcal{U}$ is
“uniformly $\mathcal{C}^{2,\gamma}$” or “uniformly smooth.”

######  Remark A.1.

It may seem strange that we restrict the range of $\phi$ to $[-r/4,r/4]$ in
(S3). For instance, this appears to rule out the hypograph $\Pi(\psi)$ of a
uniformly $\mathcal{C}^{2,\gamma}$ function
$\psi\colon\mathbb{R}^{d-1}\to\mathbb{R}$ with $\operatorname{Lip}\psi\gg 1$.
While this is true for a fixed value of $r$, we claim that such a hypograph
satisfies Definition A.1 for sufficiently small $r$.

This is due to the fact that the curvature of $\partial\Pi(\psi)$ is uniformly
bounded. Thus when $r\ll 1$, $\partial\Pi(\psi)$ appears almost flat at scale
$r$. By suitably orienting the isometry $g$, we can arrange that every local
chart $\phi$ for $\Pi(\psi)$ in (S3) satisfies $\left|\phi(0)\right|\leq r/8$
and $\left|\nabla\phi\right|\ll 1$. Thus if $r$ is small, we can impose the
additional condition $\left|\phi\right|\leq r/4$.

Next, we define a notion of smooth convergence for smooth sets. We use the
notation $\mathbb{N}_{*}\coloneqq\mathbb{N}\cup\\{\infty\\}$ for the extended
natural numbers. When we write $n\geq N$ in the following definition, we
include the limit index $n=\infty$.

######  Definition A.2.

Given a $(\gamma,r,K)$-smooth sequence $(\Omega_{n})_{n\in\mathbb{N}_{*}}$ and
$\alpha\in(0,\gamma)$, we say ${\Omega_{n}\to\Omega_{\infty}}$ locally
uniformly in $\mathcal{C}^{2,\alpha}$ as $n\to\infty$ if one of the following
holds for every $x\in\mathbb{R}^{d}$:

  1. (L1)

$B_{r/16}(x)\subset\Omega_{n}\cap\Omega_{\infty}$ when $n$ is sufficiently
large;

  2. (L2)

$B_{r/16}(x)\subset\Omega_{n}^{c}\cap\Omega_{\infty}^{c}$ when $n$ is
sufficiently large;

  3. (L3)

There exist $N\in\mathbb{N}$, $g\in\operatorname{Isom}_{0}(\mathbb{R}^{d})$,
and $(\phi_{n})_{n\geq
N}\subset\mathcal{C}^{2,\gamma}\big{(}B_{r/2}^{d-1};[-r/2,r/2]\big{)}$ such
that $g(x)=0$, ${g\Omega_{n}\cap\Gamma_{r/2}=\Pi_{r/2}(\phi_{n})}$ for all
$n\geq N,$ and $\phi_{n}\to\phi_{\infty}$ in
$\mathcal{C}^{2,\alpha}(B_{r/2}^{d-1})$ as $n\to\infty$.

Then one can naturally define locally uniform $\mathcal{C}^{2,\alpha}$
convergence for boundary parameters $\sigma_{n}$ and functions $u_{n}$ defined
on $\partial\Omega_{n}$ and $\Omega_{n}$, respectively. The details are
standard given the charts in (L1)–(L3); we omit them.

We can now state our compactness result.

######  Proposition A.1.

Let $(\Omega_{n})_{n\in\mathbb{N}}$ be a $(\gamma,r,K)$-smooth sequence of
domains. Then for each $\alpha\in(0,\gamma)$, there exist
$(\gamma,r,K)$-smooth $\Omega_{\infty}$ and a subsequence
$(\Omega_{n_{m}})_{m\in\mathbb{N}}$ of $(\Omega_{n})_{n\in\mathbb{N}}$ such
that ${\Omega_{n_{m}}\to\Omega_{\infty}}$ locally uniformly in
$\mathcal{C}^{2,\alpha}$ as $m\to\infty$.

This is essentially Arzelà–Ascoli for uniformly smooth domains.

######  Remark A.2.

Definition A.2 seems related to the notion of Cheeger–Gromov convergence from
differential geometry. For instance, if our domains are uniformly
$\mathcal{C}^{3}$, we can apply Theorem 3.1 of [1] to infer locally-uniform
subsequential convergence as in Proposition A.1.

######  Corollary A.2.

Every uniformly smooth sequence
$(\Omega_{n},\sigma_{n},u_{n})_{n\in\mathbb{N}}$ of domains $\Omega_{n}$,
boundary parameters $\sigma_{n}\colon\partial\Omega_{n}\to[0,1]$, and
functions $u_{n}\colon\Omega_{n}\to\mathbb{R}$ has a subsequence that
converges locally uniformly to a uniformly smooth limit
$(\Omega_{\infty},\sigma_{\infty},u_{\infty})$.

This follows from Proposition A.1 for $(\Omega_{n})_{n\in\mathbb{N}}$ and
Arzelà–Ascoli for $(\sigma_{n},u_{n})_{n\in\mathbb{N}}$; we omit the details.
We use Corollary A.2 incessantly throughout the paper, often without explicit
reference.

###### Proof of Proposition A.1.

Let $\big{(}B^{(m)}\big{)}_{m\in\mathbb{N}}$ be an open cover of
$\mathbb{R}^{d}$ by balls of radius $r/8$ such that every ball of radius
$r/16$ is contained in some $B^{(m)}$. Let $x^{(m)}$ denote the center of each
ball. We will construct nested subsequences

$(\Omega_{n})_{n}\supset\ldots\supset(\Omega_{m,n})_{n}\supset(\Omega_{m+1,n})_{n}\supset\ldots$

indexed by $m\in\mathbb{N}$. The subsequence $(\Omega_{m,n})_{n}$ will
determine the limit $\Omega_{\infty}$ within the ball $B^{(m)}$. Our final
subsequence $(\Omega_{n_{m}})_{m}$ will be diagonal: $(\Omega_{n,n})_{n}$. As
a subsequence of each $(\Omega_{m,n})_{n}$, it will determine a limit
$\Omega_{\infty}$ consistently within each ball $B^{(m)}$, and thus on the
whole space $\mathbb{R}^{d}$.

We proceed inductively. Given $m\in\mathbb{N}$, suppose we have constructed
$(\Omega_{m-1,n})_{n}$. (When $m=1$, we adopt the convention that
$\Omega_{0,n}=\Omega_{n}$.) We work by cases corresponding to those in
Definition A.2.

  1. (i)

If $B^{(m)}\subset\Omega_{m-1,n}$ for infinitely many $n$, let
$(\Omega_{m,n})_{n}$ be the corresponding subsequence and let
$\Omega_{\infty}\cap B^{(m)}=B^{(m)}$.

  2. (ii)

Otherwise, if $B^{(m)}\subset\Omega_{m-1,n}^{c}$ for infinitely many $n$, let
$(\Omega_{m,n})_{n}$ be the corresponding subsequence and let
$\Omega_{\infty}\cap B^{(m)}=\emptyset$.

  3. (iii)

If neither (i) nor (ii) applies, then $\partial\Omega_{m-1,n}$ intersects
$B^{(m)}=B_{r/8}\big{(}x^{(m)}\big{)}$ for all sufficiently large $n$.
Dropping the early entries, we may assume without loss of generality that this
holds for all $n$. We apply Definition A.1 to the $(\gamma,r,K)$-smooth sets
$\Omega_{m-1,n}$ around $x^{(m)}\in\mathbb{R}^{d}$. Since neither (S1) nor
(S2) holds, (S3) yields $g_{n}\in\operatorname{Isom}_{0}(\mathbb{R}^{d})$ and
${\phi_{n}\in\mathcal{C}^{2,\gamma}\big{(}B_{r}^{d-1};[-r/4,r/4]\big{)}}$ such
that $g_{n}(x^{(m)})=0$,
${\left\|\phi_{n}\right\|_{\mathcal{C}^{2,\gamma}(B_{r}^{d-1})}\leq K}$, and
$g_{n}\Omega_{m-1,n}\cap\Gamma_{r}=\Pi_{r}(\phi_{n})$ for all
$n\in\mathbb{N}$.

Now, the set of orientation-preserving isometries sending $x^{(m)}$ to $0$ is
homeomorphic to $\mathrm{SO}(d)$, and thus compact. It follows that we can
extract a subsequential limit $g_{\infty}$ of $(g_{n})_{n}$. Likewise,
Arzelà–Ascoli allows us to extract a further subsequence of $(\phi_{n})_{n}$
that converges in $\mathcal{C}^{2,\alpha}(B_{r}^{d-1})$ to some limit
${\phi_{\infty}\in\mathcal{C}^{2,\gamma}\big{(}B_{r}^{d-1};[-r/4,r/4]\big{)}}$.
We let $(\Omega_{m,n})_{n}$ be the corresponding subsequence of
$(\Omega_{m-1,n})_{n}$. We then set

$\Omega_{\infty}\cap
B^{(m)}=\big{(}g_{\infty}^{-1}\Pi_{r}(\phi_{\infty})\big{)}\cap B^{(m)}.$

We have now constructed $(\Omega_{m,n})_{n}$. Iterating, we obtain our entire
sequence of subsequences. Because the subsequences are nested, our
construction of the limit $\Omega_{\infty}$ is consistent across different
balls $B^{(m)}$. It is straightforward to check that $\Omega_{\infty}$ is
uniformly $(\gamma,r,K)$-smooth.

It remains to show that the diagonal sequence $(\Omega_{n,n})_{n}$ converges
to $\Omega_{\infty}$. Fix $x\in\mathbb{R}^{d}$ and suppose that neither (L1)
nor (L2) holds when we replace the sequence $(\Omega_{n})_{n}$ by
$(\Omega_{n,n})_{n}$. By our choice of the cover
$\big{(}B^{(m)}\big{)}_{m\in\mathbb{N}}$, the ball $B_{r/16}(x)$ is contained
in some $B^{(m)}$. We constructed $\Omega_{\infty}\cap B^{(m)}$ in Stage $m$
of the above iteration, and the failure of (L1) and (L2) implies that neither
(i) and (ii) held in that stage. Thus we used (iii) to construct
$\Omega_{\infty}\cap B^{(m)}$.

Let $g_{n}^{(m)}$ and $\phi_{n}^{(m)}$ denote the corresponding isometries and
height functions. Set $g\coloneqq g_{\infty}^{(m)}$ and
$\phi\coloneqq\phi_{\infty}^{(m)}$. The rotations
$g\big{(}g_{n}^{(m)}\big{)}^{-1}$ converge to the identity as $n\to\infty$.
Thus when $n$ is sufficiently large, the graph of $\phi_{n}^{(m)}$ over
$B_{r}^{d-1}$ is rotated by $g\big{(}g_{n}^{(m)}\big{)}^{-1}$ to the graph of
a function $\phi_{n}$ over $B_{r/2}^{d-1}$. It follows that
$g\Omega_{m,n}\cap\Gamma_{r/2}=\Pi_{r/2}(\phi_{n})$.

Moreover, because $\big{|}\phi_{n}^{(m)}\big{|}\leq r/4$, we have
$\left|\phi_{n}\right|\leq r/2$ once $n$ is large. By construction,
$\phi_{n}^{(m)}\to\phi_{\infty}^{(m)}$ in
$\mathcal{C}^{2,\alpha}(B_{r}^{d-1})$ as $n\to\infty$. It is thus easy to
check that $\phi_{n}\to\phi$ in $\mathcal{C}^{2,\alpha}(B_{r/2}^{d-1})$. Since
$(\Omega_{n,n})_{n}$ is a subsequence of $(\Omega_{m,n})_{n}$, (L3) holds and
indeed $\Omega_{n,n}\to\Omega_{\infty}$ locally uniformly in
$\mathcal{C}^{2,\alpha}$ as $n\to\infty$. ∎

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