# Numerics and analysis of Cahn–Hilliard critical points

Tobias Grafke, 111University of Warwick Sebastian Scholtes, 222RWTH Aachen
University Alfred Wagner 333RWTH Aachen University and Maria G. Westdickenberg
444RWTH Aachen University<EMAIL_ADDRESS>
###### Abstract

We explore recent progress and open questions concerning local minima and
saddle points of the Cahn–Hilliard energy in $d\geq 2$ and the critical
parameter regime of large system size and mean value close to $-1$. We employ
the String Method of E, Ren, and Vanden-Eijnden—a numerical algorithm for
computing transition pathways in complex systems—in $d=2$ to gain additional
insight into the properties of the minima and saddle point. Motivated by the
numerical observations, we adapt a method of Caffarelli and Spruck to study
convexity of level sets in $d\geq 2$.

Keywords. Complex energy landscapes, saddle points, String Method.

##  1 Introduction

The Cahn–Hilliard energy is a fundamental model of phase separation, with
wide-ranging applications in materials science and other natural sciences.
Introduced by Cahn and Hilliard to capture the mixing of a binary alloy, both
the energy and the dynamics have been the subject of extensive numerical and
analytical studies; we refer for instance to [CH, CH3, NS] and the citing
articles. Despite the large body of existing work on the model, significant
open questions remain.

In this paper we analyze critical points of the renormalized energy

$\displaystyle\mathcal{E}_{\phi}^{\xi}(u)$
$\displaystyle=\int_{\mathbb{T}_{\phi L}}\frac{\phi}{2}|\nabla
u|^{2}+\frac{1}{\phi}\Big{(}G(u)-G(-1+\phi)\Big{)}\,dx$ (1.1)

on

$\displaystyle X_{\phi}:=\Big{\\{}u\in H^{1}\cap L^{4}(\mathbb{T}_{\phi
L}):\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\mathbb{T}_{\phi
L}}u\,dx=-1+\phi\Big{\\}},$ (1.2)

where $\mathbb{T}_{\phi L}$ is the $d$-dimensional torus with side length
$\phi L$. Here $G$ is a nondegenerate double-well potential with minima $\pm
1$ (the standard choice being $G(u)=(1-u^{2})^{2}/4$); see Remark 1.1 below.
We are interested in the so-called critical regime, that is, the regime such
that

$\displaystyle L\sim\phi^{-(d+1)/d}\quad\text{and}\quad\phi\ll 1.$

We recall the origins of the problem in Section 2 below.

In [GWW] the existence of a nonconstant _local energy minimizer_ and _saddle
point_ of the energy in the critical regime was established. Moreover, the
following _constrained energy minimizers_ were introduced and analyzed:
functions $u_{\omega,\phi}$ that minimize $\mathcal{E}_{\phi}^{\xi}$ subject
to

$\displaystyle u_{\omega,\phi}\in
X_{\phi}\quad\text{and}\quad\nu(u_{\omega,\phi})=\omega,$ (1.3)

where the “volume” of a function $u\in X_{\phi}$ is defined as

$\displaystyle\nu(u):=\int_{\mathbb{T}_{\phi L}}\chi(u)\,dx,$ (1.4)

for a smooth function $\chi$ such that $\chi(s)=0$ for $s\leq 1-2\phi^{1/3}$
and $\chi(s)=1$ for $s\geq 1-\phi^{1/3}$. Since it can be shown that volume-
constrained minimizers are bounded above by $1+\phi^{1/3}$, $\nu$ roughly
measures the volume $|\\{u\approx 1\\}|$.

While much is known about the shape of minimizers on $\mathbb{R}^{d}$ (see for
instance [GNN, BL, LN]), the shape of minimizers on bounded domains is less
well understood; as seminal works on the torus, we mention [K, Br]. The Cahn-
Hilliard problem on the torus features a competition between the periodic
boundary conditions and the energetic preference for spherical level sets. We
are interested in _understanding and quantifying this competition_. Note that
this issue is not merely a matter of mathematical/geometric curiosity; the
appropriateness of the often-used sphericity assumption is of interest in
nucleation theory in physics and engineering applications; see for instance
[L].

Quantitative estimates on the energy and shape of critical points were derived
in [GWW, GWW20]. For example the Fraenkel asymmetry of the superlevel sets of
$u_{\omega,\phi}$ is shown to be bounded by $\phi^{1/6}$ [GWW]; see Section 2
below for more detail. In addition, the volume-constrained minimizers are
shown to be equal to their Steiner symmetrization [GWW20] and hence have
connected superlevel sets. It is _plausible_ that the saddle point (whose
existence was pointed out already by Cahn and Hilliard in [CH3] and is
established via a mountain-pass argument in [GWW]) is in fact equal to a
volume-constrained minimizer of appropriate volume, however this has not be
proved. Hence although bounds are obtained for the constrained minimizers,
[GWW, GWW20] stop short of establishing such bounds for the actual saddle
point.

Given the open questions in the analysis, in this article we apply the String
Method [ERV] to look numerically at the local minimum and saddle point of the
Cahn–Hilliard energy in $d=2$. The String Method is a well-known algorithm for
computing transition pathways in complex systems. Given two local minimizers
of an energy, it locates a saddle point on the boundary of the corresponding
basins of attraction. Alternatively, given two arbitrary states in distinct
basins of attraction, the string evolves to find both the local minima of the
basins and a saddle point between them. We review the method in Subsection 3.1
and summarize our numerical results in Subsection 3.3. Then in Section 4,
motivated in part by our numerical observations, we study convexity of the
sub/superlevel sets. In particular, the results of the String Method suggest
that for both the minimum and saddle point there is a value $t_{*}\in(-1,1)$
such that

$\displaystyle\text{the superlevel sets }\\{u>t\\}\text{ for $t\geq t_{*}$ and
the sublevel sets }\\{u<t\\}\text{ for $t\leq t_{*}$}$

are convex. While proving such a fact is currently out of reach, we show how
the method of Caffarelli and Spruck [CS] can be applied to obtain a partial
result in this direction; see Proposition 4.14.

######  Remark 1.1.

Our assumptions on the potential are:

  * •

$G$ is smooth,

  * •

$G>0$ for $x\neq\pm 1$ and $G(\pm 1)=0$,

  * •

$G^{\prime}\geq 0$ on $[-1,0]$, $G^{\prime}\leq 0$ on $[0,1]$, and
$G^{\prime\prime}(\pm 1)>0$.

##  2 Background and open questions

The original Cahn–Hilliard energy landscape consists of the energy

$\displaystyle E(u):=\int_{\Omega}\frac{1}{2}|\nabla u|^{2}+G(u)\,dx$ (2.1)

considered on the set

$\displaystyle\Big{\\{}u\in H^{1}\cap
L^{4}(\Omega):\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\Omega}u\,dx=m\Big{\\}},$

where the mean value $m$ is a given value strictly between the minima of $G$.
Studying the problem on a large domain $|\Omega|$ is equivalent, after a
rescaling of space, to studying the energy

$\displaystyle E_{\phi}(u):=\int_{\Omega_{1}}\frac{\phi}{2}|\nabla
u|^{2}+\frac{1}{\phi}G(u)\,dx$

for $\phi\ll 1$ on the fixed domain $\Omega_{1}$. It is well-known since the
seminal work of Modica and Mortola [MM] that for $m$ fixed and $\phi\downarrow
0$, the energy $E_{\phi}$ $\Gamma$-converges to the perimeter functional.

Droplet formation or nucleation in a low-density phase is more subtle, as is
pointed out by Biskup, Chayes, and Kotecky in [BCK]. For the Cahn-Hilliard
model, [BGLN, CCELM] probed the _competition_ between large system size and
mean value close to $-1$. To fix ideas, consider the energy (2.1) with
$\Omega=\mathbb{T}_{L}$, the torus of side-length $L$. It is shown in [BGLN,
CCELM] that mean value

$\displaystyle m:=-1+\phi$

with $\phi$ and $L$ scaling like

$\displaystyle\phi\sim L^{-d/(d+1)}$

is the _critical parameter regime_ in the following sense: There is a
dimension-dependent constant $\xi_{d}>0$ such that for $\xi<\xi_{d}$ and

$\displaystyle\phi=\xi L^{-d/(d+1)},$

_the global minimizer_ is the constant state $u\equiv-1+\phi$, while for
$\xi>\xi_{d}$, _the global minimizer_ is not the constant state and is instead
close in $L^{p}$ to a “droplet-shaped” function.

$\nu_{m}$$\nu$$c_{m}$$c_{s}$$f_{\xi}$$\nu_{s}$$f_{\xi}$$\nu_{m}$$c_{m}$$c_{s}$$\nu$$\nu_{s}$
Figure 2.1: The function $f_{\xi}$ defined in (2.2) has for
$\tilde{\xi}_{d}<\xi<\xi_{d}$ a local (but not global) minimum at $\nu_{m}>0$
(left figure) and for $\xi>\xi_{d}$ a global minimum at $\nu_{m}>0$ (right
figure). For $\xi<\tilde{\xi}_{d}$, the only local minimum is at zero (not
shown).

Comparing the energy (2.1) of a function $u$ on a large torus to the energy of
the constant state and renormalizing leads, after a rescaling of space, to the
energy (1.1) (see [GW]). The $\Gamma$-convergence of (1.1) to

$\displaystyle\mathcal{E}_{0}^{\xi}(u):=c_{0}{\rm
Per}(\\{u=1\\})-4|\\{u=1\\}|+4\frac{|\\{u=1\\}|^{2}}{\xi^{3}}$

in the critical regime is established in [GW]. This limit energy is of the
type predicted by classical nucleation theory (going back to Gibbs [G]): A
positive term penalizes the surface energy of the region of positive phase,
while a volume term reflects the reduction in bulk energy when the volume of
the positive phase is large enough. (The higher order positive term prevents
the positive phase from growing too large and is a “remnant” of the mean
constraint.)

When the limit energy $\mathcal{E}_{0}^{\xi}$ is minimized as a function of
the volume $|\\{u=1\\}|$, one obtains the function
$f_{\xi}:[0,\infty)\to\mathbb{R}$ defined by

$\displaystyle
f_{\xi}(\nu):=\bar{C}_{1}\nu^{(d-1)/d}-4\nu+4\xi^{-(d+1)}\nu^{2},\quad\text{for}\quad\bar{C}_{1}:=c_{0}\sigma_{d}^{1/d}d^{(d-1)/d},$
(2.2)

where $c_{0}$ is the minimum energy cost for an interface connecting $\pm 1$
(equal to $2\sqrt{2}/3$ for the standard potential) and $\sigma_{d}$ is the
surface area of the unit sphere in $\mathbb{R}^{d}$. The behavior of $f_{\xi}$
and its dependence on $\xi$ is illustrated in Figure 2.1. For $\xi>\xi_{d}$,
the strictly positive local minimum of $f_{\xi}$ is the global minimum. This
corresponds to the droplet-shaped global minimizer of
$\mathcal{E}_{\phi}^{\xi}$ captured in [BGLN, CCELM, GW]. For
$\tilde{\xi}_{d}<\xi<\xi_{d}$ (see [GWW] for an explicit computation of the
bifurcation point $\tilde{\xi}_{d}$), the function $f_{\xi}$ has a global
minimum at zero and a strictly positive local minimum at $\nu_{m}$ with value
$c_{m}$. This corresponds to the droplet-shaped local minimizer of
$\mathcal{E}_{\phi}^{\xi}$ whose existence is proved in [GWW]. The other
volume-constrained minimizers of $\mathcal{E}_{\phi}^{\xi}$ subject to (1.3)
are also connected to the graph of $f_{\xi}$; see Subsection 2.1.

The function $f_{\xi}$ has a local _maximum_ at $\nu_{s}\in(0,\nu_{m})$ with
value $c_{s}$. It is natural to conjecture that this fact should indicate that
there is a saddle point of $\mathcal{E}_{\phi}^{\xi}$ with volume
approximately $\nu_{s}$. For both $\xi>\xi_{d}$ and
$\tilde{\xi}_{d}<\xi\leq\xi_{d}$, a mountain pass argument leads to the
existence of a saddle point of $\mathcal{E}_{\phi}^{\xi}$ in between the
constant state and the droplet-shaped minimizer [GW, GWW]. One expects that
this saddle point is the “critical nucleus” predicted by Cahn and Hilliard
[CH3] and that it is also approximately droplet-shaped, like the minimum.
Unfortunately the mountain pass argument fails to provide much information
about the saddle point. While it does yield that the _energy_ of the saddle
point $u_{s}$ is close to $c_{s}$, it yields no information about the measure
of $\\{u_{s}\approx 1\\}$ or about the shape or properties of this set. This
gap in the analysis motivates this paper.

###  2.1 Quantitative bounds

In [GWW], the following bounds were developed for the local minimizer
$u_{m,\phi}$ and volume-constrained minimizers $u_{\omega,\phi}$. The local
minimum $u_{m,\phi}$ has energy and “volume” close to $c_{m}$ and $\nu_{m}$
with the bounds

$\displaystyle\left|\mathcal{E}_{\phi}^{\xi}(u_{m,\phi})-c_{m}\right|$
$\displaystyle\lesssim\phi^{1/3},$ (2.3)
$\displaystyle|\nu(u_{m,\phi})-\nu_{m}|$ $\displaystyle\lesssim
C(\xi)\phi^{1/6}.$ (2.4)

Here and throughout, $\lesssim$ is used to indicate an upper bound by a
constant depending at most on the dimension and the potential $G$. In
addition, $u_{m,\phi}$ is estimated in $L^{2}$ against the function $\Psi_{m}$
that is $+1$ in a ball of volume $\nu_{m}$ and $-1$ otherwise according to

$\displaystyle|u_{m,\phi}-\Psi_{m}|_{\mathbb{T}_{\phi
L}}^{2}\lesssim\begin{cases}C(\xi)\phi^{1/6}&\text{for}\quad d=2,3\\\
\phi^{1/(2d)}&\text{for}\quad d\geq 4,\end{cases}$ (2.5)

where

$\displaystyle|u-\Psi_{m}|_{\mathbb{T}_{\phi L}}$
$\displaystyle:=\inf_{x_{0}\in\mathbb{T}_{\phi L}}||u-\Psi_{m}(\cdot-
x_{0})||_{L^{2}(\mathbb{T}_{\phi L})},$

and $\Psi_{m}$ is understood in the periodic sense. In addition via elementary
arguments one obtains the following bound on the volume of the interfacial
region:

$\displaystyle\lvert\\{-1+2\phi^{\frac{1}{3}}\leq u_{m,\phi}\leq
1-2\phi^{\frac{1}{3}}\\}\rvert\lesssim\phi^{1/3}.$ (2.6)

The analogous results hold for $u_{\omega,\phi}$ with volume $\omega$ and
energy $f_{\xi}(\omega)$. Our numerical investigation in Subsection 3.3 is in
part intended to explore whether these exponents are sharp.

Similarly, let $\tilde{u}_{s}$ denote any volume-constrained minimum with
energy $c_{s}+O(\phi^{1/3})$. Then there holds

$\displaystyle|\mathcal{E}_{\phi}^{\xi}(\tilde{u}_{s})-c_{s}|$
$\displaystyle\lesssim\phi^{1/3},$ (2.7)
$\displaystyle|\nu(\tilde{u}_{s})-\nu_{s}|$ $\displaystyle\lesssim
C(\xi)\phi^{1/6},$ (2.8)

and

$\displaystyle|\tilde{u}_{s}-\Psi_{s}|_{\mathbb{T}_{\phi
L}}^{2}\lesssim\begin{cases}C(\xi)\phi^{1/6}&\text{for}\quad d=2,3\\\
\phi^{1/(2d)}&\text{for}\quad d\geq 4,\end{cases}$ (2.9)

where $\Psi_{s}$ is $+1$ in a ball of volume $\nu_{s}$ and $-1$ otherwise. The
_saddle point_ $u_{s,\phi}$ obeys (2.6) and(2.7). We present numerical results
in Subsection 3.3 that explore whether it also satisfies (2.8) and (2.9).

In [GWW20] it is also shown that the constrained minima are _equal to their
Steiner symmetrization_ with respect to some point in the torus. This yields
in particular connectedness of appropriate superlevel and sublevel sets. Open
questions include whether the constrained minima are unique and whether their
level sets are convex. The numerical results in Subsection 3.3 support these
conjectures. A partial result on convexity of superlevel sets is given in
Proposition 4.14.

##  3 Numerical results

###  3.1 The String Method

The String Method starts with a collection of states in phase space (a
“string” of states). In the first step, one evolves each of the states forward
in time. In the second step, one reparameterizes the string to keep the states
well-separated in phase space. As an example, consider the gradient flow

$\displaystyle\dot{x}(t)=-\nabla V(x).$

Suppose that $x_{*}$ and $x_{**}$ are two isolated minima of $V$. We want to
find the orbit connecting $x_{*}$ and $x_{**}$, i.e., the curve $\gamma$
connecting $x_{*}$ and $x_{**}$ such that

$\displaystyle(\nabla V)^{\perp}(\gamma)=0,$

where $(\nabla V)^{\perp}$ is the component of $\nabla V$ normal to $\gamma$:

$\displaystyle(\nabla V)^{\perp}(\gamma)=\nabla V(\gamma)-(\nabla
V(\gamma),\hat{\tau})\hat{\tau}.$

Here $\hat{\tau}$ represents the unit tangent vector to the curve $\gamma$ and
$(\cdot,\cdot)$ represents the Euclidean inner product. Note that the orbit
will pass through at least one additional critical point.

The first step is to choose a method of parameterizing paths
$\\{\phi(\alpha)\colon\alpha\in[0,1]\\}$. Parameterization by arclength is a
convenient choice (although other methods offer advantages; see [ERV2]). Given
any initial path $\gamma_{0}$ connecting $x_{*}$ and $x_{**}$, we will
generate the family of time-dependent paths $\phi(\alpha,t)$ in the following
way. In the original String Method, the evolution is prescribed as

$\displaystyle\dot{\phi}=-\nabla
V(\phi)^{\perp}+\lambda\hat{\tau},\qquad\phi(\alpha,0)=\gamma_{0},$

where $\dot{\phi}$ denotes the time derivative of $\phi$. The first term on
the right-hand side moves the string in the direction normal to the string and
the second term on the right-hand side is a Lagrange multiplier term that acts
in the tangential direction and enforces the chosen parameterization.
Numerical implementation consists of discretizing the path into the images
$\\{\phi_{i}(t),i=0,1,\ldots,M\\}$ and following a two-step method: first
evolve according to $\dot{\phi}_{i}=-\nabla V(\phi_{i})^{\perp}$, and second
reparameterize. Note that the reparameterization is essential, since otherwise
the points will cluster around the minima.

The simplified String Method is based upon the observation that we can absorb
part of $(\nabla V(\phi))^{\perp}$ in the tangential component and that doing
so simplifies the algorithm and improves both stability and accuracy [ERV2].
In this case, the evolution of the string is best conceptualized as

$\displaystyle\dot{\phi}=-\nabla V(\phi)+\bar{\lambda}\hat{\tau},$

where $\bar{\lambda}=(\nabla V(\gamma),\hat{\tau})+\lambda$. The first step is
then to evolve according to the simplified dynamics $\dot{\phi}_{i}=-\nabla
V(\phi_{i})$ and the second step is, as before, to reparameterize. See [ERV2]
for a full discussion.

We say that the string has converged when there is a point along the string
and away from the endpoints where the energy gradient vanishes (to within some
tolerance). In our application, we find only one critical point between the
local minima. We then read off the local minimizers at the endpoints of the
string and the saddle point as the interior zero of the energy gradient.

######  Remark 3.1 (Generalizations).

We note that one does not need to know $x_{*}$ a priori. If one endpoint of
the initial string lies in the basin of attraction of $x_{*}$ and is left
free, the end of the string will evolve automatically toward the stable
stationary point.

We also note that the applicability of the String Method is not limited to
gradient flows (although it has been mainly used in that context). One can
consider

$\displaystyle\dot{x}(t)=\vec{b}(x),\qquad x(0)=x_{0}$

for general vector fields $\vec{b}$ and use the String Method to find the
orbit connecting $x_{0}$ and a zero $x_{*}$ of $\vec{b}$, i.e., a path
$\gamma$ connecting $x_{0}$ and $x_{*}$ such that

$\displaystyle(\vec{b})^{\perp}(\gamma)=0.$

###  3.2 Implementation and convergence

We implemented the String Method with a simple pseudo-spectral algorithm and
the FFT in MATLAB. One string typically had a size of $512$ pictures, although
for $\phi\in\\{0.025,0.0125\\}$ we had to take reduce to $256$ to cut down the
size of the string. Computations were done in double precision. The spatial
mesh size was chosen to be

$\displaystyle
N_{x}:=2\left\lceil\frac{\xi^{\frac{3}{2}}}{\phi^{\frac{3}{2}}}\right\rceil$

and the size of the time step was set to

$\displaystyle N_{t}:=N_{x}^{2}\phi.$

When computing the saddle point for $\phi=0.05$ and $\phi=0.0125$, we were not
able to reduce the tolerance to the value set as our stopping criterion, so we
let it run for as long as we could. The tolerance was reduced relatively
quickly (approximately 439556.72 s) to approximately the level that we
achieved in the end. After that results were slow to improve.

Table 1: Size of various parameters for $\xi=2.3$ and different values of
$\phi$.

$\phi$ 0.4000 0.2000 0.1000 0.0500 0.0250 0.0125 0.00625 0.003125 gridsize
picture $28^{2}$ $78^{2}$ $222^{2}$ $624^{2}$ $1766^{2}$ $4992^{2}$
$14120^{2}$ $39936^{2}$ picture 6 KB 49 KB 0.4 MB 3.1 MB 25 MB 199 MB 1.6 GB
12.8 GB string 512 pictures 3 MB 25 MB 202 MB 1.6 GB 12.8 GB 102 GB 817 GB
6533 GB CPU time for saddle in s 15 138 970 7446510 1277578 max RAM for saddle
in MB 664 746 1405 12731 45686 CPU time for min in s 9739 125993 max RAM for
min in MB 5359 38419

###  3.3 Numerical results

Table 2: Numerical results for the local minimum for $\xi=2.3$ and different
values of $\phi$.

(a) Raw data.

$\phi$ 0.4000 0.2000 0.1000 0.0500 0.0250 0.0125 0.00625
$\lvert\\{-1+2\phi^{\frac{1}{3}}\leq u_{m,\phi}\leq
1-2\phi^{\frac{1}{3}}\\}\rvert$ 0.0000 0.0000 0.1675 0.2975 0.2459 0.1647
0.1029 $\lvert\mathcal{E}_{\phi}^{\xi}(u_{m,\phi})-c_{m}\rvert$ 3.8415 2.1848
1.1512 0.5900 0.2986 0.1502 0.0753 $\lvert\nu(u_{m,\phi})-\nu_{m}\rvert$
1.9640 0.1711 0.1514 0.1773 0.1256 0.0834 0.0518 $\lvert
u_{m,\phi}-\Psi_{m}\rvert_{\mathbb{T}_{\phi L}}^{2}$ 4.2930 1.8807 0.9145
0.4526 0.2261 0.1132 0.0564 $\mathrm{error}_{m,\phi}$ 0.0223 0.0200 0.0053
0.0037 0.0031 0.0028 0.0027

(b) Log of ratios

$\phi$ 0.2000 0.1000 0.0500 0.0250 0.0125 0.00625
$\log_{2}\left(\frac{\lvert\\{-1+2(2\phi)^{\frac{1}{3}}\leq u_{m,2\phi}\leq
1-2(2\phi)^{\frac{1}{3}}\\}\rvert}{\lvert\\{-1+2\phi^{\frac{1}{3}}\leq
u_{m,\phi}\leq 1-2\phi^{\frac{1}{3}}\\}\rvert}\right)$ -0.8287 0.2748 0.5785
0.6782
$\log_{2}\left(\frac{\lvert\mathcal{E}_{2\phi}^{\xi}(u_{m,\phi})-c_{m}\rvert}{\lvert\mathcal{E}_{\phi}^{\xi}(u_{m,\phi})-c_{m}\rvert}\right)$
0.8142 0.9244 0.9644 0.9825 0.9915 0.9957
$\log_{2}\left(\frac{\lvert\nu(u_{m,2\phi})-\nu_{m}\rvert}{\lvert\nu(u_{m,\phi})-\nu_{m}\rvert}\right)$
3.5209 0.1765 -0.2278 0.4974 0.5899 0.6891 $\log_{2}\left(\frac{\lvert
u_{m,2\phi}-\Psi_{m}\rvert_{\mathbb{T}_{2\phi L}}^{2}}{\lvert
u_{m,\phi}-\Psi_{m}\rvert_{\mathbb{T}_{\phi L}}^{2}}\right)$ 1.1907 1.0402
1.0147 1.0013 0.9985 1.0039

Although the local minimizer can be approximated by steepest descent without
the String Method, we begin by summarizing the data recovered for the
minimizer; see Table 2. Here the stopping criterion was that the spectral norm
of the difference of two consecutive approximations of the minimizer is
smaller than $5\cdot 10^{-3}$ and the energy gradient at the minimum was at
most $10^{-3}$.

Compared to (2.3), the numerical results seem to suggest the better
convergence rate $\phi^{1}$ for the energy and roughly $\phi^{2/3}$ for the
interfacial region and the droplet “volume.” The $L^{2}$ difference to the
sharp interface profile also seems to converge at rate $\phi^{1}$. These
results are consistent with the analysis but suggest that the analytical
bounds to date are not sharp. The level lines and a cross-section of the local
minimizer are shown in Figure 3.1. See also Figure 3.2 for an illustration
when $\xi<\xi_{d}$.

We now turn to the results of the String Method for the saddle point. The
numerical data is summarized in Table 3. As for the minimum, the results
suggest better convergence in energy and $L^{2}$ than reflected in (2.7) and
(2.9). A good estimate of the convergence rate for the volume and the
interfacial volume would require additional data, but the overall data support
the _conjecture that the saddle point is indeed a volume-constrained minimum
with “volume” approximately $\nu_{s}$_.

In Figure 3.3, we plot a closeup of the contours of the saddle point, which is
a droplet-shaped function with the expected properties. In addition to being
connected and Steiner symmetric about the axes, the superlevel sets for values
near $1$ appear to be nearly spherical and convex. We observe a unique minimum
point in the corner and (bearing in mind the periodicity) convexity of the
sublevel set $\\{u\leq s\\}$ for $s$ near this minimum value. Motivated by
these numerical observations, we examine convexity in Section 4.

Table 3: Numerical results for the saddle point for $\xi=2.3$ and different
values of $\phi$.

(a) Raw data. $\phi$ | 0.4000 | 0.2000 | 0.1000 | 0.0500 | 0.0250
---|---|---|---|---|---
$\lvert\\{-1+2\phi^{\frac{1}{3}}\leq u_{s,\phi}\leq 1-2\phi^{\frac{1}{3}}\\}\rvert$ | 0.0000 | 0.0000 | 0.0387 | 0.0725 | 0.0612
$\lvert\mathcal{E}_{\phi}^{\xi}(u_{s,\phi})-c_{s}\rvert$ | 0.6101 | 0.2041 | 0.0047 | 0.0182 | 0.0082
$\lvert\nu(u_{s,\phi})-\nu_{s}\rvert$ | 4.5279 | 0.4664 | 0.0724 | 0.0475 | 0.0747
$\lvert u_{s,\phi}-\Psi_{s}\rvert_{\mathbb{T}_{\phi L}}^{2}$ | 5.6429 | 2.9339 | 1.3946 | 0.7607 | 0.5849
$\mathrm{error}_{s,\phi}=\left\|\nabla E(u_{s,\phi})\right\|^{2}$ | 0.0039 | 0.0038 | 0.0008 | 0.0666 | 0.2922

(b) Log of ratios $\phi$ | 0.2000 | 0.1000 | 0.0500 | 0.0250
---|---|---|---|---
$\lvert\\{-1+2\phi^{\frac{1}{3}}\leq u_{s,\phi}\leq 1-2\phi^{\frac{1}{3}}\\}\rvert$ |  |  | -0.9055 | 0.2450
$\lvert\mathcal{E}_{\phi}^{\xi}(u_{s,\phi})-c_{s}\rvert$ | 1.5798 | 5.4405 | -1.9547 | 1.1557
$\lvert\nu(u_{s,\phi})-\nu_{s}\rvert$ | 3.2792 | 2.6875 | 0.6086 | -0.6531
$\lvert u_{s,\phi}-\Psi_{s}\rvert_{\mathbb{T}_{\phi L}}^{2}$ | 0.9436 | 1.0730 | 0.8744 | 0.3792

(a) Level curves of the local min for $\phi=0.1$, $\xi=1.9$.

(b) Cross-section of the local min at $y=-3.19$.

Figure 3.1: The local minimizer has the expected properties. The string is
converged with error of order $1e-5$.

(a) Local minimizer.

(b) Saddle point.

Figure 3.2: Local minimizer and saddle point for $\phi=0.1$ and $\xi=1.6$,
i.e., in the regime that the constant state is not the global minimizer. The
string is converged up to error $1e-5$. Figure 3.3: A close-up of the level
lines of the saddle point for $\phi=0.2$ and $\xi=1.9$. The string is
converged up to error $1e-4$. This figure motivates the analysis in section 4.

##  4 Convexity

In the following sections we consider the volume-constrained minimizers; that
is, we consider the minimizers of the energy (1.1) over the set

$\displaystyle X_{\phi,\omega}:=\left\\{u\in H^{1}\cap L^{4}(\mathbb{T}_{\phi
L}):\mathchoice{{\vbox{\hbox{$\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\scriptscriptstyle-$}}\kern-1.875pt}}\\!\int_{\mathbb{T}_{\phi
L}}u\,dx=-1+\phi,\quad\nu(u)=\omega\right\\},$ (4.1)

where the “volume” $\nu(\cdot)$ is given by (1.4). We are interested in
geometric properties of the minimizers. For simplicity of notation, we will
write $u$ instead of $u_{\omega,\phi}$.

We would like to understand convexity of the superlevel sets of the
constrained minimizers. Based on the numerics (see for instance Figure 3.3),
it seems likely that the superlevel sets are convex not only near the maximum
but for all values up to some critical value, at which point the convexity
changes and the sublevel sets around the minimum become convex. Such a result
is currently out of reach, but we will adapt a method of Caffarelli and Spruck
to demonstrate convexity of superlevel sets under Hypothesis (H2), below.
Before turning to this result, we recall some basic facts from the previous
analysis.

We begin by recalling two results from [GWW] and [GWW20]. We recall first
that, according to the mean condition, the definition of $\nu(\cdot)$, and
Lemma 2.9 from [GWW], one has the following uniform bounds.

######  Lemma 4.1.

For any $\omega_{1}>0$ there exists $\phi_{0}>0$ such that for all
$\phi\in(0,\phi_{0})$, any volume-constrained minimizer $u$ for
$\omega\in[\omega_{1},\xi^{3}/2]$ satisfies

$\displaystyle-1-\phi^{1/3}\leq\inf_{\mathbb{T}_{\phi L}}u\leq-1+\phi\leq
1-2\phi^{1/3}\leq\sup_{\mathbb{T}_{\phi L}}u\leq 1+\phi^{1/3}.$ (4.2)

The upper bound $\phi_{0}$ from Lemma 4.1 is the only restriction we will
place on $\phi>0$ for the rest of the article. As a consequence of the uniform
bounds, any constrained minimizer is smooth and satisfies the Euler Lagrange
equation

$\displaystyle-\phi\Delta
u+\frac{1}{\phi}G^{\prime}(u)+\lambda_{\phi}+\lambda_{\omega}\chi^{\prime}(u)=0\qquad\hbox{in}\>\mathbb{T}_{\phi
L},$ (4.3)

where $\lambda_{\phi}$, $\lambda_{\omega}\in\mathbb{R}$ are Lagrange
parameters corresponding to the mean and volume constraints, respectively.
Next we recall the symmetry of the constrained minimizers.

######  Theorem 4.2 (Theorem 1.1, [GWW20]).

Any volume-constrained minimizer $u$ is equal (up to a translation) to its
Steiner symmetrization about the origin. In particular, its superlevel sets
are simply connected, and $u$ is strictly decreasing along all rays starting
at the (unique) maximum of $u$. In fact for any ray $e$ starting in the
maximum, the directional derivative is strictly negative:

$\displaystyle\partial_{e}u<0$ (4.4)

on all of $\\{x\in\mathbb{T}_{\phi L}:x_{i}>0,\,i=1,\ldots,d\\}$ and the
gradient vanishes only at the (unique) maximum and minimum.

######  Remark 4.3.

The first part of the theorem is precisely Theorem 1.1 from [GWW20], and (4.4)
and the nonvanishing of the gradient follow from the Strong Maximum Principle,
as shown in the proof of Theorem 1.1 in [GWW20].

###  4.1 Adaptation of a method of Caffarelli and Spruck

We will now use the Euler Lagrange equation (4.3) and a method that Caffarelli
and Spruck introduced in [CS, Theorem 2.1] to establish convexity of the
superlevel sets of volume-constrained minimizers $u=u_{\omega,\phi}$ under an
assumption on the nonlinearity, specified in Hypothesis (H2), below. We assume
without loss of generality that the location of the unique maximum of $u$ is
$x=0$. By symmetry it is often sufficient to restrict to the first quadrant,
i.e., to

$\mathbb{T}_{\phi L}^{+}:=\\{x\in\mathbb{T}_{\phi
L}:x_{i}>0,\,i=1,\ldots,d\\}.$

######  Remark 4.4.

By symmetry of $u$ we have for $i\neq j$ that

$\displaystyle\partial_{i}u(0,\ldots,0,x_{j},0,\ldots,0)=0.$

Hence the Hessian matrix $D^{2}u(0)$ is diagonal, and we have

$\displaystyle\left(D^{2}u(0)\right)_{ij}=\lambda_{i}\delta_{ij}$

where $(\delta_{ij})_{ij}$ denotes the identity matrix in $\mathbb{R}^{d}$ and
$\lambda_{i}\in\mathbb{R}$ for $i\in\\{1,\ldots,d\\}$. Because $x=0$ is a
maximum point of $u$, we have $\lambda_{i}\leq 0$ for $i\in\\{1,\ldots,d\\}$.
More is true:

######  Lemma 4.5.

The origin is a non-degenerate critical point of any volume-constrained
minimizer $u$, and $\partial^{2}_{e}u(0)<0$ for any ray $e$ starting in the
origin.

###### Proof.

Let $e$ be any ray starting in the origin. We choose $i\in\\{1,\ldots,d\\}$
such that $e$ points into the halfspace $\mathbb{T}_{\phi
L}^{i}:=\\{x\in\mathbb{T}_{\phi L}:x_{i}>0\\}$. According to Theorem 4.2, we
have $\partial_{e}u<0$ in $\mathbb{T}_{\phi L}^{i}$. Note that by (4.3) the
function $\partial_{e}u$ satisfies the equation

$\displaystyle-\phi\Delta(\partial_{e}u)+\left(\frac{1}{\phi}G^{\prime\prime}(u)+\lambda_{\omega}\>\chi^{\prime\prime}(u)\right)\partial_{e}u=0\qquad\text{in
$\mathbb{T}_{\phi L}^{i}$. }$

For any $\rho>0$ such that ball $B_{\rho}(0)\Subset\mathbb{T}_{\phi L}$, the
function $\partial_{e}u$ takes its maximum in $\overline{\mathbb{T}_{\phi
L}^{i}\cap B_{\rho}(0)}$ in $x=0$. Hence Hopf’s maximum principle yields
$\lambda=\partial^{2}_{e}u(0)<0$. Consequently $x=0$ is a non-degenerate
maximum point. ∎

As a consequence of Lemma 4.5, the superlevel sets $S_{c}:=\\{u>c\\}$ are
strictly convex for all $c$ sufficiently close to
$u_{max}:=\sup_{\mathbb{T}_{\phi L}}u$, which according to (4.2) is close to
$1$. Thus there exists $c^{*}<u_{max}$ such that $S_{c}$ is convex for all
$c^{*}\leq c<u_{max}$.

We would now like to investigate convexity of superlevel sets of $u$ further
away from $u_{max}$. Suppose that $c_{*}>\inf_{\mathbb{T}_{\phi L}}u$ is some
value such that

> Hypothesis (H1):
>
> $\displaystyle S_{c_{*}}=\\{u>c_{*}\\}\text{ is convex and }\partial
> S_{c_{*}}\text{ is smooth.}$

We assume that $c^{*}>c_{*}$ since otherwise there is nothing to prove. We
will show that all sublevel sets $S_{c}$ for $c_{*}<c<c^{*}$ are convex.

In the remainder of this section we will write the Euler-Lagrange equation of
$u$ in the form $\Delta u=f(u)$, where

$\displaystyle
f(u):=\frac{1}{\phi^{2}}G^{\prime}(u)+\frac{\lambda_{\phi}}{\phi}+\frac{\lambda_{\omega}}{\phi}\chi^{\prime}(u).$
(4.5)

Note that because of the smoothness of $G$ and $\chi$, $f$ is also smooth. We
will work under the assumption:

> Hypothesis (H2):
>
> $\displaystyle f(c)>0\quad\text{and}\quad f^{\prime}(c)>0\quad\hbox{for
> all}\quad c_{*}<c<u_{max}.$

In the sequel we will work with three collinear points $x,y,z\in S_{c_{*}}$
such that $z=tx+(1-t)y$ for some $t\in(0,1)$, and we will write $z\in[x,y]$
for short and speak of “the triple” $(x,y,z)$. We will call a triple
_admissible_ if it is an element of

$\displaystyle{\cal{A}}:=\\{(x,y,z)\colon x,y,z\>\hbox{are collinear},\>x,y\in
S_{c_{*}},\text{and }\>u(x)>u(z)\\}.$ (4.6)

Note that ${\cal{A}}$ is not empty since $u$ is strictly decreasing along all
rays emanating from the origin.

We consider the function

$\displaystyle F(x,y,z):=u(y)-u(z),\qquad\hbox{for}\quad(x,y,z)\in{\cal{A}}.$

Because of the equivalence in Lemma 4.6 below, we will be interested in

$\displaystyle\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z).$

Let $(x_{k},y_{k},z_{k})\in{\cal{A}}$ be a maximizing sequence for $F$. By
compactness of $\overline{S_{c_{*}}}$, there exist limit points
$x_{0},y_{0},z_{0}\in\overline{S_{c_{*}}}$. By passing to suitable
subsequences, we may assume that $x_{k}\to x_{0}$, $y_{k}\to y_{0}$ and
$z_{k}\to z_{0}$. Moreover $z_{0}=t_{0}x_{0}+(1-t_{0})y_{0}$ for some
$t_{0}\in[0,1]$ and $u(x_{0})\geq u(z_{0})$. A priori it may be that
$x_{0}=z_{0}$ or $z_{0}=y_{0}$ or even $x_{0}=z_{0}=y_{0}$, however see Remark
4.8.

Our plan in order to show convexity of $S_{c}=\\{u>c\\}$ for all
$c_{*}<c<u_{max}$ is as follows.

  * •

In Lemma 4.6, we show that convexity of $S_{c}$ for all $c_{*}<c<u_{max}$ is
equivalent to

$\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z)\leq 0.$

  * •

We then argue by contradiction. Thus we assume that
$\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z)>0$ and derive necessary conditions for the
limit $(x_{0},y_{0},z_{0})$ of the maximizing sequence (see Lemmas 4.9 \-
4.13). The proofs rely heavily on Lemma 4.5, however they do not explicitly
use the equation for $u$. In particular (H2) is not needed.

  * •

The main result of this section is Proposition 4.14. It states that if
$S_{c_{*}}$ is convex, then the superlevel sets $S_{c}$ are convex for all
$c_{*}<c<u_{max}$. It is only now that Hypothesis (H2) is needed. The main
part of the proof is based on a domain variation argument. Specifically, using
that $(x_{0},y_{0},z_{0})$ is extremal, we perform a variation in the
independent variables, using Lemmas 4.9 \- 4.13 to show that the perturbed
point $(x^{*}_{0},y^{*}_{0},z^{*}_{0})$ satisfies

$\displaystyle F(x^{*}_{0},y^{*}_{0},z^{*}_{0})>F(x_{0},y_{0},z_{0})>0,$

which violates the extremality of $(x_{0},y_{0},z_{0})$.

######  Lemma 4.6.

The convexity of $S_{c}$ for all $c_{*}\leq c\leq u_{max}$ is equivalent to

$\displaystyle\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z)\leq 0.$ (4.7)

###### Proof.

Step 1. We begin by assuming that the sets $S_{c}$ are convex for all
$c_{*}<c<u_{max}$. Let $(x,y,z)\in{\cal{A}}$. If $u(y)>u(x)$, then convexity
of the superlevel sets gives $u(z)\geq u(x)$ for all $z\in[x,y]$ and $(x,y,z)$
is not admissible. Hence any admissible triple satisfies $u(x)\geq u(y)$.
However then the convexity of the superlevel sets gives $u(z)\geq u(y)$, so
that $F(x,y,z)\leq 0$, and taking the supremum over ${\cal{A}}$ preserves this
inequality.
Step 2. We now assume $\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z)\leq 0$ and will show
that $S_{c}$ is convex for all $c_{*}<c<u_{max}$. Let $x,y\in S_{c}$. For any
$z\in[x,y]$, if $(x,y,z)$ is admissible, then our assumption on $F$ yields
$u(z)\geq u(y)>c$. On the other hand if $z\in[x,y]$ and $(x,y,z)$ is not
admissible, then by definition of the admissible set there holds $u(z)\geq
u(x)>c$. Hence in either case $z\in S_{c}$, and thus $S_{c}$ is convex. ∎

In order to prove the convexity of $S_{c}$ for all $c_{*}\leq c\leq u_{max}$,
we will use Lemma 4.6 and argue by contradiction. Thus we assume that (4.7)
does not hold and we will discuss the extremal situation, by which we mean the
following:

######  Definition 4.7.

We will call $(x_{0},y_{0},z_{0})$ a positive extremal if there exists a
maximizing sequence $(x_{k},y_{k},z_{k})\in{\cal{A}}$ with $x_{k}\to x_{0}$,
$y_{k}\to y_{0}$ and $z_{k}\to z_{0}$ such that

  * ($P_{1}$)

$x_{0},y_{0},z_{0}\in\overline{S_{c_{*}}}$ and
$z_{0}=t_{0}x_{0}+(1-t_{0})y_{0}$ for some $t_{0}\in[0,1]$;

  * ($P_{2}$)

$u(x_{0})\geq u(z_{0})$;

  * ($P_{3}$)

there holds $0<\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z)=F(x_{0},y_{0},z_{0})$, so
that

$\displaystyle u(y_{0})>u(z_{0}).$ (4.8)

######  Remark 4.8.

Clearly for any positive extremal there holds $x_{0}\neq y_{0}$, since
otherwise $x_{0}=y_{0}=z_{0}$ (by ($P_{1}$)), which would imply
$F(x_{0},y_{0},z_{0})=0$ and hence violate ($P_{3}$). Also $y_{0}\neq z_{0}$
(i.e., $t_{0}\neq 0$), since this would also imply $F(x_{0},y_{0},z_{0})=0$.
On the other hand, the case $x_{0}=z_{0}$ cannot be excluded. (Indeed, Lemma
4.12 shows that this equality holds for any positive extremal if $f$ is
analytic.)

######  Lemma 4.9.

If the triple $(x_{0},y_{0},z_{0})$ is a positive extremal, then $y_{0}\neq
0$.

###### Proof.

For a contradiction we consider any triple $(x_{0},y_{0},z_{0})$ with
$y_{0}=0$, i.e., $y_{0}$ is the unique point where $u$ takes its maximum. We
will show that $(x_{0},y_{0},z_{0})$ is not the limit of a sequence of
admissible triples $(x_{k},y_{k},z_{k})\in{\cal{A}}$ as $k\to\infty$. Thus
assume that there exists a sequence $(x_{k},y_{k},z_{k})\in{\cal{A}}$ such
that $\lim_{k\to\infty}(x_{k},y_{k},z_{k})=(x_{0},y_{0},z_{0})$. We
distinguish the following cases.

  * (1)

If $y_{0}=0$ and $x_{0}\neq z_{0}$, then by ($P_{1}$), we have
$z_{0}=t_{0}x_{0}$ for some $t_{0}\in(0,1]$. Since by Lemma 4.5), $u$ is
strictly decreasing along any ray starting in the origin, there holds
$u(z_{0})>u(x_{0})$, which contradicts (P2).

  * (2)

Now suppose $y_{0}=0$ and $x_{0}=z_{0}$. Because $(x_{k},y_{k},z_{k})$ is
admissible, we have $u(x_{k})>u(z_{k})$. From the collinearity of
$(x_{0},y_{0},z_{0})$ and $y_{0}=0$, we deduce that

$\displaystyle\frac{x_{k}-z_{k}}{|x_{k}-z_{k}|}=\frac{x_{k}-y_{k}}{|x_{k}-y_{k}|}\to\frac{x_{0}}{|x_{0}|}=:e_{0}\quad\text{as
$k\to\infty$.}$

Hence

$\displaystyle
0<\frac{u(x_{k})-u(z_{k})}{|x_{k}-z_{k}|}=\int\limits_{0}^{1}\frac{x_{k}-z_{k}}{|x_{k}-z_{k}|}\cdot
Du(sx_{k}+(1-s)z_{k})\>ds\to e_{0}\cdot Du(x_{0})=\partial_{e_{0}}u(x_{0}),$

and we obtain in the limit $k\to\infty$ that $\partial_{e_{0}}u(x_{0})\geq 0,$
which contradicts the fact that the directional derivative is strictly
negative along rays from the origin (cf. Theorem 4.2).

∎

######  Lemma 4.10.

For any positive extremal $(x_{0},y_{0},z_{0})$, there holds

  * (i)

$y_{0}\in S_{c_{*}}$ and

  * (ii)

$z_{0}\in S_{c_{*}}$.

###### Proof.

Since necessarily $y_{0},z_{0}\in\overline{S_{c_{*}}}$, the issue is to show
$y_{0},z_{0}\notin\partial S_{c_{*}}$. We begin by considering $y_{0}$. Since
$z_{0}\in\overline{S_{c_{*}}}$, we have $u(z_{0})\geq c_{*}$. Combining this
with (4.8) implies (i).

We now turn to $z_{0}$. Let us assume $z_{0}\in\partial S_{c_{*}}$, i.e.,
$u(z_{0})=c_{*}$. By (i) we have $y_{0}\in S_{c_{*}}$. By ($P_{1}$) the points
$x_{0}$, $y_{0}$ and $z_{0}$ are collinear. Since $S_{c_{*}}$ is convex, the
ray that starts in $y_{0}$ and passes through $z_{0}$ and $x_{0}$ intersects
$\partial S_{c_{*}}$ in exactly one point. By ($P_{1}$) and ($P_{2}$) the
point $x_{0}$ must be in $\partial S_{c_{*}}$ and thus $x_{0}=z_{0}$. Now let
$(x_{k},y_{k},z_{k})\in{\cal{A}}$ be a maximizing sequence converging to
$(x_{0},y_{0},z_{0})$ as in Definition 4.7. For sufficiently large $k$ the
following holds: Since $x_{k},z_{k}\to x_{0}=z_{0}\in\partial S_{c_{*}}$,
there exists - for $k$ large - a ray that starts in $y_{k}$, passes through
$z_{k}$ and $x_{k}$ and intersects $\partial S_{c_{*}}$ in exactly one point.
This point is denoted by $\tilde{x}_{k}$. Note that
$x_{k}\in[\tilde{x}_{k},z_{k}]$ and $\tilde{x}_{k}\to x_{0}$ as $k\to\infty$.
Since $\partial S_{c_{*}}$ is smooth, Hopf’s boundary lemma applied on
$S_{c_{*}}$ then implies that $u$ is strictly increasing along
$[\tilde{x}_{k},z_{k}]$ (for $k$ large enough). However then
$u(x_{k})<u(z_{k})$ which contradicts the admissibility of
$(x_{k},y_{k},z_{k})$. As a consequence $z_{0}\notin\partial S_{c_{*}}$. ∎

Here and below, let

$\displaystyle e:=\frac{x_{0}-y_{0}}{|x_{0}-y_{0}|}$ (4.9)

be the unit vector pointing from $y_{0}$ to $x_{0}$.

######  Lemma 4.11.

For any positive extremal $(x_{0},y_{0},z_{0})$, there holds
$\partial_{e}u(z_{0})=0$.

###### Proof.

We consider separately the cases $x_{0}\neq z_{0}$ and $x_{0}=z_{0}$. If
$x_{0}\neq z_{0}$, then according to ($P_{1}$) and ($P_{3}$), $u$ has a
minimum in $z_{0}$ along the segment $[x_{0},y_{0}]$, which implies
$\partial_{e}u(z_{0})=0$. If $x_{0}=z_{0}$, then ($P_{1}$), ($P_{3}$), and the
definition of ${\cal{A}}$ imply that for any maximizing sequence
$(x_{k},y_{k},z_{k})\in{\cal{A}}$, the function $u$ has an interior minimum at
some point $\xi_{k}$ between $y_{k}$ and $x_{k}$, so that
$\partial_{e}u(\xi_{k})=0$. Necessarily $\xi_{k}$ converges to some point
$\xi_{0}\in[y_{0},x_{0}]$. If $\xi_{0}$ lies in the interior, then minimality
of $z_{0}=x_{0}$ is contradicted. Hence $\xi_{0}=x_{0}=z_{0}$ and
$\partial_{e}u(z_{0})=0$. ∎

######  Lemma 4.12.

For any positive extremal $(x_{0},y_{0},z_{0})$, there holds (i)
$\partial_{e}u(y_{0})=0$, (ii) $x_{0}=z_{0}$ or $u$ is constant on
$[x_{0},z_{0}]$, and (iii) $\partial_{e}^{2}u(z_{0})=0$.

###### Proof.

ad (i). From Lemma 4.10 we know that $y_{0}\in S_{c_{*}}$. We consider
perturbing $F$ by varying $y_{0}$ in the $e$ direction: Let $y_{0}^{s}$ be
given by

$\displaystyle y_{0}^{s}=y_{0}+s\>e\quad\hbox{where}\quad|s|<s_{0}$ (4.10)

for $s_{0}>0$ small enough that $y_{0}^{s}\in S_{c_{*}}$ and $y_{0}^{s}\neq
z_{0}$ for all $|s|<s_{0}$. Then

$\displaystyle F(x_{0},y_{0},z_{0})\geq F(x_{0},y_{0}^{s},z_{0})$ (4.11)

for all $|s|<s_{0}$, which implies $\partial_{e}u(y_{0})=0$.

ad (ii). It suffices to assume that $x_{0}\neq z_{0}$ and show that $u$ is
constant on the segment $[x_{0},z_{0}]$. We know that $u(x_{0})\geq u(z_{0})$
(this is ($P_{2}$)). If $u$ is not constant along the segment $[x_{0},z_{0}]$,
then there exists an intermediate point $\tilde{x}$ between $x_{0}$ and
$z_{0}$ such that $u(\tilde{x})\neq u(z_{0})$, and necessarily
$u(\tilde{x})>u(z_{0})$, since otherwise $(x_{0},y_{0},\tilde{x})$ is
admissible and $u(y_{0})-u(\tilde{x})>u(y_{0})-u(z_{0})$, violating optimality
of the triple $(x_{0},y_{0},z_{0})$. We now consider the triple
$(\tilde{x},y_{0},z_{0})$. Since the points are collinear and
$u(\tilde{x})>u(z_{0})$, this triple is admissible, and since
$F(\tilde{x},y_{0},z_{0})=F(x_{0},y_{0},z_{0})$, $(\tilde{x},y_{0},z_{0})$ is
a positive extremal, as well. As in Step 1, we deduce
$F(\tilde{x},y_{0},z_{0})\geq F(\tilde{x},y_{0}^{s},z_{0})$ for all
sufficiently small perturbations in the $e$ direction. Next we rotate the
segment $[\tilde{x},y_{0}]$ about an angle $\theta$ with center $z_{0}$.
Letting $X$ denote the rotation of $\tilde{x}$, we observe that
$u(X)>u(z_{0})$ if $\theta$ is small enough. Let $Y_{0}$ denote the rotation
of $y_{0}$. For $\theta$ small enough $Y_{0}\notin\partial S_{c_{*}}$ and the
triple $(X,Y_{0},z_{0})$ is admissible (see Figure 4.1). Since
$(\tilde{x},y_{0},z_{0})$ is a positive extremal, we have (for all $\theta$
small enough) that

$\displaystyle F(\tilde{x},y_{0},z_{0})\geq F(X,Y_{0},z_{0})>0.$ (4.12)

Note that all points in a sufficiently small neighbourhood of $y_{0}$ can be
described as a perturbation of $y_{0}$ in the direction $e$ combined with a
rotation about a small angle $\theta$. Hence combining shifts and rotations
allows us to deduce that $u$ has a local maximum in $y_{0}$. This is
impossible, since $y_{0}\neq 0$ by Lemma 4.9 and since by Theorem 4.2 $u$ is
strictly decreasing on all rays from the origin. From this contradiction we
deduce that if $x_{0}\neq z_{0}$, then $u$ is constant along the segment
$[z_{0},x_{0}]$.

ad (iii). Recall from Lemma 4.11 that $\partial_{e}u(z_{0})=0$. Optimality of
the triple hence implies $\partial_{e}^{2}u(z_{0})\geq 0$. Strict inequality
$\partial_{e}^{2}u(z_{0})>0$ would however allow us to combine shifts as in
(4.10) and rotations around $z_{0}$ in order to deduce (similarly to in Step 2
above) that $u$ has a local maximum in $y_{0}$, which, as in Step 2, can be
ruled out. Hence $\partial_{e}^{2}u(z_{0})=0$. ∎

$S_{c_{*}}$$S_{c^{*}}$$X$$Y_{0}$$x$$y_{0}$$z_{0}$ Figure 4.1:

######  Lemma 4.13.

For any positive extremal, $|\nabla u(y_{0})|\neq 0$, $|\nabla u(z_{0})|\neq
0$, and there holds

$\displaystyle\frac{\nabla u(z_{0})}{|\nabla u(z_{0})|}=\frac{\nabla
u(y_{0})}{|\nabla u(y_{0})|}.$ (4.13)

###### Proof.

The nonvanishing of the gradient in $y_{0}$ and $z_{0}$ follows from Theorem
4.2, Lemmas 4.9 and 4.10, and $c_{*}>u_{min}$. Now assume that (4.13) does not
hold. Then there exists a vector $v\in\mathbb{R}^{d}$ such that

$\displaystyle v\cdot\nabla u(y_{0})>0\qquad\hbox{and}\qquad v\cdot\nabla
u(z_{0})<0.$ (4.14)

Step 1. By Lemma 4.12 we have $u(x_{0})=u(z_{0})$ in the positive extremal
case. We will now construct an admissible triple that we will then use in Step
3 to contradict extremality. We begin by writing
$y_{0}=z_{0}+|y_{0}-z_{0}|\hat{e}$, for the unit vector

$\displaystyle\hat{e}:=\frac{y_{0}-z_{0}}{|y_{0}-z_{0}|}$

pointing from $z_{0}$ to $y_{0}$. For $\epsilon>0$ we perturb $y_{0}$ and
consider the set

$\displaystyle\left\\{w_{t}:=z_{0}+(1-t)|y_{0}-z_{0}|\hat{e}+(1-t)\,\epsilon
v\>:\>t\in(0,1)\right\\}.$

The above set includes a segment joining $w_{0}=y_{0}+\epsilon v$ (for $t=0$)
and $w_{1}=z_{0}$ (for $t=1$). For $t=0$ we observe that

$\displaystyle\left(w_{0}-y_{0}\right)\cdot\nabla u(y_{0})=\epsilon
v\cdot\nabla u(y_{0})\overset{\eqref{convex41}}{>}0.$

On the other hand for all $t\in[0,1)$ there holds

$\displaystyle\left(w_{t}-z_{0}\right)\cdot\nabla u(z_{0})$ $\displaystyle=$
$\displaystyle(1-t)(y_{0}-z_{0})\cdot\nabla u(z_{0})+(1-t)\epsilon
v\cdot\nabla u(z_{0})$ $\displaystyle=$ $\displaystyle(1-t)\epsilon
v\cdot\nabla u(z_{0})\overset{\eqref{convex41}}{<}0,$

where the second equality holds because of Lemma 4.11. For small $t_{0}>0$ we
now consider the triple $(w_{1+t_{0}},w_{0},w_{1-t_{0}})$. To see that this
triple is admissible for $\epsilon$ and $t_{0}$ close to zero, it suffices to
check:

  * (i)

For $t=\frac{1-t_{0}}{1+t_{0}}\in(0,1)$ we have
$w_{1-t_{0}}=tw_{1+t_{0}}+(1-t)w_{0}$. Thus the three points are collinear.

  * (ii)

Since $u(x_{0})=u(z_{0})$ (according to Lemma 4.12) and both $z_{0}$ and
$y_{0}$ lie in the interior of $S_{c_{*}}$ (by Lemma 4.10), there holds
$w_{1+t_{0}},\,w_{0}\in S_{c_{*}}$ for $t_{0}$ and $\epsilon$ sufficiently
small.

  * (iii)

Expanding $u(w_{1+t_{0}})-u(w_{1-t_{0}})$ with respect to $t_{0}$ and
$\epsilon$ and applying Lemma 4.11, there holds
$u(w_{1+t_{0}})>u(w_{1-t_{0}})$ for $t_{0}$ and $\epsilon$ sufficiently small.

Hence for $t_{0}$ and $\epsilon$ sufficiently small, there holds
$(w_{1+t_{0}},w_{0},w_{1-t_{0}})\in{\cal{A}}$.

Step 2. We will now contradict the extremality of $(x_{0},y_{0},z_{0})$ by
showing that $F(w_{1+t_{0}},w_{0},w_{1-t_{0}})>F(x_{0},y_{0},z_{0})$. Indeed,
again using Lemma 4.11, we deduce:

$\displaystyle F(w_{1+t_{0}},w_{0},w_{1-t_{0}})-F(x_{0},y_{0},z_{0})$
$\displaystyle=$ $\displaystyle
u(w_{0})-u(w_{1-t_{0}})-\left(u(y_{0})-u(z_{0})\right)$ $\displaystyle=$
$\displaystyle u(y_{0})+\epsilon v\cdot\nabla
u(y_{0})-\left(u(z_{0})+t_{0}\epsilon v\cdot\nabla
u(z_{0})\right)+o(\epsilon)$
$\displaystyle\qquad-\left(u(y_{0})-u(z_{0})\right)$ $\displaystyle=$
$\displaystyle\epsilon\left(v\cdot\nabla u(y_{0})-t_{0}\epsilon v\cdot\nabla
u(z_{0})\right)+o(\epsilon).$

From (4.14) and $t_{0}>0$ we deduce
$F(w_{1+t_{0}},w_{0},w_{1-t_{0}})>F(x_{0},y_{0},z_{0})$ for $\epsilon$ small.
∎

######  Proposition 4.14.

Let $u$ be a Steiner symmetric solution of $\Delta u=f(u)$ in
$\mathbb{T}_{\phi L}$, where $f$ is given by (4.5). Assume $u$ takes its
unique maximum point $u_{max}$ in $x=0$. Let $c_{*}<u_{max}$ be a level such
that Hypotheses (H1) and (H2) hold. Then $S_{c}$ is convex for all
$c_{*}<c<u_{max}$.

###### Proof.

We argue by contradiction. Thus we assume that there exists a
$c\in(c_{*},u_{max})$ such that $S_{c}$ is not convex. By Lemma 4.6 this is
equivalent to

$\displaystyle\sup_{(x,y,z)\in{\cal{A}}}F(x,y,z)>0.$

Let $(x_{0},y_{0},z_{0})$ be a positive extremal. Once again the idea is to
perturb $y_{0}$ and $z_{0}$ to construct an admissible triple that contradicts
optimality of $(x_{0},y_{0},z_{0})$. This is done in several steps.

Step 1. We begin by making the strategy more precise. For smooth maps
$\Phi,\hat{\Phi}:\mathbb{T}_{\phi L}\to\mathbb{T}_{\phi L}$ with
$\Phi(0)=\hat{\Phi}(0)=0$ we define

$\displaystyle u_{1}(x):=u(y_{0}+\Phi(x))\qquad\text{and}\qquad
u_{2}(x):=u(z_{0}+\hat{\Phi}(x)).$

Next we set

$\displaystyle v(x):=u_{1}(x)-u_{2}(x)-(u(y_{0})-u(z_{0})).$ (4.15)

and

$\displaystyle w(x):=\left(\nabla
u\right)(z_{0}+\hat{\Phi}(x))\cdot\left(z_{0}+\hat{\Phi}(x)-(y_{0}+\Phi(x))\right).$
(4.16)

We will find conditions on $\Phi$ and $\hat{\Phi}$ that imply:

  1. (i)

There exists $r>0$ such that $v(x)>0$ in $B_{r}(0)\setminus\\{0\\}$.

  2. (ii)

For any $r>0$ sufficiently small, there exists at least one point
$\tilde{x}\in B_{r}(0)$ such that $w(\tilde{x})>0$.

We will use these facts to deduce that

$\displaystyle y_{*}:=y_{0}+\Phi(\tilde{x}),\quad
z_{*}:=z_{0}+\hat{\Phi}(\tilde{x}),\qquad
x_{*}:=(1+\varepsilon)z_{*}-\varepsilon y_{*}$

for $\varepsilon>0$ small define an admissible triple and
$F(x_{*},y_{*},z_{*})>F(x_{0},y_{0},z_{0})$, contradicting the extremality of
$(x_{0},y_{0},z_{0})$. Clearly $v(\tilde{x})>0$ implies the latter inequality,
so it suffices to check the admissibility. Appropriate collinearity follows
from

$\displaystyle
z_{*}=\frac{1}{1+\varepsilon}x_{*}+\frac{\varepsilon}{1+\varepsilon}y_{*}\qquad\text{for
}\varepsilon\in(0,1).$

Also $\Phi(0)=\hat{\Phi}(0)=0$ and that $z_{0}$, $y_{0}$ lie in the interior
of $S_{c_{*}}$ (again by Lemma 4.10) yield $x_{*},\,y_{*}\in S_{c_{*}}$ for
$|x|$ and $\varepsilon$ small enough. Finally we remark that
$u(x_{*})>u(z_{*})$ for $\varepsilon$ sufficiently small if $w(\tilde{x})>0$.
Hence it remains only to verify the above claims (i) and (ii) for $w$ and $v$.

Step 2. We will now use Taylor expansions to find appropriate conditions on
$\Phi$ and $\hat{\Phi}$. For $v$ near the origin we obtain

$\displaystyle v(x)$ $\displaystyle=$ $\displaystyle
x\cdot\left(D\Phi(0)\,\nabla
u(y_{0})\right)-x\cdot\left(D\hat{\Phi}(0)\,\nabla u(z_{0})\right)$
$\displaystyle+\frac{1}{2}\>\partial_{i}u(y_{0})\,\left(D^{2}\Phi_{i}(0)\,x\right)\cdot
x-\frac{1}{2}\>\partial_{i}u(z_{0})\,\left(D^{2}\hat{\Phi}_{i}(0)\,x\right)\cdot
x$
$\displaystyle+\frac{1}{2}\,\cdot\left(D^{2}u(y_{0})\,\left(D\Phi(0)x\right)\right)\cdot\left(D\Phi(0)x\right)-\frac{1}{2}\,\cdot\left(D^{2}u(y_{0})\,\left(D\hat{\Phi}(0)x\right)\right)\cdot\left(D\hat{\Phi}(0)x\right)$
$\displaystyle+o(|x|^{2}),$

where we sum over repeated indices and $\frac{o(|x|^{2})}{|x|^{2}}\to 0$ as
$|x|\to 0$. Similarly for $w$ we get

$\displaystyle w(x)$ $\displaystyle=$
$\displaystyle\partial_{i}u(z_{0})\,\left(D\left(\hat{\Phi}_{i}-\Phi_{i}\right)(0)\,x\right)+\left(D^{2}u(z_{0})\,\left(D\hat{\Phi}(0)\,x\right)\right)\cdot(z_{0}-y_{0})$
$\displaystyle+\frac{1}{2}\,\partial_{i}u(z_{0})\,\left(D^{2}\left(\hat{\Phi}_{i}-\Phi_{i}\right)(0)\,x\right)\cdot
x+\left(D^{2}u(z_{0})\,\left(D\hat{\Phi}(0)x\,\right)\right)\cdot\left(D(\hat{\Phi}-\Phi)(0)\,x\right)$
$\displaystyle+\frac{1}{2}\,\left(D^{2}\hat{\Phi}_{i}(0)\,x\right)\cdot
x\>\partial_{i}\partial_{j}u(z_{0})\,(z_{0}-y_{0})_{j}+\frac{1}{2}\,(z_{0}-y_{0})_{i}\,\left(D^{2}\partial_{i}u(z_{0})\,\left(D\hat{\Phi}(0)x\right)\right)\cdot\left(D\hat{\Phi}(0)x\right)$
$\displaystyle+o(|x|^{2}).$

Our goal is to choose the coefficients of the Taylor expansion of $\Phi$ and
$\hat{\Phi}$ such that for some $\mu>0$ and some $r>0$, there holds

$\displaystyle
v(x)\geq\mu|x|^{2}\quad\hbox{in}\>B_{r}(0)\qquad\hbox{and}\qquad
w(x)>0\>\hbox{for at least one}\>x\in{B_{r}(0)}.$ (4.19)

Setting

$\displaystyle a:=\frac{1}{|\nabla u(y_{0})|}\qquad\text{and}\qquad
b:=\frac{1}{|\nabla u(z_{0})|},$

we will show that it suffices to choose $\Phi$ and $\hat{\Phi}$ so that
$D\Phi(0)_{ij}=a\,\delta_{ij}$ and $D\hat{\Phi}(0)_{ij}=b\,\delta_{ij}.$ Note
that Lemma 4.13 implies

$\displaystyle x\cdot\left(D\Phi(0)\,\nabla
u(y_{0})\right)-x\cdot\left(D\hat{\Phi}(0)\,\nabla
u(z_{0})\right)=a\,x\cdot\nabla u(y_{0})-b\,x\cdot\nabla u(z_{0})=0,$

so that (4.1) and (4.1) improve to

$\displaystyle v(x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\>\partial_{i}u(y_{0})\,\left(D^{2}\Phi_{i}(0)\,x\right)\cdot
x-\frac{1}{2}\>\partial_{i}u(z_{0})\,\left(D^{2}\hat{\Phi}_{i}(0)\,x\right)\cdot
x$ $\displaystyle+\frac{a^{2}}{2}\,\left(D^{2}u(y_{0})\,x\right)\cdot
x-\frac{b^{2}}{2}\,\left(D^{2}u(y_{0})\,x\right)\cdot x+o(|x|^{2}),$

and

$\displaystyle w(x)$ $\displaystyle=$ $\displaystyle(b-a)\,x\cdot\nabla
u(z_{0})+b\,\left(D^{2}u(z_{0})\,x\right)\cdot(z_{0}-y_{0})$
$\displaystyle+\frac{1}{2}\,\partial_{i}u(z_{0})\,\left(D^{2}\left(\hat{\Phi}_{i}-\Phi_{i}\right)(0)\,x\right)\cdot
x+b(b-a)\,\left(D^{2}u(z_{0})\,x\right)\cdot x$
$\displaystyle+\frac{1}{2}\>\partial_{i}\partial_{j}u(z_{0})\,(z_{0}-y_{0})_{j}\,\left(D^{2}\hat{\Phi}_{i}(0)\,x\right)\cdot
x+\frac{b^{2}}{2}\,(z_{0}-y_{0})_{i}\,\left(D^{2}\partial_{i}u(z_{0})\,x\right)\cdot
x$ $\displaystyle+o(|x|^{2}),$

respectively.

We set $D^{2}\hat{\Phi}_{i}(0)=0$. In view of (4.19) a natural choice for
$D^{2}\Phi(0)$ is then

$\displaystyle\left(D^{2}\Phi_{i}(0)\right)_{jk}=a\,\partial_{i}u(y_{0})\left(-a^{3}\,\partial_{j}\partial_{k}u(y_{0})+a\,b^{2}\,\partial_{j}\partial_{k}u(z_{0})+4\,a\,\mu\,\delta_{jk}\right).$

for some $\mu>0$. This implies

$\displaystyle\frac{1}{2}\>\partial_{i}u(y_{0})\,\left(D^{2}\Phi_{i}(0)\,x\right)\cdot
x:=-\frac{a^{2}}{2}\,\left(D^{2}u(y_{0})\,x\right)\cdot
x+\frac{b^{2}}{2}\,\left(D^{2}u(z_{0})\,x\right)\cdot x+2\,\mu\,|x|^{2}.$

Substituting into (4.1) yields $v(x)=2\mu|x|^{2}+o(|x|^{2})$ and hence the
first statement in (4.19).

Step 3. It remains to prove the second part of (4.19). From Lemma 4.13 we
deduce

$\displaystyle\frac{1}{2}\>\partial_{i}u(z_{0})\,\left(D^{2}\Phi_{i}(0)\,x\right)\cdot
x=-\frac{a^{3}}{2b}\,\left(D^{2}u(y_{0})\,x\right)\cdot
x+\frac{a\,b}{2}\,\left(D^{2}u(z_{0})\,x\right)\cdot
x+2\,\mu\,\frac{a}{b}\,|x|^{2}.$

We plug this into (4.1). Rearranging terms gives

$\displaystyle w(x)$ $\displaystyle=$ $\displaystyle(b-a)\,x\cdot\nabla
u(z_{0})+b\,\left(D^{2}u(z_{0})\,x\right)\cdot(z_{0}-y_{0})+\frac{a^{3}}{2b}\,\left(D^{2}u(y_{0})\,x\right)\cdot
x$ $\displaystyle-\frac{a\,b}{2}\,\left(D^{2}u(z_{0})\,x\right)\cdot
x-2\frac{a}{b}\,\mu|x|^{2}+b(b-a)\,\left(D^{2}u(z_{0})\,x\right)\cdot x$
$\displaystyle+\frac{b^{2}}{2}\,(z_{0}-y_{0})_{i}\,\left(D^{2}\partial_{i}u(z_{0})\,x\right)\cdot
x+o(|x|^{2}),$

and a straightforward computation yields

$\displaystyle\Delta w=\frac{a^{3}}{b}\,\Delta u(y_{0})-a\,b\,\Delta
u(z_{0})-4d\frac{a}{b}\,\mu+2b(b-a)\,\Delta
u(z_{0})+b^{2}\,\Delta\partial_{i}u(z_{0})\,(z_{0}-y_{0})_{i}+o(1).$

Recall that according to Lemma 4.11, there holds

$\displaystyle\partial_{i}\Delta
u(z_{0})\,(z_{0}-y_{0})_{i}=f^{\prime}(u(z_{0}))\,\nabla
u(z_{0})\cdot(z_{0}-y_{0})=0.$

Thus

$\displaystyle\Delta w=\frac{a^{3}}{b}\,\Delta
u(y_{0})+(2\,b^{2}-3\,a\,b)\,\Delta u(z_{0})-4d\frac{a}{b}\,\mu+o(1).$ (4.22)

We will now show that $\Delta w>0$ for $\mu$ and $|x|$ small enough. Combining
this fact with $w(0)=0$ and the Strong Maximum Principle, we then obtain the
second part of (4.19).

We first consider the case $a=b$, in which equation (4.22) yields

$\displaystyle\Delta w=a^{2}\left(\Delta u(y_{0})-\Delta
u(z_{0})\right)-4d\,\mu+o(1)=a^{2}\left(f(u(y_{0}))-f(u(z_{0}))\right)-4d\,\mu+o(1).$

We use the equation and Hypothesis (H2). Since $u(y_{0})>u(z_{0})$, the strict
monotonicity of $f$ implies $f(u(y_{0}))>f(u(z_{0}))$. Hence for $\mu$ and
$|x|$ small enough, we get the existence of a constant $c_{0}>0$ such that
$\Delta w\geq c_{0}.$

We now consider the case $a\neq b$ and rewrite (4.22) as

$\displaystyle\Delta
w=\frac{a^{3}}{b}\,f(u(y_{0}))+\left(2b^{2}-3\,a\,b\right)\,f(u(z_{0}))-4\,d\,\frac{a}{b}\,\mu+o(1).$

Again using the strict monotonicity of $f$, we deduce

$\displaystyle\Delta w$ $\displaystyle\geq$
$\displaystyle\left(\frac{a^{3}}{b}+2\,b^{2}-3\,a\,b\right)\,f(u(z_{0}))-4d\frac{a}{b}\,\mu+o(1)$
$\displaystyle=$ $\displaystyle
b^{2}\left(\frac{a}{b}+2\right)\,\left(\frac{a}{b}-1\right)^{2}\,f(u(z_{0}))-4d\frac{a}{b}\,\mu+o(1).$

Since $f(u(z_{0}))>0$ by Hypothesis (H2), we can again choose $\mu$ and $|x|$
small enough so that there exists $c_{0}>0$ such that $\Delta w\geq c_{0}$. ∎

## References

## Acknowledgements

We thank Eric Vanden–Eijnden for helpful discussions and for encouraging our
interaction. Sebastian Scholtes was partially supported by DFG Grant WE
5760/1-1.


# An Orthogonal Equivalence Theorem for Third Order Tensors111To appear in:
Journal of Industrial and Management Optimization.

Liqun Qi, Chen Ling, Jinjie Liu , and Chen Ouyang Department of Mathematics,
School of Science, Hangzhou Dianzi University, Hangzhou 310018 China;
(liqun.qi@polyu.edu.hk). Department of Mathematics, Hangzhou Dianzi
University, Hangzhou 310018, China; (macling@hdu.edu.cn). This author’s work
was supported by Natural Science Foundation of China (No. 11971138) and
Natural Science Foundation of Zhejiang Province (No. LY19A010019,
LD19A010002).School of Mathematical Sciences, Shanghai Jiao Tong University,
Shanghai, 200240, China;(jinjie.liu@sjtu.edu.cn). This author’s work was
supported by Natural Science Foundation of China (No. 12001366, No.
11801479).Corresponding Author. School of Computer Science and Technology,
Dongguan University of Technology, Dongguan, 523000, China;(oych26@163.com).
This author’s work was supported by Natural Science Foundation of China
(No.11971106) and Guangdong Universities’ Special Projects in Key Fields of
Natural Science(No. 2019KZDZX1005).

###### Abstract

In 2011, Kilmer and Martin proposed tensor singular value decomposition
(T-SVD) for third order tensors. Since then, T-SVD has applications in low
rank tensor approximation, tensor recovery, multi-view clustering, multi-view
feature extraction, tensor sketching, etc. By going through the Discrete
Fourier Transform (DFT), matrix SVD and inverse DFT, a third order tensor is
mapped to an f-diagonal third order tensor. We call this a Kilmer-Martin
mapping. We show that the Kilmer-Martin mapping of a third order tensor is
invariant if that third order tensor is taking T-product with some orthogonal
tensors. We define singular values and T-rank of that third order tensor based
upon its Kilmer-Martin mapping. Thus, tensor tubal rank, T-rank, singular
values and T-singular values of a third order tensor are invariant when it is
taking T-product with some orthogonal tensors. Some properties of singular
values, T-rank and best T-rank one approximation are discussed.

Key words. Third order tensors, orthogonal equivalence, singular value,
T-rank, the best T-rank one approximation.

AMS subject classifications. 15A69, 15A18

##  1 Introduction

The T-product operation, T-SVD factorization and tensor tubal ranks were
introduced by Kilmer and her collaborators in [3, 4, 5, 18]. They are now
widely used in engineering [1, 8, 9, 7, 11, 12, 13, 14, 15, 16, 17, 19, 20].
In particular, Kilmer and Martin [4] proposed T-SVD factorization. By going
through the Discrete Fourier Transform (DFT), matrix SVD and inverse DFT, a
third order tensor is diagonalized to an f-diagonal third order tensor. The
tensor tubal rank is defined based upon such an f-diagonal tensor. The matrix
SVD should follow the standard non-increasing ordering for the singular values
of the matrices involved. If a different ordering is used, the diagonalization
result would be different.

We call the above particular diagonalization the Kilmer-Martin mapping, and
say that two third order tensors are orthogonally equivalent if one of them
can be obtained by the product of another with some orthogonal tensors. We
show that if two third order tensors are orthogonally equivalent, then their
Kilmer-Martin mappings are the same. Thus, the real f-diagonal tensor obtained
by the Kilmer-Martin mapping of a third order tensor extracts the main
features of that third order tensor. We call the absolute values of the
diagonal entries of the f-diagonal tensor as the singular values of the
original third order tensor, and the number of the nonzero singular values as
the T-rank of the third order tensor. Some properties of singular values and
T-ranks are studied.

The largest singular value of a real matrix is always greater than or equal to
the absolute value of any entry of that matrix. We show that this is also true
for third order tensors. Furthermore, based on this property, we show that the
best T-rank one approximation of a third order tensor can be given by its
largest singular value and related orthogonal tensors.

The remaining of this paper is distributed as follows. In the next section,
some preliminary knowledge on T-product of third order tensors is reviewed.
The Kilmer-Martin mapping and orthogonal equivalence are defined in Section 3.
We show there that the Kilmer-Martin mappings of two orthogonally equivalent
tensors are the same. Singular values and T-ranks are defined in Section 4.
Their properties are also studied there. In Section 5, we study the best
T-rank one approximation of a third order tensor. Some further discussion is
made in Section 6.

##  2 Preliminaries

In this paper, real matrices are denoted by capital Roman letters
$A,B,\cdots$, complex matrices are denoted by capital Greek letters
$\Delta,\Sigma,\cdots$, and tensors are denoted by Euler script letters
$\mathcal{A},\mathcal{B},\cdots$. We use $\mathbb{R}$ to denote the real
number field, and $\mathbb{C}$ to denote the complex number field. For a third
order tensor $\mathcal{A}\in\mathbb{R}^{m\times n\times p}$, its $(i,j,k)$-th
element is represented by $a_{ijk}$, and use the Matlab notation
$\mathcal{A}(i,:,:)$, $\mathcal{A}(:,i,:)$ and $\mathcal{A}(:,:,i)$
respectively to represent the $i$-th horizontal, lateral and frontal slice of
the $\mathcal{A}$. The frontal slice $\mathcal{A}(:,:,i)$ is represented by
$A^{(i)}$. Define $\|\mathcal{A}\|_{F}:=\sqrt{\sum_{ijk}|a_{ijk}|^{2}}$.

For a third order tensor ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$, as
in [3, 4], define

${\rm bcirc}({\mathcal{A}}):=\left(\begin{aligned} A^{(1)}\
&A^{(p)}&A^{(p-1)}&\cdots&A^{(2)}\ \\\
A^{(2)}&A^{(1)}&A^{(p)}&\cdots&A^{(3)}\\\ \cdot\ \ \ &\ \cdot&\cdot\ \
&\cdots&\cdot\ \ \ \\\ \cdot\ \ \ &\ \cdot&\cdot\ \ &\cdots&\cdot\ \ \ \\\
A^{(p)}&A^{(p-1)}&A^{(p-2)}&\cdots&A^{(1)}\end{aligned}\right),$

and bcirc${}^{-1}($bcirc$({\mathcal{A}})):={\mathcal{A}}$.

For a third order tensor ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$, its
transpose is defined as

${\mathcal{A}}^{\top}={\rm bcirc}^{-1}[({\rm birc}({\mathcal{A}}))^{\top}].$

This will be the same as the definition in [3, 4]. The identity tensor
$\mathcal{I}_{nnp}$ may also be defined as

$\mathcal{I}_{nnp}={\rm bcirc}^{-1}(I_{np}),$

where $I_{np}$ is the identity matrix in $\mathbb{R}^{np\times np}$.

A third order tensor $\mathcal{S}$ in $\mathbb{R}^{m\times n\times p}$ is
f-diagonal in the sense of [3, 4] if all of its frontal slices
$S^{(1)},\cdots,S^{(p)}$ are diagonal. We call the diagonal entries of
$S^{(1)},\cdots,S^{(p)}$ as diagonal entries of $\mathcal{S}$.

For a third order tensor ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$, it
is defined [4] that

${\rm unfold}({\mathcal{A}}):=\left(\begin{aligned} A^{(1)}\\\ A^{(2)}\\\
\cdot\ \ \\\ \cdot\ \ \\\ \cdot\ \ \\\
A^{(p)}\end{aligned}\right)\in\mathbb{R}^{mp\times n},$

and fold$($unfold$({\mathcal{A}})):={\mathcal{A}}$. For
${\mathcal{A}}\in\mathbb{R}^{m\times s\times p}$ and
$\mathcal{B}\in\mathbb{R}^{s\times n\times p}$, the T-product of
${\mathcal{A}}$ and $\mathcal{B}$ is defined as ${\mathcal{A}}*\mathcal{B}:=$
fold$($bcirc$({\mathcal{A}})$unfold$(\mathcal{B}))\in\mathbb{R}^{m\times
n\times p}$. Then, we see that

${\mathcal{A}}*\mathcal{B}={\rm bcirc}^{-1}({\rm bcirc}({\mathcal{A}}){\rm
bcirc}(\mathcal{B})).$ (2.1)

Thus, the bcirc and bcirc-1 operations not only form a one-to-one relationship
between third order tensors and block circulant matrices, but also their
product operation is reserved. By [4], the T-product operation (2.1) can be
done by applying the fast Fourier transform (FFT). The computational cost for
this is $O(mnsp)$ flops.

A tensor ${\mathcal{A}}\in\mathbb{R}^{n\times n\times p}$ has an inverse
${\mathcal{A}}^{-1}:=\mathcal{B}\in\mathbb{R}^{n\times n\times p}$ if

${\mathcal{A}}*\mathcal{B}=\mathcal{B}*{\mathcal{A}}=\mathcal{I}_{nnp}.$

If $\mathcal{Q}^{-1}=\mathcal{Q}^{\top}$ for
$\mathcal{Q}\in\mathbb{R}^{n\times n\times p}$, then $\mathcal{Q}$ is called
an orthogonal tensor.

###### Definition 2.1

Suppose that ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$. The smallest
integer $r$ such that

${\mathcal{A}}=\mathcal{B}*\mathcal{C},$ (2.2)

where $\mathcal{B}\in\mathbb{R}^{m\times r\times p}$ and
$\mathcal{C}\in\mathbb{R}^{r\times n\times p}$, is called the tensor tubal
rank of ${\mathcal{A}}$.

This definition was implicitly raised by Kilmer and Martin [4] in 2011. In
[10], this definition was formally used.

##  3 The Kilmer-Martin Mapping and Orthogonal Equivalence

Suppose that ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$. By (3.1) of
[4], we may block-diagonalize bcirc$({\mathcal{A}})$ as

$\Delta({\mathcal{A}}):=(F_{p}\otimes I_{m}){\rm
bcirc}({\mathcal{A}})(F_{p}^{*}\otimes
I_{n})=\begin{bmatrix}\Delta^{(1)}&&&\\\ &\Delta^{(2)}&&\\\ &&\ddots&\\\
&&&\Delta^{(p)}\\\ \end{bmatrix},$ (3.3)

where $F_{p}$ is the $p\times p$ DFT matrix, $F_{p}^{*}$ is its conjugate
transpose, $\otimes$ denotes the Kronecker product,
$\Delta^{(k)}\in\mathbb{C}^{m\times n}$ for $k=1,\cdots,p$. For each matrix
$\Delta^{(k)}$, compute its SVD

$\Delta^{(k)}=\Phi^{(k)}\Sigma^{(k)}{\Psi^{(k)}}^{*},$

where $\Phi^{(k)}\in\mathbb{C}^{m\times m}$ and
$\Psi^{(k)}\in\mathbb{C}^{n\times n}$ are unitary matrices,
$\Sigma^{(k)}\in\mathbb{C}^{m\times n}$ is a diagonal matrix, the singular
values of $\Delta^{(k)}$ follow the standard non-increasing order. Denote

$\Sigma({\mathcal{A}}):=\begin{bmatrix}\Sigma^{(1)}&&&\\\ &\Sigma^{(2)}&&\\\
&&\ddots&\\\ &&&\Sigma^{(p)}\\\ \end{bmatrix}.$ (3.4)

Let

$\mathcal{S}=\mathcal{S}({\mathcal{A}}):={\rm
bcirc}^{-1}\left((F_{p}^{*}\otimes I_{m})\Sigma({\mathcal{A}})(F_{p}\otimes
I_{n})\right).$ (3.5)

Then $\mathcal{S}({\mathcal{A}})\in\mathbb{R}^{m\times n\times p}$ is a real
f-diagonal tensor. We call $\mathcal{S}(\cdot)$ the Kilmer-Martin mapping. In
particular, by Eq. (3.5), we have

$\mathcal{S}(i,i,k)={1\over
p}\sum_{l=1}^{p}\bar{\omega}^{(k-1)(l-1)}\Sigma(i,i,l),$ (3.6)

for $i=1,\cdots,\min\\{m,n\\}$, $k=1,\cdots,p$, and
$\omega=e^{-2\pi\sqrt{-1}/p}$. Note that ${\mathcal{A}}$ has $mnp$ entries,
and $\mathcal{S}=\mathcal{S}({\mathcal{A}})$ has $p\min\\{m,n\\}$ diagonal
entries. In a certain sense, the main features of ${\mathcal{A}}$ are
extracted in the diagonal entries of $\mathcal{S}$.

As noticed in [4], the particular diagonalization $\mathcal{S}({\mathcal{A}})$
was achieved using the standard non-increasing ordering for the singular
values of each $\Delta^{(k)}$. If a different ordering is used, a different
diagonalization $\mathcal{S}_{1}({\mathcal{A}})$ would be achieved. Then the
set of the diagonal entries of $\mathcal{S}_{1}({\mathcal{A}})$ can be
different from the set of the diagonal entries of
$\mathcal{S}({\mathcal{A}})$.

Let

$\Phi({\mathcal{A}}):=\begin{bmatrix}\Phi^{(1)}&&&\\\ &\Phi^{(2)}&&\\\
&&\ddots&\\\ &&&\Phi^{(p)}\\\ \end{bmatrix},$
$\Psi({\mathcal{A}}):=\begin{bmatrix}\Psi^{(1)}&&&\\\ &\Psi^{(2)}&&\\\
&&\ddots&\\\ &&&\Psi^{(p)}\\\ \end{bmatrix},$
$\mathcal{U}=\mathcal{U}({\mathcal{A}})={\rm
bcirc}^{-1}\left((F_{p}^{*}\otimes I_{m})\Phi({\mathcal{A}})(F_{p}\otimes
I_{n})\right),$ $\mathcal{V}=\mathcal{V}({\mathcal{A}})={\rm
bcirc}^{-1}\left((F_{p}^{*}\otimes I_{m})\Psi({\mathcal{A}})(F_{p}\otimes
I_{n})\right).$

Then $\mathcal{U}\in\mathbb{R}^{m\times m\times p}$ and
$\mathcal{V}\in\mathbb{R}^{n\times n\times p}$ are orthogonal tensors, and
${\mathcal{A}}$ has its T-SVD

${\mathcal{A}}=\mathcal{U}*\mathcal{S}*\mathcal{V}^{\top}.$ (3.7)

Theorem 4.3 of [4] showed that an Eckart-Young like theorem holds for the
tensor tubal rank of ${\mathcal{A}}$ here. For the Kilmer-Martin T-SVD
factorization (3.7), by [4, 5], we have

$\sum_{k=1}^{p}\mathcal{S}(1,1,k)^{2}\geq\sum_{k=1}^{p}\mathcal{S}(2,2,k)^{2}\geq\cdots\geq\sum_{k=1}^{p}\mathcal{S}(\min\\{m,n\\},\min\\{m,n\\},k)^{2}.$
(3.8)

Recently, Qi and Yu [10] defined the $i$th largest T-singular value of
${\mathcal{A}}$ as

$\lambda_{i}:=\sqrt{\sum_{k=1}^{p}\mathcal{S}(i,i,k)^{2}},$

for $i=1,\cdots,\min\\{m,n\\}$, and used T-singular values to define the tail
energy for the error estimate of a proposed tensor sketching algorithm.
T-singular values are nonnegative numbers. The number of the nonzero
T-singular values of ${\mathcal{A}}$ is equal to the tensor tubal rank of
${\mathcal{A}}$.

###### Definition 3.1

Suppose that ${\mathcal{A}},\mathcal{B}\in\mathbb{R}^{m\times n\times p}$. If
there are orthogonal tensors ${\mathcal{Y}}\in\mathbb{R}^{m\times m\times p}$
and $\mathcal{Z}\in\mathbb{R}^{n\times n\times p}$ such that

${\mathcal{A}}={\mathcal{Y}}*\mathcal{B}*\mathcal{Z}^{\top}.$

Then we say that ${\mathcal{A}}$ and $\mathcal{B}$ are orthogonally
equivalent.

###### Theorem 3.2

Suppose that ${\mathcal{A}},\mathcal{B}\in\mathbb{R}^{m\times n\times p}$ are
orthogonally equivalent,
${\mathcal{A}}={\mathcal{Y}}*\mathcal{B}*\mathcal{Z}^{\top}$, where
${\mathcal{Y}}\in\mathbb{R}^{m\times m\times p}$ and
$\mathcal{Z}\in\mathbb{R}^{n\times n\times p}$ are orthogonal tensors. Then

$\mathcal{S}({\mathcal{A}})=\mathcal{S}(\mathcal{B}).$ (3.9)

Proof We have

${\rm bicrc}({\mathcal{A}})={\rm bcirc}({\mathcal{Y}}){\rm
bcirc}(\mathcal{B}){\rm bcirc}(\mathcal{Z}^{\top}).$

Apply $(F_{p}\otimes I_{m})$ to the left and $(F_{p}^{*}\otimes I_{n})$ to the
right of each of the block circulant matrices in the above expression, where
$F_{p}$ is the $p\times p$ DFT matrix, $F^{*}_{p}$ is its conjugate transpose,
$\otimes$ denotes the Kronecker product. For each diagonal block, by SVD of
matrices, we have

$\Delta({\mathcal{A}})^{(k)}=\Xi^{(k)}\Delta(\mathcal{B})^{(k)}(\Theta^{(k)})^{*},$

where $\Xi^{(k)}\in\mathbb{C}^{m\times m}$ and
$\Theta^{(k)}\in\mathbb{C}^{n\times n}$ are unitary matrices for
$k=1,\cdots,p$. Then $\Delta({\mathcal{A}})^{(k)}$ and
$\Delta(\mathcal{B})^{(k)}$ have the same set of singular values for
$k=1,\cdots,p$. This implies that

$\Sigma({\mathcal{A}})^{(k)}=\Sigma(\mathcal{B})^{(k)},$

for $k=1,\cdots,p$, i.e.,

$\Sigma({\mathcal{A}})=\Sigma(\mathcal{B}),$

which implies (3.9).

###### Corollary 3.3

Suppose that ${\mathcal{A}},\mathcal{B}\in\mathbb{R}^{m\times n\times p}$ are
orthogonally equivalent. Then they have the same tensor tubal rank and
T-singular value set.

Suppose that ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$ is f-diagonal.
It is possible that $\mathcal{S}({\mathcal{A}})$ is very different from
${\mathcal{A}}$. For example, let
${\mathcal{A}}=(a_{ijk})\in\mathbb{R}^{3\times 3\times 3}$ be f-diagonal with
$a_{221}=6$, $a_{112}=5$, $a_{332}=9$, $a_{333}=9$. The other entries of
${\mathcal{A}}$ are zero. Let $\mathcal{S}=\mathcal{S}({\mathcal{A}})$. Then
we have

$\mathcal{S}(1,1,1)=12,\ \mathcal{S}(2,2,1)=6,\
\mathcal{S}(3,3,1)={5},\mathcal{S}(1,1,2)=\mathcal{S}(1,1,3)=3.$

The other entries of $\mathcal{S}$ are zero. We see that $\mathcal{S}$ and
${\mathcal{A}}$ are very different.

In [6, Theorem 2.1], the following result was proved.

###### Proposition 3.4

Suppose that ${\mathcal{A}},\mathcal{B}\in\mathbb{R}^{m\times n\times p}$ are
orthogonally equivalent,
${\mathcal{A}}={\mathcal{Y}}*\mathcal{B}*\mathcal{Z}^{\top}$, where
${\mathcal{Y}}\in\mathbb{R}^{m\times m\times p}$ and
$\mathcal{Z}\in\mathbb{R}^{n\times n\times p}$ are orthogonal tensors. If
$\mathcal{B}=(b_{ijk})$ is f-diagonal, then

$\max\\{|b_{ijk}|:1\leq i\leq m,1\leq j\leq n,1\leq k\leq
p\\}\leq\sigma_{1}({\mathcal{A}}),$

where $\sigma_{1}({\mathcal{A}})$ is defined in Definition 4.1.

By Theorem 3.2 and Proposition 3.4, we have the following proposition.

###### Proposition 3.5

Suppose that ${\mathcal{A}}=(a_{ijk})\in\mathbb{R}^{m\times n\times p}$ is
f-diagonal. Then

$\max\\{|a_{ijk}|:1\leq i\leq m,1\leq j\leq n,1\leq k\leq
p\\}\leq\sigma_{1}({\mathcal{A}}),$

where $\sigma_{1}({\mathcal{A}})$ is defined in Definition 4.1.

##  4 Singular Values and T-Rank

###### Definition 4.1

Suppose that ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$. Let (3.7) be
the Kilmer-Martin T-SVD factorization of ${\mathcal{A}}$, where
$\mathcal{S}=\mathcal{S}({\mathcal{A}})$. The absolute values of the diagonal
entries of the frontal slices of $\mathcal{S}$ are called the singular values
of ${\mathcal{A}}$. Let $s$ be a positive integer such that $1\leq s\leq
p\min\\{m,n\\}$. The $s$th largest singular value of ${\mathcal{A}}$ is
denoted as $\sigma_{s}({\mathcal{A}})$. The number of the nonzero singular
values of ${\mathcal{A}}$ is called the T-rank of ${\mathcal{A}}$. Let
$\mathcal{S}_{s}\in\mathbb{R}^{m\times n\times p}$ be an f-diagonal tensor,
such that its entries are the same as the entries of $\mathcal{S}$, where
$\sigma_{i}({\mathcal{A}})$ for $1\leq i\leq s$ are located, and its other
entries are zero. Denote
${\mathcal{A}}_{s}=\mathcal{U}*\mathcal{S}_{s}*\mathcal{V}^{\top}$.

By Theorem 3.2 and Definition 4.1, we have the following corollary.

###### Corollary 4.2

Suppose that ${\mathcal{A}},\mathcal{B}\in\mathbb{R}^{m\times n\times p}$ are
orthogonally equivalent. Then they have the same T-rank and singular value
set.

Zhang and Aeron [17] defined singular values of a third order tensor
${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$. Suppose that ${\mathcal{A}}$
has a T-SVD

${\mathcal{A}}=\mathcal{U}*\mathcal{S}*\mathcal{V}^{\top},$

where $\mathcal{U}$ and $\mathcal{V}$ are orthogonal tensors, $\mathcal{S}$ is
an f-diagonal tensor. Then they call the entries of $\mathcal{S}$ the singular
values of ${\mathcal{A}}$ [17, Definition II.7]. First, if not specifying
$\mathcal{S}\not=\mathcal{S}({\mathcal{A}})$, this definition is not well-
defined. Second, off-diagonal entries of $\mathcal{S}$ are zeros. They are not
needed to be involved. Third, some diagonal entries of $\mathcal{S}$ may be
negative. Hence, our definition is different from theirs.

###### Proposition 4.3

Suppose that ${\mathcal{A}}=(a_{ijk})\in\mathbb{R}^{m\times n\times p}$ has
only one nonzero entry. Then the T-rank of ${\mathcal{A}}$ is equal to one.

Proof Assume that $a_{i_{0}j_{0}k_{0}}=a\not=0$, and the other entries of
${\mathcal{A}}$ are zero. By writing out $F_{p}$ and $F^{*}_{p}$ explicitly,
in (3.3), for $i=1,\cdots,m$, $j=1,\cdots,n$ and $k=1,\cdots,p$, we have

$\Delta^{(k)}(i,j)=\sum_{l=1}^{p}\omega^{(k-1)(l-1)}A^{(l)}(i,j),$

where $\omega=e^{-2\pi\sqrt{-1}/p}$. Then for $k=1,\cdots,p$,

$\Delta^{(k)}(i_{0},j_{0})=\omega^{(k-1)(k_{0}-1)}A^{(k_{0})}(i_{0},j_{0}).$

The other entries of $\Delta({\mathcal{A}})$ are zero. Consider the SVD of
$\Delta^{(k)}$. The singular values of $\Delta^{(k)}$ are the square roots of
eigenvalues of $(\Delta^{(k)})^{*}\Delta^{(k)}$. Then,
$(\Delta^{(k)})^{*}\Delta^{(k)}$ only has a nonzero entry

$(\Delta^{(k)})^{*}\Delta^{(k)}(j_{0},j_{0})=a_{i_{0}j_{0}k_{0}}^{2},$

for $k=1,\cdots,p$. Then, for $k=1,\cdots,p$, $\Delta^{(k)}$ has only one
nonzero singular value $|a|$. This implies that
$\Sigma^{(k)}(1,1)=|a_{i_{0}j_{0}k_{0}}|=|a|$ and $\Sigma^{(k)}(i,i)=0$ for
$k=1,\cdots,p$ and $i=2,\cdots,\min\\{m,n\\}$. By the inverse DFT, we have

$\mathcal{S}(1,1,1)={1\over p}\sum_{k=1}^{p}|a|=|a|,$
$\mathcal{S}(1,1,k)={1\over
p}\sum_{l=1}^{p}\bar{\omega}^{(k-1)(l-1)}\Sigma^{(l)}(1,1)={|a|\over
p}\sum_{l=1}^{p}\bar{\omega}^{(k-1)(l-1)}=0,$

as $\sum_{l=1}^{p}\bar{\omega}^{(k-1)(l-1)}=0$, for $k=2,\cdots,p$. We also
have

$\mathcal{S}(i,i,k)=0$

for $i\geq 2$ and $k=1,\cdots,p$. Then $\mathcal{S}({\mathcal{A}})$ has only
one nonzero entry $\mathcal{S}(1,1,1)=|a|$.

###### Proposition 4.4

Suppose ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$,
${\mathcal{A}}={\mathcal{A}}^{\prime}+{\mathcal{A}}^{\prime\prime}$. Then

$\sigma_{1}({\mathcal{A}})\leq\sigma_{1}({\mathcal{A}}^{\prime})+\sigma_{1}({\mathcal{A}}^{\prime\prime}).$

Proof Denote $A^{(i)}={\mathcal{A}}(:,:,i)$,
$A^{\prime(i)}={\mathcal{A}}^{\prime}(:,:,i)$ and
$A^{\prime\prime(i)}={\mathcal{A}}^{\prime\prime}(:,:,i)$, then
$A^{(i)}=A^{\prime(i)}+A^{\prime\prime(i)}$. We have

$\mbox{bcirc}({\mathcal{A}})=\mbox{bcirc}({\mathcal{A}}^{\prime})+\mbox{bcirc}({\mathcal{A}}^{\prime\prime}).$

Applying FFT to both sides, the above equation is transformed to the
following:

$\begin{bmatrix}\Delta^{(1)}&&&\\\ &\Delta^{(2)}&&\\\ &&\ddots&\\\
&&&\Delta^{(p)}\\\ \end{bmatrix}=\begin{bmatrix}\Delta^{\prime(1)}&&&\\\
&\Delta^{\prime(2)}&&\\\ &&\ddots&\\\ &&&\Delta^{\prime(p)}\\\
\end{bmatrix}+\begin{bmatrix}\Delta^{\prime\prime(1)}&&&\\\
&\Delta^{\prime\prime(2)}&&\\\ &&\ddots&\\\ &&&\Delta^{\prime\prime(p)}\\\
\end{bmatrix},$

where
$\Delta^{(i)}=\sum_{l=1}^{p}\omega^{(l-1)(i-1)}A^{(i)}=\sum_{l=1}^{p}\omega^{(l-1)(i-1)}(A^{\prime(i)}+A^{\prime\prime(i)})=\Delta^{\prime(i)}+\Delta^{\prime\prime(i)}$.

Denote $\sigma_{1}(A)$ as the largest singular value of a matrix $A$. Then for
each $i=1,\cdots,p$, we have

$\sigma_{1}(\Delta^{(i)})\leq\sigma_{1}(\Delta^{\prime(i)})+\sigma_{1}(\Delta^{\prime\prime(i)}).$

Thus,

$\displaystyle\max_{i=1,\cdots,p}\\{\sigma_{1}(\Delta^{(i)})\\}$
$\displaystyle\leq$
$\displaystyle\max_{i=1,\cdots,p}\\{\sigma_{1}(\Delta^{\prime(i)})\\}+\max_{i=1,\cdots,p}\\{\sigma_{1}(\Delta^{\prime\prime(i)})\\}$

Hence,

$\displaystyle\sigma_{1}({\mathcal{A}})$ $\displaystyle\leq$
$\displaystyle\sigma_{1}({\mathcal{A}}^{\prime})+\sigma_{1}({\mathcal{A}}^{\prime\prime}).$

##  5 The Best T-Rank One Approximation to a Third Order Tensor

In this section, we propose the best T-rank one approximation to a third order
tensor. Before that, we first give a theorem to show the relation between the
largest singular value and the maximum absolute value of elements in a third
order tensor.

Recall that by (3.6), for any ${\mathcal{A}}=(a_{ijk})\in\mathbb{R}^{m\times
n\times p}$, we have

$a_{ijk}={1\over p}\sum_{l=1}^{p}\bar{\omega}^{(k-1)(l-1)}\Delta(i,j,l),$
(5.10)

for $i=1,\cdots,m$, $j=1,\cdots,n$, $k=1,\cdots,p$. Furthermore, according to
(3.6), we have

$\mathcal{S}(1,1,1)={1\over p}\sum_{l=1}^{p}\Sigma(1,1,l).$ (5.11)

Thus, if not all $\Sigma(1,1,l)$ are equal, i.e.,
$\min_{l}\\{\Sigma(1,1,l)\\}<\max_{l}\\{\Sigma(1,1,l)\\}$, then

$\mathcal{S}(1,1,1)<\max_{l}\\{\Sigma(1,1,l)\\}.$

Hence, we have the following theorem.

###### Theorem 5.1

Suppose ${\mathcal{A}}=(a_{ijk})\in\mathbb{R}^{m\times n\times p}$. Then

$\sigma_{1}({\mathcal{A}})\equiv\mathcal{S}(1,1,1)\geq\max\\{|a_{ijk}|:1\leq
i\leq m,1\leq j\leq n,1\leq k\leq p\\}.$ (5.12)

Proof By the property of matrix singular values [2, (2.3.8) and Corollary
2.4.3]222[2, (2.3.8) and Corollary 2.4.3] is for real matrices, but they can
be extended to complex matrices easily., we have

$|\Delta(i,j,k)|\leq\Sigma(1,1,k),$ (5.13)

for $i=1,\cdots,m$, $j=1,\cdots,n$, $k=1,\cdots,p$. Then by (5.10) and (5.11),
we have

$\displaystyle|a_{ijk}|$ $\displaystyle\leq$ $\displaystyle{1\over
p}\sum_{l=1}^{p}|\bar{\omega}^{(k-1)(l-1)}\Delta(i,j,l)|$ $\displaystyle\leq$
$\displaystyle{1\over p}\sum_{l=1}^{p}|\Delta(i,j,l)|$ $\displaystyle\leq$
$\displaystyle{1\over p}\sum_{l=1}^{p}\Sigma(1,1,l)$ $\displaystyle=$
$\displaystyle\mathcal{S}(1,1,1),$

for $i=1,\cdots,m$, $j=1,\cdots,n$, $k=1,\cdots,p$, i.e,

$\sigma_{1}({\mathcal{A}})\equiv\mathcal{S}(1,1,1)\geq\max\\{|a_{ijk}|:1\leq
i\leq m,1\leq j\leq n,1\leq k\leq p\\}.$

Such a property holds in the matrix case [2]. Our theorem shows that it holds
as well in the third order tensor case. Based on Theorem 5.1, we have the
following theorem.

###### Theorem 5.2

For any ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$, ${\mathcal{A}}_{1}$
is the best T-rank one approximation of ${\mathcal{A}}$, where
${{\mathcal{A}}}_{1}$ is defined by in Definition 4.1.

Proof Let ${\mathcal{A}}\in\mathbb{R}^{m\times n\times p}$. We have

$\|{\mathcal{A}}-{\mathcal{A}}_{1}\|_{F}^{2}=\|\mathcal{S}-\mathcal{S}_{1}\|_{F}^{2}=\sum_{i=2}^{p\min\\{m,n\\}}\sigma_{i}^{2}.$

Now, assume that $\mathcal{B}\in\mathbb{R}^{m\times n\times p}$ has T-rank
one. Suppose that $\mathcal{B}$ has a Kilmer-Martin T-SVD factorization

$\mathcal{B}={\mathcal{Y}}*\mathcal{D}*\mathcal{Z}^{\top},$

where ${\mathcal{Y}}\in\mathbb{R}^{m\times m\times p}$ and
$\mathcal{Z}\in\mathbb{R}^{n\times n\times p}$ are orthogonal,
$\mathcal{D}\in\mathbb{R}^{m\times n\times p}$ is f-diagonal and has at most
$1$ nonzero elements. Let

${\mathcal{A}}^{\prime}={\mathcal{Y}}^{\top}*{\mathcal{A}}*\mathcal{Z}.$

Then

$\|{\mathcal{A}}-\mathcal{B}\|_{F}^{2}=\|{\mathcal{A}}^{\prime}-\mathcal{D}\|_{F}^{2}\geq\|{\mathcal{A}}^{\prime}\|_{F}^{2}-\sigma_{1}({\mathcal{A}}^{\prime})^{2}.$

where the inequality follows from Theorem 5.1 applied to
${\mathcal{A}}^{\prime}$.

By Theorem 3.2, $\sigma_{i}({\mathcal{A}}^{\prime})=\sigma_{i}({\mathcal{A}})$
for $i=1,\cdots,\min\\{m,n\\}$.

Then we have

$\|{\mathcal{A}}-\mathcal{B}\|_{F}^{2}\geq\|{\mathcal{A}}^{\prime}\|_{F}^{2}-\sigma_{1}({\mathcal{A}}^{\prime})^{2}=\|{\mathcal{A}}\|_{F}^{2}-\sigma_{1}({\mathcal{A}})^{2}=\sum_{i=2}^{p\min\\{m,n\\}}\sigma_{i}^{2}=\|{\mathcal{A}}-{\mathcal{A}}_{1}\|_{F}^{2}.$

This shows that ${{\mathcal{A}}}_{1}$ is the best T-rank one approximation of
${\mathcal{A}}$.

##  6 Further Discussion

  1. 1.

Comparing with the tensor tubal rank, the T-rank is simple in the best rank
one approximation to third order tensors.

  2. 2.

The T-rank is not sub-additive. Let
$\mathcal{B}=(b_{ijk}),\mathcal{C}=(c_{ijk}),\mathcal{D}=(d_{ijk}),\mathcal{E}=(e_{ijk})\in\mathbb{R}^{3\times
3\times 3}$, and each of them has exactly one nonzero entry as $b_{221}=6$,
$c_{112}=5$, $d_{332}=9$, $e_{333}=9$. Their other entries are zero. Then by
Proposition 4.3, they are all T-rank one tensors. Let
${\mathcal{A}}=\mathcal{B}+\mathcal{C}+\mathcal{D}+\mathcal{E}$. Then we find
T-rank$({\mathcal{A}})=5>$ T-rank$(\mathcal{B})+$ T-rank$(\mathcal{C})+$
T-rank$(\mathcal{D})+$ T-rank$(\mathcal{E})=4$. In fact, for
$\mathcal{S}=\mathcal{S}({\mathcal{A}})$, we have

$\mathcal{S}(1,1,1)=12,\ \mathcal{S}(2,2,1)=6,\
\mathcal{S}(3,3,1)={5},\mathcal{S}(1,1,2)=\mathcal{S}(1,1,3)=3.$

The other entries of $\mathcal{S}$ are zero.

  3. 3.

Kilmer and Martin [4, Theorem 4.3] showed that an Eckart-Young like theorem
holds for the tensor tubal rank of third order tensors. Does another Eckart-
Young like theorem hold for the T-rank of third order tensors? This may be an
interesting point for further exploration.

Acknowledgement

We are thankful to Mr. Xin Chen for his comments.

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