# On the improvement of Hölder seminorms in superquadratic Hamilton-Jacobi
equations

Marco Cirant

###### Abstract

We show in this paper that maximal $L^{q}$-regularity for time-dependent
viscous Hamilton-Jacobi equations with unbounded right-hand side and
superquadratic $\gamma$-growth in the gradient holds in the full range
$q>(N+2)\frac{\gamma-1}{\gamma}$. Our approach is based on new
$\frac{\gamma-2}{\gamma-1}$-Hölder estimates, which are consequence of the
decay at small scales of suitable nonlinear space and time Hölder quotients.
This is obtained by proving suitable oscillation estimates, that also give in
turn some Liouville type results for entire solutions.

##  1 Introduction

The goal of the present paper is to address a problem of maximal
$L^{q}$-regularity for semilinear equations of Hamilton-Jacobi (HJ) type

$-\partial_{t}u-\Delta u+h(x,t)|Du|^{\gamma}=f(x,t)\qquad\text{on
$Q=\Omega\times(0,T)$}$ (1.1)

where $\Omega\subset{\mathbb{R}}^{N}$ is a bounded domain, $N\geq 1$, $h$ is a
continuous function which does not change sign (and which will be assumed here
to be strictly positive), $f$ is given in the Lebesgue space $L^{q}(Q)$, and
the nonlinear gradient term is superquadratic, i.e. $\gamma>2$. We will show
that for strong solutions $u$ in the parabolic Sobolev space $W^{2,1}_{q}$, it
is possible to control (locally) the $L^{q}$-norms of $\partial_{t}u$,
$D^{2}u$ and $|Du|^{\gamma}$ with respect to the $L^{q}$-norm of $f$.

Equation (1.1) appears in several contexts, such as stochastic optimal
control, where it is naturally backward in time. The regularity result we
obtain here sits in between the realms of weak solutions (developed mostly
below the so-called natural growth $\gamma=2$), and viscosity solutions. It is
impossible to summarize here the many, many contributions on the study of
(1.1) and its quasi-linear generalizations, but it is rather well known how to
deal with weak solutions, which involve the information $|Du|^{\gamma}\in
L^{1}$, or viscosity ones, that structurally require $f$ to be at least
continuous. The problem of maximal regularity in $L^{q}$, which has a genuine
flavor of strong solutions, has been recently considered in [5]. In the
superquadratic case $\gamma>2$, that will be addressed here, previous results
are not complete, since maximal regularity is expected to hold up to

$q>q_{0}=(N+2)\frac{\gamma-1}{\gamma},$

while $q\geq(N+2)\frac{\gamma-1}{2}$ only has been covered in [5] (see also
[12] for results on nonlocal HJ equations). Here, we deal with the full range
$q>q_{0}$. This extension requires a substantial improvement of the methods
developed previously.

There is an elliptic counterpart of maximal regularity for HJ equations, which
has been addressed in [6, 8, 11, 13]. For $q>N(\gamma-1)/\gamma$, that was
conjectured to hold by P.-L. Lions a decade ago, and it has been achieved
using nonlinear methods (such as the Bernstein method) [6, 11, 13] or blow-up
techniques [8]. The stationary problem has deep differences with respect to
the parabolic one, and ideas developed for the former have by no means obvious
adaptations to the latter. For example, blow-up techniques are based on the
validity of Liouville theorems. For stationary HJ equations, these are known
from the eighties [17, 19], and they are so powerful that they hold without
any information on the growth of solutions at infinity. On the other hand,
Liouville theorems for parabolic problems are known under rather restrictive
conditions on the growth of the solutions [22], and they are even false for
solutions having a linear behavior at infinity (see for example Remark 4.2).
The problem is even more delicate beyond the natural growth $\gamma>2$, which
is addressed here, where the diffusive part of the equation is typically
regarded as negligible (indeed, several “unnatural” phenomena which are
typical of first-order equations arise, see e.g. [9, 21]). Heuristically, many
difficulties come from the very different behavior at small scales of the two
operators $-\partial_{t}u-\Delta u$ and $-\partial_{t}u-|Du|^{\gamma}$, and
that is why one usually “chooses” between one of them, and regards the rest of
the equation as a perturbation. Nevertheless, we wish to show that their
effects are somehow intertwined at the $\alpha_{0}$-Hölder scale, where
$\alpha_{0}=\frac{\gamma-2}{\gamma-1}$.

Here, our way to maximal regularity goes indeed through the study of Hölder
regularity, which is intimately connected with Liouville theorems, and in all
of the three aspects our results are new, up to our knowledge. A few works,
such as [3, 24], addressed the problem of Hölder regularity for parabolic
problems with $\gamma>2$ and unbounded $f$, although with a perturbative
viewpoint which excluded possible regularization effects coming from the
diffusion, and under the sign condition $f\geq 0$ (see Remark 6.2 on further
comments on the issues related to the unboundedness of $f$ from below). Let us
also mention [1], where (first-order) counterexamples suggest that without any
control on the modulus of continuity of $h$, there might be limitations on the
Hölder regularity of solutions. Since our goal is to achieve
$\alpha_{0}$-Hölder (which might be very close to Lipschitz, for large
$\gamma$), we will admit estimates to depend on the modulus of continuity of
$h$. First-order problems are also considered in [2], where Sobolev regularity
of $Du$ is studied with unbounded $f$; it is worth noting that even without
diffusion, the HJ equation works at different scales in space and time (that
is why [2] uses techniques a là DiBenedetto, where cylinders depending on the
solution itself are employed). Here, we also need to take care of the effects
of the diffusion.

Let us now state the main result. Throughout the paper,

$\gamma>2,\qquad h\in C(\overline{\Omega}\times[0,T]),\qquad 0<h_{0}\leq
h(x,t)\leq h_{1}.$

######  Theorem 1.1.

Let $q_{0}<q<N+2$. For every $K>0$ and $Q^{\prime}\subset\joinrel\subset Q$,
there exists $C$ depending on $K,Q^{\prime},Q,h,q_{0}$ such that if $u\in
W^{2,1}_{q}(Q)$ solves (1.1) in the strong sense, with
$\|u\|_{L^{\infty}(Q)},\|f\|_{L^{q}(Q)}\leq K$, then

$\|\partial_{t}u\|_{L^{q}(Q^{\prime})}+\|D^{2}u\|_{L^{q}(Q^{\prime})}+\||Du|^{\gamma}\|_{L^{q}(Q^{\prime})}\leq
C.$

The statement is also true when $q\geq N+2$, see Remark 6.1. As we previously
announced, the proof of this result is a consequence of some Hölder regularity
properties, which we state here as follows:

######  Theorem 1.2.

Let $q_{0}<q<N+2$. For every $K>0$ and $Q^{\prime}\subset\joinrel\subset Q$,
there exists $C$ depending on $K,Q^{\prime},Q,h,q_{0}$ such that if $u\in
W^{2,1}_{q}(Q)$ solves (1.1) in the strong sense, with
$\|u\|_{L^{\infty}(Q)},\|f\|_{L^{q}(Q)}\leq K$, then

$\frac{|u(x,t)-u(x^{\prime},t^{\prime})|}{|x-x^{\prime}|^{\alpha}+|t-t^{\prime}|^{\nicefrac{{\alpha}}{{2}}}}\leq
C$

for all $(x,t),(x^{\prime},t^{\prime})\in Q^{\prime}$, where
$\alpha=2-\frac{N+2}{q}$.

If $q=q_{0}$, then the same conclusion holds, but $K$ might depend on $f$ not
only through its norm (in fact, $K$ is independent of $f$ whenever $f$ varies
in a uniformly integrable subset of $L^{q}(Q)$).

This theorem will be proven in two steps. First, for $q=q_{0}$ in Theorem 3.1.
Then, for $q>q_{0}$ in Theorem 5.1. While the general strategy will be
presented in detail in the second part of the introduction, we already mention
that the first step is the core of the paper. When $q=q_{0}$, then
$\alpha=\alpha_{0}=\frac{\gamma-2}{\gamma-1}$. We may then regard Theorem 1.2
as a parabolic version of the $\alpha_{0}$-Hölder regularity obtained in [9]
for stationary problems (actually for general divergence-form equations, for
subsolutions, and up to the boundary…). The intermediate passage between the
two steps $q=q_{0}$ and $q>q_{0}$ will be the Liouville Theorem 4.1.

The techniques used here have their roots in [5] (see also [4, 14]), in [8]
and in [3]. However, new and crucial ingredients are introduced, such as the
use of “nonlinear” Hölder seminorms, and the study of their decay (or
improvement) in smaller and smaller cylinders. All the estimates obtained here
are local in nature, but they could be extended up to the parabolic boundary
of $Q$ under suitable boundary conditions; we will not pursue their derivation
here. Moreover, more general nonlinear functions of $Du$ are admissible, see
for instance Remark 3.4. On the other hand, the presence of $(x,t)$-dependent
diffusions cause additional difficulties, and they will not be treated here.

Let us mention that we do not know whether maximal regularity holds below the
exponent $q_{0}$, as no counterexamples like in the stationary case are at
this stage available. Nevertheless, we strongly believe that it is not
possible to have maximal regularity below $q_{0}$, and $L^{\infty}$ estimates
might even fail ($f$ need not be bounded below here). Besides the fact that
$q_{0}$ is the parabolic analogue of the sharp stationary exponent, $q_{0}$ is
“critical” (with respect to scaling properties of (1.1)), and Hölder estimates
“see” properties of $f$ beyond its norm… Finally, we do not know at the moment
whether it is possible or not to obtain universal estimates, as in parabolic
problems with nonlinear zero-th order term [20], or for stationary HJ
equations [8]; our Hölder and maximal regularity bounds depend indeed the
$L^{\infty}$-norm of $u$.

Before entering into the details of the proofs, let us say a few words on the
structure of the paper. Section 2 will be devoted to the proof of crucial
oscillation estimates for HJ equations. In Section 3, these will be used to
obtain the first step of $\alpha_{0}$-Hölder regularity, while in Section 4 we
will show how to derive a Liouville type result. In Section 5 the full range
of Hölder regularity will be obtained (that is, for any $\alpha>\alpha_{0}$),
and in Section 6 we will finally reach the maximal regularity result. The
Appendix A contains several useful results on linear equations which will be
used thoroughly.

Strategy of the proofs. The first main idea to achieve $W^{2,1}_{q}$ estimates
can be described as follows; standard parabolic embeddings read

$W^{2,1}_{q}\hookrightarrow W^{1,0}_{p}\hookrightarrow
C^{\alpha,\nicefrac{{\alpha}}{{2}}},$

where $\frac{1}{p}=\frac{1}{q}-\frac{1}{N+2}$ and $\alpha=2-\frac{N+2}{q}$
(for $q>\frac{N+2}{2}$). In fact, since we consider solutions in
$W^{2,1}_{q}$, we look for a priori estimates (at all scales, from Hölder to
$W^{2,1}_{q}$). If $q>q_{0}$, then a priori bounds can be obtained by
following the arrows in the “opposite way”; if one is able indeed to control
$|Du|^{\gamma}$ in $L^{q}$, then standard parabolic regularity yields
$W^{2,1}_{q}$ estimates. Furthermore, if one has $\alpha$-Hölder bounds, then
an additional interpolation argument yields back $W^{2,1}_{q}$ estimates. This
step is carried out by Proposition A.4. We stress that this is possible only
if $q>q_{0}$. In other words, the ground $\alpha$-Hölder control allows to
treat the HJ equation in a perturbative manner, and the effect of the
nonlinear Hamiltonian term $|Du|^{\gamma}$ can be absorbed by the usual gain
of regularity coming from the linear part of the equation.

The second main ingredient is the improvement of the $\alpha_{0}$-Hölder
bounds to $\alpha$-Hölder bounds, which is done by Theorem 5.1. Here, we
follow an idea developed in [8] for stationary problems: if there are
sequences of solutions whose $\alpha$-Hölder seminorm explodes, then a
suitable rescaling argument allows to construct a nonconstant “entire”
solution (on ${\mathbb{R}}^{N}\times(0,\infty)$), which is in contradiction
with a Liovuille Theorem, forcing in fact such a solution to be constant.
There are a few points that are worth noticing: first, the rescaling here
preserves the linear structure ( $(x,t)\mapsto(rx,r^{2}t)$ ), but the
Hamiltonian might not vanish in the limit problem. For this reason we need a
Liouville theorem for homogeneous HJ equations. Such a Liouville theorem
cannot hold if the solution has linear growth in the $x$ variable (see Remark
4.2), that is why we need to start with $\alpha_{0}$-Hölder bounds, that give
in the limit solutions with sublinear growth.

The Liouville property (Theorem 4.1) and the $\alpha_{0}$-Hölder bounds
(Theorem 3.1) are the main novelties in this paper, and they are deduced as
follows. It is classical that Hölder regularity can be obtained whenever the
oscillation of $u$ in a cylinder is smaller (by a fixed factor) than the
oscillation of $u$ itself in a larger cylinder (at all scales). This
principle, which has been used in the context of HJ equations in [3],
typically leads to Hölder regularity with rather implicit exponent, which
might be actually very close to zero. Here, we need the precise exponent
$\alpha_{0}$ (which approaches $1$ as $\gamma\to\infty$). Therefore, we focus
on the following principle:

the $\alpha_{0}$-Hölder seminorm in a cylinder is strictly smaller than the
$\alpha_{0}$-Hölder seminorm in a larger cylinder,

which can be seen as an “improvement of $\alpha_{0}$-Hölder” regularity.

Practically, we will rescale sequences of solutions having (by contradiction)
growing $\alpha_{0}$-Hölder seminorm in a way that it is exactly $1$ in a
cylinder $Q_{0}$ of radius one, while the global $\alpha_{0}$-Hölder seminorm
remains bounded by a constant. We then prove that the control on the
$\alpha_{0}$-Hölder seminorm at the boundary of a suitable large cylinder
$Q_{1}$ implies that the oscillation of $u$ on $Q_{0}$ needs to be strictly
smaller than $1$, which is impossible. To perform this fundamental step, we
will need to introduce a nonstandard “nonlinear” $\alpha_{0}$-Hölder seminorm,
which will be denoted below by $\llbracket\cdot\rrbracket_{\alpha_{0}}$;
indeed, if we rescale in a way that the full operator

$-\partial_{t}u-\Delta u+|Du|^{\gamma},$

is preserved, then the standard $\alpha_{0}$-Hölder parabolic seminorm remains
invariant. We will then rescale in a way that the transport part
$-\partial_{t}u+|Du|^{\gamma}$ is kept unchanged, while the space-time Hölder
quotients

$\llbracket
u\rrbracket_{\alpha_{0}}\approx\max\left\\{\,\frac{|u(x,t)-u(\bar{x},t)|}{|x-\bar{x}|^{\alpha}},\,\left(\frac{|u(x,t)-u(x,\bar{t})|}{|t-\bar{t}|^{\nicefrac{{\alpha}}{{2}}}}\right)^{\nicefrac{{2}}{{\gamma}}}\right\\}$
(1.2)

are renormalized. Note that in this way the rescaled equation has vanishing
viscosity. The oscillation core estimate which is then used to conclude is
contained in Proposition 2.1. Since the proof of the proposition is rather
technical, we try to sketch here the main ideas involved (and briefly describe
how to handle the vanishing viscosity, which deteriorates second-order
regularization effects).

As $u$, and its scaling $w$, solve an HJ equation, we have at our disposal a
representation formula which arises in stochastic optimal control theory: if
for simplicity $h$ is constant, then

$w(x,0)=\inf_{b_{s}}\,\mathbb{E}\int_{0}^{\tau}\ell|b_{s}|^{\gamma^{\prime}}+f(X_{s},s)ds+\mathbb{E}w(X_{\tau},\tau),$

where the infimum is taken among stochastic trajectories
$dX_{s}=b_{s}ds+\sqrt{2\sigma}dB_{s}$ originating from $(x,0)$ and $\tau$ is
the first exit time from the cylinder $Q=B_{R}\times(0,T)$. In PDE terms, the
previous formula reads

$w(x,0)=\iint_{Q}\ell|b|^{\gamma^{\prime}}m+fm\operatorname{d\\!}y\operatorname{d\\!}s+\int_{B_{R}}w(T)m(T)\,\operatorname{d\\!}y-\iint_{\partial
B_{R}\times(0,T)}wDm\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s,$ (1.3)

where $b=-h\gamma|Dw|^{\gamma-2}Dw$, $\nu$ is the outward normal vector to
$\partial B_{R}$, and

$\begin{cases}\partial_{s}m-\sigma\Delta m-\operatorname{div}(bm)=0&\text{on
$Q$},\\\ m|_{\partial B_{R}\times(0,\tau)}=0,\ \ m(0)=\delta_{x}.\end{cases}$
(1.4)

Formula (1.3) has been used extensively to deduce regularity properties of
$w$. Besides its interpretation in optimal control, it can be obtained using a
sort of duality between (1.1) and (1.4); such a duality is at the core of the
so-called (nonlinear) adjoint method, which allows to derive regularity of $w$
by regularity properties of $m$. This strategy has been proposed in [10] for
HJ equations. In particular, Hölder seminorms of $u$ are related to variations
with respect to $x$ in (1.3), that necessarily involve variations of $m$ with
respect to $x$, by the presence of $\iint fm$. Unfortunately, the only
available information on $m$ is a bound in $L^{q_{0}^{\prime}}$ (see Lemma
A.2); bounds on the derivatives of $m$ are possible in smaller spaces only,
and that is why Hölder and maximal regularity are not achieved up to $q=q_{0}$
in [5]. For the reader who is experienced with the adjoint method for HJ
equations, the main challenge here is to “differentiate” $u$ in (1.3) with
respect to $x$ without “differentiating” $m$ in $\iint fm$ ! This is a
substantial step which is developed in the present paper.

To compare $w(x,0)$ and $w(x+h,0)$, we just “bend” $m$ to a new $\tilde{m}$,
which starts from $\delta_{x+h}$ and coincides with $m$ at final time. Such
new $\tilde{m}$ is used in the representation formula for $w(x+h,0)$ (since
the new $\tilde{m}$ is “suboptimal”, the equality becomes an inequality in
(1.3)), which is then compared with (1.3) for $w(x,0)$. Note that
$w(x+h,0)-w(x,0)$ is renormalized to be $1$, and there is no need to divide by
$h$. The discrepancies in the two formulas mainly involve the difference
between the two Lagrangian terms $\iint|b|^{\gamma^{\prime}}m$, the so-called
running costs $\iint fm$, and the boundary contributions $\iint_{\partial
B_{R}\times(0,T)}wDm\cdot\nu$. These depend on the dimensions $R,T$ of the
large cylinder; loosely speaking, Lagrangian terms can be made small by
choosing $T$ large, and $\iint fm$ vanishes since the $L^{q_{0}}$-norm of $f$
can be made small at small scales. The boundary terms are definitely the most
delicate ones; the key point is to realize that $\iint_{\partial
B_{R}\times(0,T)}Dm\cdot\nu$, which represent the total density leaving the
cylinder from its lateral boundary, is small if $R$ is large, and it can
compensate $w$, which might be large (if the large cylinder is chosen
carefully enough). Similar arguments allow to control the oscillation of $w$
in time.

A few comments are now in order. As we said, the scaled operator has vanishing
viscosity

$-\partial_{t}u-\sigma\Delta u+|Du|^{\gamma},\qquad\sigma=o(1),$

which allows to exploit the first order regularization effects, but of course
brings a deterioration of linear regularity. Nevertheless, by the particular
way in which space-time is scaled we can still use the (small) linear
contribution: the deterioration of the $L^{q_{0}^{\prime}}$-norm of $m$ is
compensated by rapid vanishing of the $L^{q_{0}}$-norm of $f$, so that the
whole term can $\iint fm$ be controlled in the scaling process. To carry out
the whole procedure, we stress again that we need to work with ad-hoc Hölder
quotients, and that both the linear and nonlinear part of the equation play a
major role in regularization; the argument is by no means perturbative.
Finally, the reader will see that we need an additional parameter $z$ in the
“nonlinear” Hölder seminorm to fine tune the argument. This is mainly because
we are able to shrink the time Hölder seminorm from $z$ to $z/2$ only if $z$
is large, while $z$ plays basically no role when shrinking seminorms in space
(provided that cylinders are fat enough). Since our goal is to obtain local
estimates, suitable distance functions will appear also in the computations,
but this will be just an annoying technical point.

Notations.

  * •

$q_{0}=\frac{N+2}{\gamma^{\prime}}$, $\alpha_{0}=\frac{\gamma-2}{\gamma-1}$.
Throughout the paper, $\gamma>2$, $q_{0}\leq q<N+2$ and
$\alpha=2-\frac{N+2}{q}$, unless otherwise specified. Note that $q$ is always
larger than $1+\frac{N}{2}$.

  * •

$|\cdot|$ will denote the usual Euclidean distance in ${\mathbb{R}}^{N}$;
$B_{r}(x)=\\{y\in{\mathbb{R}}^{N}\mathrel{\mathop{\mathchar
58\relax}}|x-y|<r\\}$. $d(x,\Omega)$ be the Euclidean distance from
$x\in{\mathbb{R}}^{N}$ to the set $\Omega\subset{\mathbb{R}}^{N}$. For a
cylinder $Q=\Omega\times(a,b)$, we will use two different parabolic distances
from the (backward parabolic) boundary
$\partial^{+}Q=\partial\Omega\times(a,b)\cup\Omega\times\\{b\\}$:

$d{\big{(}(x,t),\partial^{+}Q\big{)}}=d(x,\partial\Omega)+|b-t|^{\nicefrac{{1}}{{2}}},\qquad{\mathcal{d}}_{\alpha}{\big{(}(x,t),\partial^{+}Q\big{)}}=d^{\alpha}(x,\partial\Omega)+|b-t|^{\nicefrac{{\alpha}}{{\gamma}}}$

(${\mathcal{d}}_{\alpha}$ will be often used without the subscript
$\alpha\in(0,1)$ for brevity).

  * •

Parabolic cylinders: $Q_{r,t}=B_{r}(0)\times(0,t)$,
$Q_{r}=B_{r}(0)\times(0,r^{2})$.

  * •

For $Q=\Omega\times(a,b)$, $0<\alpha<1$, $c>0$, we define the usual parabolic
Hölder seminorm

$[u]_{\alpha;Q}=\sup_{(x,t),(\bar{x},\bar{t})\in
Q}\frac{|u(x,t)-u(\bar{x},\bar{t})|}{(|x-\bar{x}|+|t-\bar{t}|^{\nicefrac{{1}}{{2}}})^{\alpha}}\,,$

and its wighted version

$[u]^{c}_{\alpha;Q}=\sup_{(x,t),(\bar{x},\bar{t})\in
Q}[\min\\{d((x,t),\partial^{+}Q),d((\bar{x},\bar{t}),\partial^{+}Q)\\}]^{c}\frac{|u(x,t)-u(\bar{x},\bar{t})|}{(|x-\bar{x}|+|t-\bar{t}|^{\nicefrac{{1}}{{2}}})^{\alpha}}.$

Finally, for any $z>0$,

$\displaystyle\llbracket u\rrbracket^{x}_{\alpha;Q}=$
$\displaystyle\sup_{(x,t),(\bar{x},t)\in
Q}\min\\{{\mathcal{d}}_{\alpha}{((x,t),\partial^{+}Q)},{\mathcal{d}}_{\alpha}{((\bar{x},t),\partial^{+}Q)}\\}\frac{|u(x,t)-u(\bar{x},t)|}{|x-\bar{x}|^{\alpha}},$
$\displaystyle\llbracket u\rrbracket^{t}_{\nicefrac{{\alpha}}{{2}};Q}=$
$\displaystyle\sup_{(x,t),(x,\bar{t})\in
Q}[\min\\{{\mathcal{d}}_{\alpha}{((x,t),\partial^{+}Q)},{\mathcal{d}}_{\alpha}{((x,\bar{t}),\partial^{+}Q)}\\}]^{\nicefrac{{\gamma}}{{2}}}\frac{|u(x,t)-u(x,\bar{t})|}{|t-\bar{t}|^{\nicefrac{{\alpha}}{{2}}}},$
$\displaystyle\llbracket u\rrbracket_{\alpha,z;Q}=$
$\displaystyle\max\left\\{\llbracket
u\rrbracket^{x}_{\alpha;Q}\,,\,(z^{-1}\llbracket
u\rrbracket^{t}_{\nicefrac{{\alpha}}{{2}};Q})^{\nicefrac{{2}}{{\gamma}}}\right\\}.$

  * •

$\|\cdot\|_{L^{q}(X)}$ be the usual norm on the Lebesgue space $L^{q}(X)$.
Parabolic Sobolev spaces:

$W^{2,1}_{q}(X)=\\{\text{measurable $u$}\mathrel{\mathop{\mathchar
58\relax}}\partial^{r}_{t}D^{\beta}_{x}u\in L^{q}(X)\text{ for any
}|\beta|+2r\leq 2\\}.$

  * •

Recall that a set ${\mathcal{F}}\subset L^{q}(X)$ is $L^{q}$-uniformly
integrable if for any $\varepsilon>0$, there exists $\delta>0$ such that

$\|f\|_{L^{q}(E\cap X)}<\varepsilon\qquad\text{whenever $f\in{\mathcal{F}}$
and $\int_{E\cap X}1<\delta$.}$

  * •

$C,C_{1},...$ will denote positive constants, whose value may vary from line
to line. Unless otherwise specified, they may depend on $N,\alpha,\gamma,q,Q$
and $h$ (in particular on $h_{0},h_{1}$ and the modulus of continuity of $h$),
but not on $u$ and $f$.

Acknowledgements. The author was partially supported by the King Abdullah
University of Science and Technology (KAUST) project CRG2021-4674 “Mean-Field
Games: models, theory, and computational aspects”, by the INDAM-GNAMPA grant
“Proprietà quantitative e qualitative per EDP non lineari con termini di
gradiente” (2022).

##  2 Oscillation estimates

This section is devoted to the proof of the oscillation estimates, which will
be used to show that certain Hölder seminorms decay in smaller and smaller
cylinders. These will be also crucial to deduce the Liouville theorem for
entire solutions to HJ equations.

Throughout this section, $w\in W^{2,1}_{q}(Q_{R+1,\tau})$ (for some $q\geq
q_{0}$) solves on $Q_{R+1,\tau}$ the differential inequalities

$\displaystyle-\partial_{s}w-\sigma\Delta w+h_{0}|Dw|^{\gamma}\leq g,$ (2.1)
$\displaystyle-\partial_{s}w-\sigma\Delta w+h_{1}|Dw|^{\gamma}\geq g$ (2.2)

in the strong sense, for a diffusion factor $0\leq\sigma\leq 1$, and
$0<\underline{h}\leq h_{0},h_{1}\leq\bar{h}$. The function $g$ is given in
$L^{q_{0}}(Q_{R+1,\tau})$.

######  Proposition 2.1.

Assume that $R^{2}\geq\sigma\tau$, and that for some $0<\alpha<1$, $z\geq 1$,

$\sup_{(y,s),(y^{\prime},s)\in\overline{Q}_{R,\tau}}\frac{|w(y,s)-w(y^{\prime},s)|}{|y-y^{\prime}|^{\alpha}}\leq
3,\qquad\sup_{(y,s),(y,s^{\prime})\in\overline{Q}_{R,\tau}}\frac{|w(y,s)-w(y,s^{\prime})|}{|s-s^{\prime}|^{\alpha/2}}\leq
3^{\frac{\gamma}{2}}z.$

Then there exist $f^{0},c_{1},C_{2},C_{3}$ depending on
$\underline{h},\bar{h},\alpha,q_{0}$, and ${\mathcal{K}}\geq 0$, such that if

$\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g\|_{L^{q_{0}}(Q_{R,\tau})}\leq
f^{0},\qquad z\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R}\leq c_{1},$ (2.3)

then for any $|y_{0}|\leq 1$,

$\displaystyle|w(0,0)-w(0,\tau)|+{\mathcal{K}}\leq
C_{2}\left(\tau^{\frac{\alpha}{2}}+\tau^{\frac{\alpha_{0}}{2}}+\tau^{\frac{\alpha}{\gamma-\alpha(\gamma-1)}}+\tau\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R}z\right),$
(2.4) $\displaystyle w(y_{0},0)-w(0,0)\leq
C_{3}\Big{(}\frac{{\mathcal{K}}^{1/\gamma}}{\tau^{1/\gamma}}+\frac{1}{\tau^{\gamma^{\prime}-1}}+$
$\displaystyle\qquad\qquad+\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}({\mathcal{K}}+\tau^{\alpha_{0}/2})\|g\|_{L^{q_{0}}(Q_{R+1,\tau})}+\frac{\tau^{1/\gamma}{\mathcal{K}}^{1/\gamma^{\prime}}}{R}+\frac{\tau}{R^{2}}+(\ell_{0}-\ell_{1}){\mathcal{K}}\Big{)},$
(2.5)

where $\ell_{i}=\frac{h_{i}(\gamma-1)}{(h_{i}\gamma)^{\gamma^{\prime}}}$,
$i=1,2$

The proof will be obtained in several steps. In what follows, will always
assume that $R^{2}\geq\sigma\tau$.

######  Lemma 2.2.

There exists $C_{0}$ depending on $\alpha,q_{0}$ such that

$w(0,0)-w(0,\tau)\leq
C_{0}\left(\tau^{\frac{\alpha}{2}}+\tau\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R^{2}}z+\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g\|_{L^{q_{0}}(Q_{R,\tau})}\tau^{{\alpha_{0}}/2}\right).$

###### Proof.

Let $\mu$ be the solution of

$\begin{cases}\partial_{s}\mu-\sigma\Delta\mu=0&\text{on
$B_{R}\times(0,\tau)$}\\\ \mu(y,s)=0&\text{on $\partial
B_{R}\times(0,\tau)$}\\\ \mu(0)=\delta_{0}\end{cases}$

(the existence of such $\mu$ and its regularity are detailed in Appendix A).
Testing (2.1) by $\mu$ and the equation for $\mu$ by $w$, and integrating by
parts on $Q_{R,\tau}$ yields

$w(0,0)+h_{0}\iint_{B_{R}\times(0,\tau)}|Dw|^{\gamma}\mu\operatorname{d\\!}y\operatorname{d\\!}s\leq\\\
+\iint_{B_{R}\times(0,\tau)}g\mu\operatorname{d\\!}y\operatorname{d\\!}s+\int_{B_{R}}w(y,\tau)\mu(y,\tau)\,\operatorname{d\\!}y-\sigma\iint_{\partial
B_{R}\times(0,\tau)}wD\mu\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$

Since $\int_{B_{R}}\mu(y,\tau)-\sigma\iint_{\partial
B_{R}\times(0,s)}D\mu\cdot\nu=1$,

$w(0,0)-w(0,\tau)\leq\int_{B_{R}}[w(y,\tau)-w(0,\tau)]\mu(y,\tau)\,\operatorname{d\\!}y\\\
+\iint_{B_{R}\times(0,\tau)}g\mu\operatorname{d\\!}y\operatorname{d\\!}s-\sigma\iint_{\partial
B_{R}\times(0,\tau)}[w-w(0,\tau)]D\mu\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$

First, by the growth assumptions on $w$ and Lemma A.1

$\int_{B_{R}}[w(y,\tau)-w(0,\tau)]\mu(y,\tau)\,\operatorname{d\\!}y\leq
3\int_{B_{R}}|y|^{\alpha}\mu(y,\tau)\leq
3C\left(\sigma\tau\right)^{\alpha/2}.$

Secondly, applying Lemma A.2

$\iint_{B_{R}\times(0,\tau)}g\mu\operatorname{d\\!}y\operatorname{d\\!}s\leq\|g\|_{L^{q_{0}}(Q_{R,\tau})}\|\mu\|_{L^{q_{0}^{\prime}}(Q_{R,\tau})}\leq
C\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g\|_{L^{q_{0}}(Q_{R,\tau})}\sigma^{\gamma^{\prime}/2}\tau^{\alpha_{0}/2}.$

Finally, again by the growth assumptions on $w$ and (A.5)

$\sigma\iint_{\partial
B_{R}\times(0,\tau)}[w(y,s)-w(0,\tau)](-D\mu(y,s)\cdot\nu(y))\operatorname{d\\!}y\operatorname{d\\!}s\leq\\\
\sigma 3^{\gamma/2}z(R^{\alpha}+\tau^{\frac{\alpha}{2}})\iint_{\partial
B_{R}\times(0,\tau)}-D\mu(y,s)\cdot\nu(y)\operatorname{d\\!}y\operatorname{d\\!}s\leq
Cz(R^{\alpha}+\tau^{\frac{\alpha}{2}})\frac{\tau}{R^{2}},$

which, together with the previous inequalities, yields the assertion. ∎

Let us now define

$b(y,s)=h_{1}\gamma|Dw(y,s)|^{\gamma-2}Dw(y,s),$

and $m$ be the solution of the dual problem

$\begin{cases}\partial_{s}m-\sigma\Delta m-\operatorname{div}(bm)=0&\text{on
$B_{R}\times(0,\tau)$}\\\ m(y,s)=0&\text{on $\partial B_{R}\times(0,\tau)$}\\\
m(0)=\delta_{0}.\end{cases}$

######  Lemma 2.3.

There exists $c_{0},c_{1},C_{1},f^{0}>0$ depending on $\bar{h},\alpha,q_{0}$
such that if

$\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g\|_{L^{q_{0}}(Q_{R,\tau})}\leq
f^{0},\qquad z\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R}\leq c_{1},$ (2.6)

then

$w(0,0)-w(0,\tau)\geq
c_{0}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s+\Phi,$

where

$\Phi\geq-
C_{1}\left(\tau^{\frac{\alpha}{2}}+\tau^{\frac{\alpha_{0}}{2}}+\tau^{\frac{\alpha}{\gamma-\alpha(\gamma-1)}}+\tau\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R}z\right).$

###### Proof.

Testing (2.2) by $m$ and the equation for $m$ by $w$, and integrating by parts
on $Q_{R,\tau}$ yields

$w(0,0)\geq
h_{1}(\gamma-1)\iint_{B_{R}\times(0,\tau)}|Dw|^{\gamma}m\operatorname{d\\!}y\operatorname{d\\!}s+\\\
+\iint_{B_{R}\times(0,\tau)}gm\operatorname{d\\!}y\operatorname{d\\!}s+\int_{B_{R}}w(y,\tau)m(y,\tau)\,\operatorname{d\\!}y-\sigma\iint_{\partial
B_{R}\times(0,\tau)}wDm\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$
(2.7)

whence

$w(0,0)-w(0,\tau)\geq
c_{0}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s+\Phi,$

where
$2c_{0}=\frac{\bar{h}(\gamma-1)}{(\bar{h}\gamma)^{\gamma^{\prime}}}\leq\frac{h_{1}(\gamma-1)}{(h_{1}\gamma)^{\gamma^{\prime}}}=\mathrel{\mathop{\mathchar
58\relax}}\ell_{1}$, and

$\Phi=c_{0}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s+\int_{B_{R}}[w(y,\tau)-w(0,\tau)]m(y,\tau)\,\operatorname{d\\!}y\\\
+\iint_{B_{R}\times(0,\tau)}gm\operatorname{d\\!}y\operatorname{d\\!}s-\sigma\iint_{\partial
B_{R}\times(0,\tau)}[w-w(0,\tau)]Dm\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$

First, by the growth assumptions on $w$, inequality (A.6), and Young’s
inequality (notice that
$\frac{\alpha}{\gamma}\big{(}\frac{\gamma^{\prime}}{\alpha}\big{)}^{\prime}=\frac{\alpha}{\gamma-\alpha(\gamma-1)}$),

$\int_{B_{R}}[w(y,\tau)-w(0,\tau)]m(y,\tau)\,\operatorname{d\\!}y\geq-3\int_{B_{R}}|y|^{\alpha}m(y,\tau)\\\
\geq-C\tau^{\alpha/\gamma}\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s\right)^{\alpha/\gamma^{\prime}}-C\left(\sigma\tau\right)^{\alpha/2}\\\
\geq-\frac{c_{0}}{3}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s-C(\tau^{\alpha/2}+\tau^{\frac{\alpha}{\gamma-\alpha(\gamma-1)}}).$

Secondly, applying Lemma A.2 and assuming $f^{0}$ small enough (depending on
$c_{0}$ and $C$ below),

$\iint_{B_{R}\times(0,\tau)}gm\operatorname{d\\!}y\operatorname{d\\!}s\geq-\|g\|_{L^{q_{0}}(Q_{R,\tau})}\|m\|_{L^{q_{0}^{\prime}}(Q_{R,\tau})}\geq-f^{0}\sigma^{\gamma^{\prime}\frac{N+1}{N+2}}\|m\|_{L^{q_{0}^{\prime}}(Q_{R,\tau})}\\\
\geq-
Cf^{0}\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t+\sigma^{\gamma^{\prime}/2}\tau^{\alpha_{0}/2}\right)\\\
\geq-\frac{c_{0}}{3}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s-C\sigma^{\gamma^{\prime}/2}\tau^{\alpha_{0}/2}.$

Finally, by the growth assumptions on $w$, (A.5) and Young’s inequality,

$\sigma\iint_{\partial
B_{R}\times(0,\tau)}[w(y,s)-w(0,\tau)](-Dm(y,s)\cdot\nu(y))\operatorname{d\\!}y\operatorname{d\\!}s\geq\\\
\geq-\sigma 3^{\gamma/2}z(R^{\alpha}+\tau^{\frac{\alpha}{2}})\iint_{\partial
B_{R}\times(0,\tau)}-Dm(y,s)\cdot\nu(y)\operatorname{d\\!}y\operatorname{d\\!}s\\\
\geq-
Cz(R^{\alpha}+\tau^{\frac{\alpha}{2}})\left(\frac{\tau^{1/\gamma}}{R}\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t\right)^{1/\gamma^{\prime}}+\frac{\tau}{R^{2}}\right)\\\
\geq-\overline{C}z(R^{\alpha}+\tau^{\frac{\alpha}{2}})\left(\frac{1}{R}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t+\frac{\tau}{R}+\frac{\tau}{R^{2}}\right)\\\
\stackrel{{\scriptstyle\eqref{cond0}}}{{\geq}}-\frac{c_{0}}{3}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s-\overline{C}z(R^{\alpha}+\tau^{\frac{\alpha}{2}})\left(\frac{\tau}{R}+\frac{\tau}{R^{2}}\right),$

where $c_{1}$ in (2.6) is chosen to be $c_{1}=c_{0}/(3\overline{C})$. The
three estimates above yield the statement. ∎

######  Corollary 2.4.

Under the assumption (2.6), for some $c_{0},C_{2}$ depending on
$\bar{h},\alpha,q_{0}$,

$|w(0,0)-w(0,\tau)|+\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s\leq
C_{2}\left(\tau^{\frac{\alpha}{2}}+\tau^{\frac{\alpha_{0}}{2}}+\tau^{\frac{\alpha}{\gamma-\alpha(\gamma-1)}}+\tau\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R}z\right).$
(2.8)

###### Proof.

The previous lemma states that

$c_{0}\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s\leq
w(0,0)-w(0,\tau)-\Phi,$

hence by Lemma 2.2 one concludes (using that $R\geq 1$ and
$\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g\|_{L^{q_{0}}(Q_{R,\tau})}\leq
f^{0}$) . ∎

The previous corollary allows to control the oscillation of $w$ in time. We
now take care of the oscillation of $w$ in space. Let $y_{0}$ be any unit
vector, $R,\tau\geq 1$, $R^{2}\geq\sigma\tau$ as before, and

$\xi_{s}=\frac{\tau-s}{\tau}y_{0}.$

######  Lemma 2.5.

The following inequality holds.

$w(y_{0},0)\leq\ell_{0}\iint_{B_{R}\times(0,\tau)}|b-\xi^{\prime}_{s}|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s+\iint_{B_{R}\times(0,\tau)}g(y+\xi_{s},s)m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s+\\\
+\int_{B_{R}}w(y,\tau)m(y,\tau)\,\operatorname{d\\!}y-\sigma\iint_{\partial
B_{R}\times(0,\tau)}w(y+\xi_{s},s)Dm(y,s)\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$
(2.9)

###### Proof.

Since

$h_{0}|p|^{\gamma}=\sup_{q\in{\mathbb{R}}^{N}}\left\\{p\cdot
q-\ell_{0}|q|^{\gamma^{\prime}}\right\\},\qquad\text{where
$\ell_{0}=\frac{h_{0}(\gamma-1)}{(h_{0}\gamma)^{\gamma^{\prime}}}$},$

for any vector field $\tilde{b}$ we have

$-\partial_{s}w-\sigma\Delta
w+Dw\cdot\tilde{b}-\ell_{0}|\tilde{b}|^{\gamma^{\prime}}\leq g\qquad\text{on
$Q_{R+1,\tau}$}.$ (2.10)

Let us now choose

$\tilde{b}(\tilde{y},s)=b(\tilde{y}-\xi_{s},s)-\xi^{\prime}_{s}\,=h_{1}\gamma|Dw(\tilde{y}-\xi_{s},s)|^{\gamma-2}Dw(\tilde{y}-\xi_{s},s)+\frac{y_{0}}{\tau}\quad\text{on
$B_{R}(\xi_{s})\times(0,\tau)$}.$

Test now the inequality (2.10) by $m(\tilde{y}-\xi_{s},s)$, and integrate by
parts on $B_{R}(\xi_{s})\times(0,\tau)\ni(\tilde{y},s)$ as follows:

$-\iint_{B_{R}(\xi_{s})\times(0,\tau)}\partial_{s}w(\tilde{y},s)m(\tilde{y}-\xi_{s},s)\operatorname{d\\!}\tilde{y}\operatorname{d\\!}s=-\iint_{B_{R}\times(0,\tau)}\partial_{s}w(y+\xi_{s},s)m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s\\\
=-\iint_{B_{R}\times(0,\tau)}\partial_{s}(w(y+\xi_{s},s))m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s+\iint_{B_{R}\times(0,\tau)}Dw(y+\xi_{s},s)\cdot\xi_{s}^{\prime}m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s\\\
=\iint_{B_{R}\times(0,\tau)}w(y+\xi_{s},s)\partial_{s}m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s+\iint_{B_{R}\times(0,\tau)}Dw(y+\xi_{s},s)\cdot\xi_{s}^{\prime}m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s\\\
-\int_{B_{R}}w(y+\xi_{\tau},\tau)m(y,\tau)\,\operatorname{d\\!}y+w(\xi_{0},0).$

Secondly,

$-\iint_{B_{R}(\xi_{s})\times(0,\tau)}\sigma\Delta
w(\tilde{y},s)m(\tilde{y}-\xi_{s},s)\operatorname{d\\!}\tilde{y}\operatorname{d\\!}s=\\\
=-\iint_{B_{R}\times(0,\tau)}\sigma\Delta
w(y+\xi_{s},s)m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s\\\
=-\iint_{B_{R}\times(0,\tau)}\sigma\Delta
m(y,s)w(y+\xi_{s},s)\operatorname{d\\!}y\operatorname{d\\!}s+\sigma\iint_{\partial
B_{R}\times(0,\tau)}w(y+\xi_{s},s)Dm\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$

Regarding the third term,

$\iint_{B_{R}(\xi_{s})\times(0,\tau)}Dw(\tilde{y},s)\cdot\tilde{b}(\tilde{y},s)m(\tilde{y}-\xi_{s},s)\operatorname{d\\!}\tilde{y}\operatorname{d\\!}s=\\\
=\iint_{B_{R}\times(0,\tau)}Dw(y+\xi_{s},s)\cdot[b(y,s)-\xi^{\prime}_{s}]m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s\\\
=-\iint_{B_{R}\times(0,\tau)}w(y+\xi_{s},s)\operatorname{div}\big{(}b(y,s)m(y,s)\big{)}\operatorname{d\\!}y\operatorname{d\\!}s\\\
-\iint_{B_{R}\times(0,\tau)}Dw(y+\xi_{s},s)\cdot\xi^{\prime}_{s}m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s$

Adding up the three previous inequalities, one finds the equation for $m$
which is being multiplied by $w(y+\xi_{s},s)$, and the two terms $\iint
Dw(y+\xi_{s},s)\cdot\xi^{\prime}_{s}m$ cancel out, so

$-\int_{B_{R}}w(y+\xi_{\tau},\tau)m(y,\tau)\,\operatorname{d\\!}y+w(\xi_{0},0)+\sigma\iint_{\partial
B_{R}\times(0,\tau)}w(y+\xi_{s},s)Dm\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s+\\\
-\ell_{0}\iint_{B_{R}(\xi_{s})\times(0,\tau)}|\tilde{b}|^{\gamma^{\prime}}m(\tilde{y}-\xi_{s},s)\operatorname{d\\!}\tilde{y}\operatorname{d\\!}s\leq\iint_{B_{R}(\xi_{s})\times(0,\tau)}gm(\tilde{y}-\xi_{s},s)\operatorname{d\\!}\tilde{y}\operatorname{d\\!}s.$

After a further change of variables $y=\tilde{y}-\xi_{s}$, and the fact that
$\xi_{\tau}=0$ and $\xi_{0}=y_{0}$ we obtain the assertion. ∎

######  Lemma 2.6.

Let
${\mathcal{K}}=\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s$.
Then, for some $C_{3}$ depending on $\underline{h},\alpha,q_{0}$,

$w(y_{0},0)-w(0,0)\leq
C_{3}\Big{(}\frac{{\mathcal{K}}^{1/\gamma}}{\tau^{1/\gamma}}+\frac{1}{\tau^{\gamma^{\prime}-1}}+\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}({\mathcal{K}}+\tau^{\alpha_{0}/2})\|g\|_{L^{q_{0}}(Q_{R+1,\tau})}+\\\
+\frac{\tau^{1/\gamma}{\mathcal{K}}^{1/\gamma^{\prime}}}{R}+\frac{\tau}{R^{2}}+(\ell_{0}-\ell_{1}){\mathcal{K}}\Big{)},$
(2.11)

where $\ell_{i}=\frac{h_{i}(\gamma-1)}{(h_{i}\gamma)^{\gamma^{\prime}}}$,
$i=1,2$.

###### Proof.

Recall (2.7), which reads

$w(0,0)\geq\ell_{1}{\mathcal{K}}+\iint_{B_{R}\times(0,\tau)}gm\operatorname{d\\!}y\operatorname{d\\!}s+\\\
+\int_{B_{R}}w(y,\tau)m(y,\tau)\,\operatorname{d\\!}y-\sigma\iint_{\partial
B_{R}\times(0,\tau)}wDm\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s.$

Subtracting (2.9) and the previous equality yields

$\displaystyle w(y_{0},0)-w(0,0)\leq\ell_{0}$
$\displaystyle\iint_{B_{R}\times(0,\tau)}[|b-\xi^{\prime}_{s}|^{\gamma^{\prime}}-|b|^{\gamma^{\prime}}]m\operatorname{d\\!}y\operatorname{d\\!}s$
$\displaystyle+$
$\displaystyle\iint_{B_{R}\times(0,\tau)}[g(y+\xi_{s},s)-g(y,s)]m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s$
$\displaystyle-$ $\displaystyle\sigma\iint_{\partial
B_{R}\times(0,\tau)}[w(y+\xi_{s},s)-w(y,s)]Dm(y,s)\cdot\nu\operatorname{d\\!}y\operatorname{d\\!}s$
$\displaystyle+(\ell_{0}-\ell_{1}){\mathcal{K}}.$

To estimate the first term, we make use of the inequality (A.11), that
$|\xi^{\prime}|=\tau^{-1}$ and the Hölder inequality:

$\ell_{0}\iint_{B_{R}\times(0,\tau)}[|b-\xi^{\prime}_{s}|^{\gamma^{\prime}}-|b|^{\gamma^{\prime}}]m\operatorname{d\\!}y\operatorname{d\\!}s\stackrel{{\scriptstyle\eqref{Ldiff}}}{{\leq}}\\\
\leq
C|\xi^{\prime}_{s}|\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}-1}m\operatorname{d\\!}y\operatorname{d\\!}s+C|\xi^{\prime}_{s}|^{\gamma^{\prime}}\iint_{B_{R}\times(0,\tau)}m\operatorname{d\\!}y\operatorname{d\\!}s\\\
\leq\frac{C}{\tau}\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}y\operatorname{d\\!}s\right)^{\frac{1}{\gamma}}\left(\iint_{B_{R}\times(0,\tau)}m\operatorname{d\\!}y\operatorname{d\\!}s\right)^{\frac{1}{\gamma^{\prime}}}+\frac{C}{\tau^{\gamma^{\prime}}}\iint_{B_{R}\times(0,\tau)}m\operatorname{d\\!}y\operatorname{d\\!}s\\\
\stackrel{{\scriptstyle\eqref{massloss}}}{{\leq}}\frac{C{\mathcal{K}}^{1/\gamma}}{\tau^{1/\gamma}}+\frac{C}{\tau^{\gamma^{\prime}-1}}.$

As for the second term, Lemma (A.2) implies

$\iint_{B_{R}\times(0,\tau)}[g(y+\xi_{s},s)-g(y,s)]m(y,s)\operatorname{d\\!}y\operatorname{d\\!}s\leq
2\|g\|_{L^{q_{0}}(Q_{R+1,\tau})}\|m\|_{L^{q_{0}^{\prime}}(Q_{R,\tau})}\\\ \leq
C\sigma^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g\|_{L^{q_{0}}(Q_{R+1,\tau})}\left({\mathcal{K}}+\sigma^{\gamma^{\prime}/2}\tau^{\alpha_{0}/2}\right)$

Finally, by the growth assumptions on $w$,

$\sigma\iint_{\partial
B_{R}\times(0,\tau)}[w(y+\xi_{s},s)-w(y,s)](-Dm\cdot\nu)\operatorname{d\\!}y\operatorname{d\\!}s\leq\\\
\leq 3\sigma\iint_{\partial
B_{R}\times(0,\tau)}(-Dm\cdot\nu)\operatorname{d\\!}y\operatorname{d\\!}s\stackrel{{\scriptstyle\eqref{boundaryloss}}}{{\leq}}\frac{3\tau^{1/\gamma}}{R}{\mathcal{K}}^{1/\gamma^{\prime}}+\frac{C\tau}{R^{2}},$

and by collecting the three estimate we conclude. ∎

To conclude the proof of Proposition 2.1, it just suffices to collect the
estimates (2.8) and (2.11).

##  3 $\alpha_{0}$-Hölder regularity

In this section we prove the $\alpha_{0}$-Hölder estimates, which are at the
base of further regularity results.

######  Theorem 3.1.

Let ${\mathcal{F}}\subset L^{q_{0}}(Q)$ be a uniformly integrable set in
$L^{q_{0}}(Q)$ and assume that $\|u\|_{L^{\infty}(Q)}\leq K$. Then there
exists $C,z$ depending on ${\mathcal{F}},K,Q,h,q_{0}$ such that

$\llbracket u\rrbracket_{\alpha_{0},z;Q}\leq C.$

1\. Set-up of the contradiction argument. Fix $z>0$, which will be specified
below. Assume by contradiction that for some sequence
$\\{f_{n}\\}\subset{\mathcal{F}}$ and $\\{u_{n}\\}$ solving (1.1),

$\llbracket u\rrbracket_{\alpha_{0},z;Q}\to\infty\quad\text{as $n\to\infty$}.$

Setting $L_{n}\mathrel{\mathop{\mathchar 58\relax}}=\llbracket
u\rrbracket_{\alpha_{0},z;Q}/2$, we will distinguish two cases:

(a) $2L_{n}=\llbracket u\rrbracket^{x}_{\alpha_{0};Q}\geq(z^{-1}\llbracket
u\rrbracket^{t}_{\nicefrac{{\alpha_{0}}}{{2}};Q})^{\frac{2}{\gamma}}$,
---
(b) $2L_{n}=(z^{-1}\llbracket
u\rrbracket^{t}_{\nicefrac{{\alpha_{0}}}{{2}};Q})^{\frac{2}{\gamma}}\geq\llbracket
u\rrbracket^{x}_{\alpha_{0};Q}$ .

In case (a), there are sequences
$\\{(x_{n},\bar{t}_{n})\\},\\{(\bar{x}_{n},\bar{t}_{n})\\}\subset Q$ such that
for every $n$, $x_{n}\neq\bar{x}_{n}$ and

$L_{n}\leq\min\\{{\mathcal{d}}((x_{n},\bar{t}_{n}),\partial
Q),{\mathcal{d}}((\bar{x}_{n},\bar{t}_{n}),\partial
Q)\\}\frac{|u_{n}(x_{n},\bar{t}_{n})-u(\bar{x}_{n},\bar{t}_{n})|}{|x_{n}-\bar{x}_{n}|^{\alpha_{0}}}\leq
2L_{n}.$

If we assume w.l.o.g. that ${\mathcal{d}}((x_{n},\bar{t}_{n}),\partial
Q)\geq{\mathcal{d}}((\bar{x}_{n},\bar{t}_{n}),\partial Q)$ and
$u(\bar{x}_{n},\bar{t}_{n})=0$, setting

$d_{n}=d(\bar{x}_{n},\partial\Omega),\qquad
M_{n}=|u_{n}(x_{n},\bar{t}_{n})|,\qquad r_{n}=|x_{n}-\bar{x}_{n}|,$

then the previous inequalities read

$L_{n}\leq\big{(}d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}\big{)}\frac{M_{n}}{r_{n}^{\alpha_{0}}}\leq
2L_{n}.$ (3.1)

In the other case (b), there are sequences
$\\{(\bar{x}_{n},t_{n})\\},\\{(\bar{x}_{n},\bar{t}_{n})\\}\subset Q$ such that
for every $n$, $t_{n}>\bar{t}_{n}$ and

$L_{n}\leq
z^{-\frac{2}{\gamma}}\min\\{{\mathcal{d}}{((\bar{x}_{n},t_{n}),\partial
Q)},{\mathcal{d}}{((\bar{x}_{n},\bar{t}_{n}),\partial
Q)}\\}\frac{|u(\bar{x}_{n},t_{n})-u(\bar{x}_{n},\bar{t}_{n})|^{\frac{2}{\gamma}}}{|t_{n}-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}}\leq
2L_{n}.$

If we let

$d_{n}=d(\bar{x}_{n},\partial\Omega),\qquad
M_{n}=\frac{|u(\bar{x}_{n},t_{n})|}{z},\qquad
r_{n}=(t_{n}-\bar{t}_{n})^{\frac{1}{\gamma}}M_{n}^{\frac{\gamma-1}{\gamma}},$

we have (as before $u(\bar{x}_{n},\bar{t}_{n})=0$)

$L_{n}\leq\big{(}d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}\big{)}\frac{M_{n}^{\frac{2}{\gamma}+\alpha_{0}\frac{\gamma-1}{\gamma}}}{r_{n}^{\alpha_{0}}}=\big{(}d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}\big{)}\frac{M_{n}}{r_{n}^{\alpha_{0}}}\leq
2L_{n}.$ (3.2)

In both cases, since $M_{n}\leq\|u_{n}\|_{\infty}\leq K$ and $Q$ is bounded,
we infer from (3.1) and (3.2) that, as $n\to\infty$,

$\frac{d_{n}}{r_{n}}\to+\infty,\qquad r_{n}\to
0,\qquad\frac{M_{n}^{\gamma-1}}{r_{n}^{\gamma-2}}=\left(\frac{M_{n}}{r_{n}^{\alpha_{0}}}\right)^{\gamma-1}\to+\infty,$
(3.3)

as well as

$\displaystyle\begin{split}(T-\bar{t}_{n})\frac{M_{n}^{\gamma-1}}{r_{n}^{\gamma}}&\geq(T-\bar{t}_{n})\frac{M_{n}^{\frac{\gamma}{\alpha_{0}}}}{r_{n}^{\gamma}}K^{-\frac{2}{\alpha_{0}}}\to+\infty,\\\
\frac{M_{n}^{\frac{\alpha_{0}}{\gamma^{\prime}}}}{r_{n}^{\alpha_{0}}}(d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}})&\geq\frac{M_{n}}{r_{n}^{\alpha_{0}}}(d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}})K^{\frac{\alpha_{0}}{\gamma^{\prime}}-1}\to+\infty.\end{split}$
(3.4)

2\. Scaling. We now define the blow-up sequence

$w_{n}(y,s)=\frac{1}{M_{n}}\,u_{n}\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s\right),\\\
(y,s)\in\frac{\Omega-\bar{x}_{n}}{r_{n}}\times\left(-\bar{t}_{n}\frac{M_{n}^{\gamma-1}}{r_{n}^{\gamma}},(T-\bar{t}_{n})\frac{M_{n}^{\gamma-1}}{r_{n}^{\gamma}}\right)=Q_{n}.$

In case (a), let $y_{0}$ be such that $x_{n}=\bar{x}_{n}+r_{n}y_{0}$. By the
definition of $r_{n}$, $|y_{0}|=1$. Recall also that
$u(\bar{x}_{n},\bar{t}_{n})=0$, so

$|w_{n}(y_{0},0)-w_{n}(0,0)|=\frac{1}{M_{n}}|u_{n}(x_{n},\bar{t}_{n})-u_{n}(\bar{x}_{n},\bar{t}_{n})|=1.$
(3.5)

On the other hand, in case (b),

$|w_{n}(0,1)-w_{n}(0,0)|=\frac{1}{M_{n}}|u_{n}(\bar{x}_{n},t_{n})|=z.$ (3.6)

The sequence $w_{n}$ solves the following HJ equation

$-\partial_{s}w_{n}-\sigma_{n}\Delta
w_{n}+h\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s\right)|Dw_{n}|^{\gamma}=g_{n}\qquad\text{on
$Q_{n}$},$ (3.7)

where

$\sigma_{n}=\frac{r_{n}^{\gamma-2}}{M_{n}^{\gamma-1}},\qquad
g_{n}(y,s)=\frac{r_{n}^{\gamma}}{M_{n}^{\gamma}}f_{n}\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s\right)$

Note that $\sigma_{n}\to 0$ (see (3.3)). Moreover, for any $R,\tau>0$, a
straightforward computation yields

$\|g_{n}\|_{L^{q_{0}}(Q_{R,\tau})}=\sigma_{n}^{\gamma^{\prime}\frac{N+1}{N+2}}\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R,\tau})},\quad\widetilde{Q}_{n,R,\tau}=B_{Rr_{n}}(\bar{x}_{n})\times(\bar{t}_{n},\bar{t}_{n}+r_{n}^{\gamma}M_{n}^{1-\gamma}\tau)$

Therefore, since $f_{n}$ is uniformly integrable in $L^{q_{0}}(Q)$,

$\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R,\tau})}\to 0\qquad\text{as
$n\to\infty$},$ (3.8)

because $r_{n},r_{n}^{\gamma}M_{n}^{1-\gamma}\to 0$ in view of (3.3), (3.4).
Since $h$ is continuous on $\overline{Q}$, for any $\varepsilon>0$ (smaller
than $h_{0}/2$), for $n$ large enough we have on on $Q_{n}$

$\begin{gathered}-\partial_{s}w_{n}-\sigma_{n}\Delta
w_{n}+(h_{n}-\varepsilon)|Dw_{n}|^{\gamma}\leq g_{n},\\\
-\partial_{s}w_{n}-\sigma_{n}\Delta
w_{n}+(h_{n}+\varepsilon)|Dw_{n}|^{\gamma}\geq g_{n},\end{gathered}$ (3.9)

where $h_{0}\leq h_{n}=h(\bar{x}_{n},\bar{t}_{n})\leq h_{1}$.

In both cases, we have the following control on Hölder seminorms of $w_{n}$.

######  Lemma 3.2.

For any $R,\tau>1$, we have for $n$ large enough that
$\overline{Q}_{R,\tau}=\overline{B_{R}}\times[0,\tau]\subset Q_{n}$ and

$\sup_{(y,s),(y^{\prime},s)\in\overline{Q}_{R,\tau}}\frac{|w_{n}(y,s)-w_{n}(y^{\prime},s)|}{|y-y^{\prime}|^{\alpha_{0}}}\leq
3,\qquad\sup_{(y,s),(y,\bar{s})\in\overline{Q}_{R,\tau}}\frac{|w_{n}(y,s)-w_{n}(y,s^{\prime})|}{|s-s^{\prime}|^{\alpha_{0}/2}}\leq
3^{\frac{\gamma}{2}}z.$

###### Proof.

First,

$d(0,\partial\Omega_{n})=d\left(0,\frac{\partial\Omega-\bar{x}_{n}}{r_{n}}\right)=\frac{1}{r_{n}}d(\bar{x}_{n},\partial\Omega)\to+\infty$

by (3.3). Moreover, (3.4) guarantees that
$(T-\bar{t}_{n})\frac{M_{n}^{\gamma-1}}{r_{n}^{\gamma}}\to+\infty$, therefore
$\overline{B_{R}}\times[0,\tau]\subset Q_{n}$ whenever $n$ is large.

Let now $(y,s),(y^{\prime},s)\in\overline{Q}_{R,\tau}$ and assume that
${\mathcal{d}}((y^{\prime},s),\partial Q_{n})\geq{\mathcal{d}}((y,s),\partial
Q_{n})$. Then, writing

$(x,t)=\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s\right),\quad(x^{\prime},t)=\left(\bar{x}_{n}+r_{n}y^{\prime},\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s\right)$

we get

$\frac{|u_{n}(x,t)-u_{n}(x^{\prime},t)|}{|x-x^{\prime}|^{\alpha_{0}}}\leq\frac{[u]^{x}_{\alpha_{0};Q}}{d^{\alpha_{0}}(x,\partial\Omega)+|T-t|^{\frac{\alpha_{0}}{\gamma}}}\leq\frac{2L_{n}}{d^{\alpha_{0}}(x,\partial\Omega)+|T-t|^{\frac{\alpha_{0}}{\gamma}}}.$
(3.10)

Now,

$\displaystyle\begin{split}d^{\alpha_{0}}_{n}+(T-\bar{t}_{n})^{\frac{\alpha_{0}}{\gamma}}&\leq
d^{\alpha_{0}}(x,\partial\Omega)+(T-t)^{\frac{\alpha_{0}}{\gamma}}+|x-\bar{x}_{n}|^{\alpha_{0}}+|t-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}\\\
&\leq
d^{\alpha_{0}}(x,\partial\Omega)+(T-t)^{\frac{\alpha_{0}}{\gamma}}+r_{n}^{\alpha_{0}}R^{\alpha_{0}}+\frac{r_{n}^{\alpha_{0}}}{M_{n}^{\frac{\alpha_{0}}{\gamma^{\prime}}}}\tau^{\frac{\alpha_{0}}{\gamma}}\\\
&\leq
d^{\alpha_{0}}(x,\partial\Omega)+(T-t)^{\frac{\alpha_{0}}{\gamma}}+C\frac{r_{n}^{\alpha_{0}}}{M_{n}^{\frac{\alpha_{0}}{\gamma^{\prime}}}},\end{split}$
(3.11)

for some $C$ depending on $K,R,\tau$. Therefore,

$\frac{|w_{n}(y,s)-w_{n}(y^{\prime},s)|}{|y-y^{\prime}|^{\alpha_{0}}}=\frac{|u_{n}(x,t)-u_{n}(x^{\prime},t)|}{|x-x^{\prime}|^{\alpha_{0}}}\frac{r_{n}^{\alpha_{0}}}{M_{n}}\stackrel{{\scriptstyle\eqref{Lna},\eqref{eq001}}}{{\leq}}\frac{2L_{n}}{d^{\alpha_{0}}(x,\partial\Omega)+|T-t|^{\frac{\alpha_{0}}{\gamma}}}\frac{d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}}{L_{n}}\\\
\stackrel{{\scriptstyle\eqref{eq0015}}}{{\leq}}2\left(1-C\frac{r_{n}^{\alpha_{0}}}{M_{n}^{\frac{\alpha_{0}}{\gamma^{\prime}}}(d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}})}\right)^{-1}\leq
3,$

the last inequality being true for large $n$.

Similarly, $(y,s),(y,s^{\prime})\in\overline{Q}_{R,\tau}$ be such that
${\mathcal{d}}((y,s^{\prime}),\partial Q_{n})\geq{\mathcal{d}}((y,s),\partial
Q_{n})$. Then, for

$(x,t)=\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s\right),\quad(x,t^{\prime})=\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+\frac{r_{n}^{\gamma}}{M_{n}^{\gamma-1}}s^{\prime}\right)$

we have

$\frac{|u_{n}(x,t)-u_{n}(x,t^{\prime})|}{|t-t^{\prime}|^{\frac{\alpha_{0}}{2}}}\leq\frac{[u_{n}]^{t}_{\frac{\alpha_{0}}{2};Q}}{[d^{\alpha_{0}}(x,\partial\Omega)+|T-t|^{\frac{\alpha_{0}}{\gamma}}]^{\frac{\gamma}{2}}}\leq
z2^{\frac{\gamma}{2}}\frac{L_{n}^{\frac{\gamma}{2}}}{[d^{\alpha_{0}}(x,\partial\Omega)+|T-t|^{\frac{\alpha_{0}}{\gamma}}]^{\frac{\gamma}{2}}}.$
(3.12)

Thus,

$\frac{|w_{n}(y,s)-w_{n}(y,s^{\prime})|}{|s-s^{\prime}|^{\frac{\alpha_{0}}{2}}}=\frac{|u_{n}(x,t)-u_{n}(x,t^{\prime})|}{|t-t^{\prime}|^{\frac{\alpha_{0}}{2}}}\frac{r_{n}^{\alpha_{0}\frac{\gamma}{2}}}{M_{n}^{\frac{\gamma}{2}}}\\\
\stackrel{{\scriptstyle\eqref{Lna},\eqref{eq002}}}{{\leq}}\frac{2^{\frac{\gamma}{2}}zL^{\frac{\gamma}{2}}_{n}}{[d^{\alpha_{0}}(x,\partial\Omega)+|T-t|^{\frac{\alpha_{0}}{\gamma}}]^{\frac{\gamma}{2}}}\frac{[d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}}]^{\frac{\gamma}{2}}}{L_{n}^{\frac{\gamma}{2}}}\\\
\stackrel{{\scriptstyle\eqref{eq0015}}}{{\leq}}2^{\frac{\gamma}{2}}z\left(1-C\frac{r_{n}^{\alpha_{0}}}{M_{n}^{\frac{\alpha_{0}}{\gamma^{\prime}}}(d^{\alpha_{0}}_{n}+|T-\bar{t}_{n}|^{\frac{\alpha_{0}}{\gamma}})}\right)^{-\frac{\gamma}{2}}\leq
3^{\frac{\gamma}{2}}z.$

∎

3\. Conclusion.  We now have to choose $z,\tau,R,\varepsilon$, and pick $n$
large enough so that a contradiction is reached. To avoid circular
dependencies, the general idea will be to choose first $z$ large, then $\tau$,
then $R$ large enough, and finally $\varepsilon$ small. The smallness of
$\|f_{n}\|$ will also play a crucial role.

In view of Lemma 3.2, we can employ Proposition 2.1 with $\alpha=\alpha_{0}$
(note that in this case
$\frac{\alpha_{0}}{\gamma-\alpha_{0}(\gamma-1)}=\frac{\alpha_{0}}{2}$ and
$\sigma_{n}^{-\gamma^{\prime}\frac{N+1}{N+2}}\|g_{n}\|_{L^{q_{0}}(Q_{R,\tau})}=\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R,\tau})}$),
which yields

$|w_{n}(0,0)-w_{n}(0,\tau)|+{\mathcal{K}}\leq
C_{2}\left(\tau^{\frac{\alpha_{0}}{2}}+\tau\frac{R^{\alpha_{0}}+\tau^{\frac{\alpha_{0}}{2}}}{R}z\right)$
(3.13)

and

$|w_{n}(y_{0},0)-w_{n}(0,0)|\leq
C_{3}\Big{(}\frac{{\mathcal{K}}^{1/\gamma}}{\tau^{1/\gamma}}+\frac{1}{\tau^{\gamma^{\prime}-1}}+({\mathcal{K}}+\tau^{\alpha_{0}/2})\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R+2,\tau})}\\\
+\frac{\tau^{1/\gamma}{\mathcal{K}}^{1/\gamma^{\prime}}}{R}+\frac{\tau}{R^{2}}+\eta_{n}{\mathcal{K}}\Big{)},$
(3.14)

provided that

$\overline{B_{R+2}}\times[0,\tau]\subset
Q_{n},\qquad\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R,\tau})}\leq f^{0},\qquad
R^{2}\geq\sigma_{n}\tau,\qquad
z\frac{R^{\alpha_{0}}+\tau^{\frac{\alpha_{0}}{2}}}{R}\leq c_{1},$

where, since $h_{n}-\varepsilon\geq h_{0}/2$,

$\eta_{n}=\frac{(\gamma-1)}{\gamma^{\gamma^{\prime}}}\left(\frac{1}{(h_{n}-\varepsilon)^{\gamma^{\prime}}}-\frac{1}{(h_{n}+\varepsilon)^{\gamma^{\prime}}}\right)\leq
C_{4}\varepsilon.$

Here $f^{0},c_{1},C_{2},C_{3}$ are universal constants. The first and the
second condition are always satisfied for large $n$ (see Lemma 3.2 and (3.8)).
Note that the absolute value of $|w_{n}(y_{0},0)-w_{n}(0,0)|$ appears in
(3.14); this is a consequence of (2.5), and (2.5) again if one exchanges the
roles of $0$ and $y_{0}$ (which can be done eventually enlarging the domain
from $Q_{R,\tau}$ to $Q_{R+1,\tau}$). The third and the fourth condition will
be verified below.

We first aim to rule out case (b). To do so, we fix

$z=4C_{2},\qquad\tau=1$

and $R>1$ large enough so that

$\quad 4C_{2}\frac{R^{\alpha_{0}}+1}{R}\leq\min\\{1,c_{1}\\},$

hence (3.13) applies, yielding

$|w_{n}(0,0)-w_{n}(0,1)|\leq
C_{2}\left(1+4C_{2}\frac{R^{\alpha_{0}}+1}{R}\right)\leq 2C_{2}=\frac{z}{2}.$

For large $n$, $Q_{R,1}\subset\tilde{Q}_{n}$, hence (3.6) is impossible.

We now treat case (a), with the same choice of $z$ as before, but possibly
with larger $\tau,R$. First, pick $\tau\geq 1$ so that

$C_{2}\tau^{\frac{\alpha_{0}}{2}}\leq\left(\frac{1}{8C_{3}}\right)^{\gamma}\tau,\qquad\frac{1}{\tau^{\gamma^{\prime}-1}}\leq\frac{1}{8C_{3}},$

and $R$ large such that

$C_{2}\left(\tau\frac{R^{\alpha_{0}}+\tau^{\frac{\alpha_{0}}{2}}}{R}z\right)\leq\left(\frac{1}{8C_{3}}\right)^{\gamma}\tau,\quad\left(\
2\left(\frac{1}{8C_{3}}\right)^{\gamma}\tau\right)^{1/\gamma^{\prime}}\frac{\tau^{1/\gamma}}{R}\leq\frac{1}{8C_{3}},\quad\frac{\tau}{R^{2}}\leq\frac{1}{8C_{3}}.$

Then, $n$ be large so that

$\left(2\left(\frac{1}{8C_{3}}\right)^{\gamma}\tau+\tau^{\alpha_{0}/2}\right)\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R+2,\tau})}\leq\frac{1}{8C_{3}},\qquad\text{and
$Q_{R+2,\tau}\subset Q_{n}$}.$

With these choices, (3.13) reads

${\mathcal{K}}\leq
C_{2}\left(\tau^{\frac{\alpha_{0}}{2}}+\tau\frac{R^{\alpha_{0}}+\tau^{\frac{\alpha_{0}}{2}}}{R}z\right)\leq
2\left(\frac{1}{8C_{3}}\right)^{\gamma}\tau,$

whence

$\frac{{\mathcal{K}}^{1/\gamma}}{\tau^{1/\gamma}}\leq\frac{2^{1/\gamma}}{8C_{3}},\qquad\frac{\tau^{1/\gamma}{\mathcal{K}}^{1/\gamma^{\prime}}}{R}\leq\frac{1}{8C_{3}},\qquad({\mathcal{K}}+\tau^{\alpha_{0}/2})\|f_{n}\|_{L^{q_{0}}(\widetilde{Q}_{n,R+2,\tau})}\leq\frac{1}{8C_{3}}.$

We can eventually pick $\varepsilon>0$ small (and consequently increase $n$,
if needed) so that

$\eta_{n}{\mathcal{K}}\leq
2C_{4}\left(\frac{1}{8C_{3}}\right)^{\gamma}\tau\varepsilon\leq\frac{1}{8C_{3}}.$

Finally, we plug the previous inequalities into (3.14) to get

$|w_{n}(y_{0},0)-w_{n}(0,0)|\leq\frac{2^{1/\gamma}+5}{8}\leq\frac{7}{8},$

contradicting (3.5), so case (a) is also impossible, and this proves Theorem
3.1.

######  Remark 3.3 (On the role of $h$).

The constant $C$ in Theorem 3.1 depends on $h$ just through its lower and
upper bounds $h_{0},h_{1}$, and its modulus of continuity on $\overline{Q}$.

######  Remark 3.4 (On the assumptions).

The proof of Theorem 3.1 can be carried out analogously for solutions of

$-\partial_{t}u-\Delta D^{2}u+H(x,t,Du)=0$

under the following assumption on $H$: for some $0<h_{0}\leq h_{1}$, for all
$\varepsilon>0$ there exists $\delta=\delta(\varepsilon)$ such that if
$|Q|<\delta$, then

$(h_{Q}-\varepsilon)|\xi|^{\gamma}-f_{Q}(x,t)\leq
H(x,t,\xi)\leq(h_{Q}+\varepsilon)|\xi|^{\gamma}+f_{Q}(x,t)\qquad\text{on
$Q\times{\mathbb{R}}^{N}$}$

for some $h_{0}\leq h_{Q}\leq h_{1}$ and
$\|f_{Q}\|_{L^{q_{0}}(Q)}\leq\varepsilon$. Indeed, this is enough to guarantee
(3.8), (3.9), and the rest of the proof remains unchanged. For example,

$H(x,t,\xi)=h|\xi|^{\gamma}+b(x,t)\cdot\xi,$

where $b$ varies in a set of uniformly $L^{N+2}$-integrable function,
satisfies the previous assumption. Note finally that we actually need (3.9)
for some $\varepsilon$ small enough, hence discontinuous $h$ are allowed,
provided that the local oscillation is smaller than some universal constant
$\varepsilon_{0}>0$.

##  4 A Liouville theorem

Let $q>q_{0}$, $v\in W_{q,loc}^{2,1}({\mathbb{R}}^{N}\times(0,\infty))$ be a
strong solution of

$-\partial_{t}v-\Delta v+h|Dv|^{\gamma}=0\qquad\text{on
${\mathbb{R}}^{N}\times(0,\infty)$},$ (4.1)

$h\neq 0$. The purpose of this section is to prove the following Liouville-
type theorem for “far future” solutions $v$ (which would be just “ancient”
solutions if the equation was forward in time). Note that a standard bootstrap
procedure shows that $v$ is actually smooth, up to $C^{\infty}$.

######  Theorem 4.1.

Suppose that for some $0<\alpha<1$,

$\sup_{(x,t)\neq(x^{\prime},t^{\prime})\in{\mathbb{R}}^{N}\times(0,\infty)}\frac{|v(x,t)-v(x^{\prime},t^{\prime})|}{|x-x^{\prime}|^{\alpha}+|t-t^{\prime}|^{\alpha/2}}<\infty.$

Then, $v$ is constant on ${\mathbb{R}}^{N}\times(0,\infty)$.

###### Proof.

The linear case $h=0$ is proved in [23, Theorem 1.2(b)]. Otherwise, w.l.o.g.
we may assume that $h>0$ (if $h<0$ it is sufficient to replace $v$ by $-v$).
We may also assume that

$\sup_{(y,s),(y^{\prime},s)\in{\mathbb{R}}^{N}\times(0,\infty)}\frac{|v(y,s)-v(y^{\prime},s)|}{|y-y^{\prime}|^{\alpha}}\leq
3,\qquad\sup_{(y,s),(y,\bar{s})\in{\mathbb{R}}^{N}\times(0,\infty)}\frac{|v(y,s)-v(y,s^{\prime})|}{|s-s^{\prime}|^{\alpha/2}}\leq
3,$

which can be always verified after dividing $v$ by a suitable positive
constant (and changing $h$ accordingly).

Then, using Proposition 2.1 with $g\equiv 0$ and $h_{0}=h_{1}=h$,

$\displaystyle v(x_{0},0)-v(0,0)\leq
C_{3}\left(\frac{{\mathcal{K}}^{1/\gamma}}{\tau^{1/\gamma}}+\frac{1}{\tau^{\gamma^{\prime}-1}}+\frac{\tau^{1/\gamma}{\mathcal{K}}^{1/\gamma^{\prime}}}{R}+\frac{\tau}{R^{2}}\right),$
$\displaystyle\qquad\qquad{\mathcal{K}}\leq
C_{2}\left(\tau^{\frac{\alpha}{2}}+\tau^{\frac{\alpha_{0}}{2}}+\tau^{\frac{\alpha}{\gamma-\alpha(\gamma-1)}}+\tau\frac{R^{\alpha}+\tau^{\frac{\alpha}{2}}}{R}\right),$

for any $R\geq R_{0}=R_{0}(\tau)$ and $|x_{0}|\leq 1$. If we let
$R\to+\infty$, we obtain

$v(x_{0},0)-v(0,0)\leq
C_{3}\left(\left(\frac{\tau^{\frac{\alpha}{2}}+\tau^{\frac{\alpha_{0}}{2}}+\tau^{\frac{\alpha}{\gamma-\alpha(\gamma-1)}}}{\tau}\right)^{1/\gamma}+\frac{1}{\tau^{\gamma^{\prime}-1}}\right).$

Since $\alpha/2,\alpha_{0}/2,\alpha/(\gamma-\alpha(\gamma-1))<1$, as
$\tau\to\infty$ we conclude that $v(x_{0},0)-v(0,0)\leq 0$ for any
$|x_{0}|\leq 1$. By exchanging the roles of $x_{0}$ and $0$ we get that
$v(\cdot,0)$ is constant. The same argument applies to $v(\cdot,t)$ for any
$t>0$, which allows to conclude that $v(\cdot,t)$ is constant for any $t>0$.
Since now $\partial_{t}v\equiv 0$ on ${\mathbb{R}}^{N}\times(0,\infty)$, the
conclusion is reached. ∎

######  Remark 4.2 (On the behavior of $v$).

We note that the main requirement on $v$ is its behavior as $(|x|,t)$ goes to
infinity. We could indeed get the same result assuming only, for $0<\alpha<1$
and $\beta>0$,

$\sup_{(x,t),(x^{\prime},t^{\prime})\in{\mathbb{R}}^{N}\times(0,\infty)}\frac{|v(x,t)-v(x^{\prime},t^{\prime})|}{|x-x^{\prime}|^{\alpha}+|t-t^{\prime}|^{\beta}+1}<\infty,$

using the arguments developed in the previous section. Note that the statement
would be false if $\alpha=1$ was allowed, just consider $v(x,t)=ht+x$.

######  Remark 4.3 (On the role of the diffusion).

Since the right-hand side in (4.1) is zero, there is no real need of the
regularizing effect of a nondegenerate diffusion. In other words, we can
replace $\Delta v$ by $\operatorname{tr}AD^{2}v$, $A\geq 0$, and the proof of
the Liouville theorem works, at least formally. To make it rigorous, one has
then to be careful on the notion of solution for (4.1), and the dual Fokker-
Planck equation which is used in Proposition 2.1.

##  5 $\alpha$-Hölder regularity

######  Theorem 5.1.

Assume that $\|u\|_{L^{\infty}(Q)},[u]_{\alpha_{0};Q},\|f\|_{L^{q}(Q)}\leq K$.
Then, there exists $C$ depending on $K,Q,h,q,N$ such that

$[u]^{\alpha-\alpha_{0}}_{\alpha;Q}\leq C,$

where $\alpha=2-\frac{N+2}{q}$.

1\. Set-up of the contradiction argument. Assume by contradiction that for
some sequence $\\{f_{n}\\}$ and $\\{u_{n}\\}$ solving (1.1),

$[u]^{\alpha-\alpha_{0}}_{\alpha;Q}\to\infty\quad\text{as $n\to\infty$}.$

Setting $L_{n}\mathrel{\mathop{\mathchar
58\relax}}=[u]^{\alpha-\alpha_{0}}_{\alpha;Q}/2$ there are sequences
$\\{(x_{n},t_{n})\\},\\{(\bar{x}_{n},\bar{t}_{n})\\}\subset Q$ such that for
every $n$, $(x_{n},t_{n})\neq(\bar{x}_{n},\bar{t}_{n})$ and

$L_{n}\leq[\min\\{d((x,t),\partial Q),d((\bar{x},\bar{t}),\partial
Q)\\}]^{\alpha-\alpha_{0}}\frac{|u(x,t)-u(\bar{x},\bar{t})|}{(|x-\bar{x}|+|t-\bar{t}|^{\frac{1}{2}})^{\alpha}}\leq
2L_{n}.$

If we assume w.l.o.g. that $d((x_{n},t_{n}),\partial Q)\geq
d((\bar{x}_{n},\bar{t}_{n}),\partial Q)$ and $u(\bar{x}_{n},\bar{t}_{n})=0$,
setting

$d_{n}=d((\bar{x}_{n},\bar{t}_{n}),\partial Q),\qquad
M_{n}=|u_{n}(x_{n},t_{n})|,\qquad
r_{n}=|x_{n}-\bar{x}_{n}|+|t_{n}-\bar{t}_{n}|^{\frac{1}{2}}.$

the previous inequality reads

$L_{n}\leq d^{\alpha-\alpha_{0}}_{n}\frac{M_{n}}{r_{n}^{\alpha}}\leq 2L_{n}.$
(5.1)

Since $\frac{M_{n}}{r_{n}^{\alpha_{0}}}\leq K$ and $Q$ is bounded, we infer
from (5.1) that, as $n\to\infty$,

$\frac{d_{n}}{r_{n}}\to+\infty,\qquad r_{n}\to
0,\qquad\frac{r_{n}^{\alpha}}{M_{n}}\to 0.$ (5.2)

2\. Scaling. We now define the blow-up sequence

$w_{n}(y,s)=\frac{1}{M_{n}}\,u_{n}\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+r_{n}^{2}s\right),\qquad(y,s)\in\frac{\Omega-\bar{x}_{n}}{r_{n}}\times\left(-\frac{\bar{t}_{n}}{r_{n}^{2}},\frac{T-\bar{t}_{n}}{r_{n}^{2}}\right)=Q_{n}.$

Let $(y_{0},s_{0})$ be such that
$(x_{n},t_{n})=(\bar{x}_{n}+r_{n}y_{0},\bar{t}_{n}+r_{n}^{2}s_{0})$. By the
definition of $r_{n}$, $|y_{0}|+|s_{0}|^{1/2}=1$. Recall also that
$u(\bar{x}_{n},\bar{t}_{n})=0$, so

$|w_{n}(y_{0},s_{0})-w_{n}(0,0)|=\frac{1}{M_{n}}|u_{n}(x_{n},t_{n})-u_{n}(\bar{x}_{n},\bar{t}_{n})|=1.$
(5.3)

The sequence $w_{n}$ solves the following HJ equation

$-\partial_{s}w_{n}-\Delta
w_{n}+\theta_{n}h(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+r_{n}^{2}s)|Dw_{n}|^{\gamma}=g_{n}\qquad\text{on
$Q_{n}$},$ (5.4)

where

$\theta_{n}=\frac{M_{n}^{\gamma-1}}{r_{n}^{\gamma-2}},\qquad
g_{n}(y,s)=\frac{r_{n}^{2}}{M_{n}}f_{n}\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+r_{n}^{2}s\right)$

Note that by the assumptions,

$\theta_{n}^{\frac{1}{\gamma-1}}=\frac{|u_{n}(x_{n},t_{n})-|u_{n}(\bar{x}_{n},\bar{t}_{n})|}{(|x_{n}-\bar{x}_{n}|+|t_{n}-\bar{t}_{n}|^{\frac{1}{2}})^{\alpha_{0}}}\leq
K.$ (5.5)

Moreover, for any $R,\tau>0$, a straightforward computation yields

$\|g_{n}\|_{L^{q}(Q_{R,\tau})}=\frac{r_{n}^{2-\frac{N+2}{q}}}{M_{n}}\|f_{n}\|_{L^{q}(\widetilde{Q}_{n,R,\tau})},\quad\widetilde{Q}_{n,R,\tau}=B_{Rr_{n}}(\bar{x}_{n})\times(\bar{t}_{n},\bar{t}_{n}+r_{n}^{2}\tau).$

Therefore, since $\|f_{n}\|_{q}\leq K$ and
$\frac{r^{2-\frac{N+2}{q}}}{M_{n}}=\frac{r^{\alpha}}{M_{n}}\to 0$ by (5.2),
then

$\|g_{n}\|_{L^{q}(Q_{R,\tau})}\to 0\qquad\text{as $n\to\infty$}.$ (5.6)

######  Lemma 5.2.

For any $R>1$, we have for $n$ large enough that
$\overline{Q}_{R}=\overline{B_{R}}\times[0,R^{2}]\subset Q_{n}$ and

$\sup_{(y,s),(y^{\prime},s^{\prime})\in\overline{Q}_{R}}\frac{|w_{n}(y,s)-w_{n}(y^{\prime},s^{\prime})|}{(|y-y^{\prime}|+|s-s^{\prime}|^{\frac{1}{2}})^{\alpha}}\leq
3.$

###### Proof.

First,

$d((0,0),\partial^{+}Q_{n})=d\left(0,\frac{\partial\Omega-\bar{x}_{n}}{r_{n}}\right)+\frac{(T-\bar{t}_{n})^{1/2}}{r_{n}}=\frac{d_{n}}{r_{n}}\to+\infty$

by (5.2).

Let now $(y,s),(y^{\prime},s^{\prime})\in\overline{Q}_{R}$ and assume that
$d((y^{\prime},s^{\prime}),\partial^{+}Q_{n})\geq d((y,s),\partial^{+}Q_{n})$.
Then, writing

$(x,t)=\left(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+r_{n}^{2}s\right),\quad(x^{\prime},t^{\prime})=\left(\bar{x}_{n}+r_{n}y^{\prime},\bar{t}_{n}+r_{n}^{2}s^{\prime}\right)$

we get

$\frac{|u_{n}(x,t)-u_{n}(x^{\prime},t^{\prime})|}{(|x-x^{\prime}|+|t-t^{\prime}|^{1/2})^{\alpha}}\leq\frac{[u]^{x}_{\alpha;Q}}{d^{\alpha-\alpha_{0}}((x,t),\partial^{+}Q)}\leq\frac{2L_{n}}{d^{\alpha-\alpha_{0}}((x,t),\partial^{+}Q)}.$
(5.7)

Now,

$\displaystyle\begin{split}d_{n}=d((\bar{x}_{n},\bar{t}_{n}),\partial^{+}Q)&\leq
d(x,\partial\Omega)+(T-t)^{\frac{1}{2}}+|x-\bar{x}_{n}|+|t-\bar{t}_{n}|^{\frac{1}{2}}\\\
&\leq d(x,\partial\Omega)+(T-t)^{\frac{1}{2}}+2r_{n}R\\\
&=d((x,t),\partial^{+}Q)+2r_{n}R.\end{split}$ (5.8)

Therefore,

$\frac{|w_{n}(y,s)-w_{n}(y^{\prime},s^{\prime})|}{(|y-y^{\prime}|+|s-s^{\prime}|^{1/2})^{\alpha}}=\frac{|u_{n}(x,t)-u_{n}(x^{\prime},t^{\prime})|}{(|x-x^{\prime}|+|t-t^{\prime}|^{1/2})^{\alpha}}\frac{r_{n}^{\alpha}}{M_{n}}\stackrel{{\scriptstyle\eqref{Ln2},\eqref{eq004}}}{{\leq}}\frac{2L_{n}}{d^{\alpha-\alpha_{0}}((x,t),\partial^{+}Q)}\frac{d_{n}^{\alpha-\alpha_{0}}}{L_{n}}\\\
\stackrel{{\scriptstyle\eqref{eq0016}}}{{\leq}}2\left(1-2R\frac{r_{n}}{d_{n}}\right)^{-(\alpha-\alpha_{0})}\leq
3,$

the last inequality being true for large $n$. ∎

######  Lemma 5.3.

For any $R>1$, there exists $C$ such that

$\|\partial_{t}w_{n}\|_{q,Q_{R}}+\|D^{2}w_{n}\|_{q;Q_{R}}\leq C.$

for $n$ large enough.

###### Proof.

First, $n$ need to be large so that $Q_{2R}\subset Q_{n}$ (which is possible
in view of the previous lemma). Moreover, by (5.6), again for large $n$ one
has $\|g_{n}\|_{L^{q}(Q_{2R})}\leq 1$. Since $w_{n}$ solves the HJ equation
(5.4),

$|-\partial_{s}w_{n}-\Delta
w_{n}|=|-\theta_{n}h(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+r_{n}^{2}s)|Dw_{n}|^{\gamma}+g_{n}|\stackrel{{\scriptstyle\eqref{thetabound}}}{{\leq}}h_{1}K^{\gamma-1}|Dw_{n}|^{\gamma}+1,$

the conclusion follows by invoking Proposition A.4. ∎

3\. The limit problem. We now pass to the limit in the equation (5.4). Limits
below are up to subsequences, and they are constructed via a standard diagonal
procedure. First, by Lemmata 5.2 and 5.3, $w_{n}$ has (local) weak limits in
$W^{2,1}_{q}$ on $R^{N}\times(0,\infty)$.

We may assume that $\bar{x}_{n}$ and $\theta_{n}$ converge to
$\bar{x}_{\infty}$ and $\theta_{\infty}$ respectively. Note that standard
parabolic embeddings guarantee that $Dw_{n}$ converge (locally) strongly in
$L^{p}$ for any $p<\frac{(N+2)q}{N+2-q}$. Therefore, as $\gamma
q<\frac{(N+2)q}{N+2-q}$,

$\theta_{n}h(\bar{x}_{n}+r_{n}y,\bar{t}_{n}+r_{n}^{2}s)|Dw_{n}|^{\gamma}\to\theta_{\infty}h(\bar{x}_{\infty},\bar{t}_{\infty})|Dw|^{\gamma}$

locally strongly in $L^{q}$, and the same holds for $g_{n}$ (see (5.6)).
Finally, $w_{n}$ converges locally uniformly to some $w$ which cannot be
constant on $Q_{2}$ in view of (5.3).

Hence, the limit $w\in W^{2,1}_{q,loc}({\mathbb{R}}^{N}\times(0,\infty))$ is a
nonconstant, strong solution of

$-\partial_{s}w-\Delta
w+\theta_{\infty}h(\bar{x}_{\infty},\bar{t}_{\infty})|Dw|^{\gamma}=0\qquad\text{on
${\mathbb{R}}^{N}\times(0,\infty)$}.$

It also satisfies

$\sup_{(y,s),(y^{\prime},s^{\prime})}\frac{|w(y,s)-w(y^{\prime},s^{\prime})|}{(|y-y^{\prime}|+|s-s^{\prime}|^{\frac{1}{2}})^{\alpha}}\leq
3,$

hence the Liouville Theorem 4.1 applies, contradicting the fact that $w$
cannot be constant.

##  6 Proof of Theorem 1.1 and final remarks

We now conclude with the proof the main theorem on maximal regularity, and
list a few remarks.

###### Proof of Theorem 1.1.

Fix any $Q^{\prime\prime},Q^{\prime\prime\prime}$ such that
$Q^{\prime}\subset\joinrel\subset Q^{\prime\prime}\subset\joinrel\subset
Q^{\prime\prime\prime}\subset\joinrel\subset Q$. Note that since $q>q_{0}$ and
$Q$ is bounded, functions satisfying $\|f\|_{q;Q}\leq K$ belong to a uniformly
integrable set of $L^{q_{0}}(Q)$. Therefore, Theorem 3.1 yields the existence
of $C$ such that

$\frac{|u(x,t)-u(\bar{x},t)|}{|x-\bar{x}|^{\alpha}}+\left(\frac{|u(x,t)-u(x,\bar{t})|}{|t-\bar{t}|^{\nicefrac{{\alpha}}{{2}}}}\right)^{\nicefrac{{2}}{{\gamma}}}\leq
C$

for any $(x,t),(x,\bar{t}),(\bar{x},t)\in Q^{\prime\prime\prime}$. Hence
Theorem 5.1 applies on $Q^{\prime\prime\prime}$, and for some $C^{\prime}>0$,

$[u]_{\alpha;Q^{\prime\prime}}\leq C^{\prime}.$

Finally, one achieves the desired estimate by Proposition A.4. ∎

######  Remark 6.1 ($q\geq N+2$ and Lipschitz estimates).

If $q\geq N+2$, then the same conclusion of Theorem 1.1 holds using a standard
bootstrap argument. Choose indeed any
$(N+2)\frac{\gamma+1}{\gamma}<\tilde{q}<N+2$, and apply Theorem 1.1 to control
$u$ in $W^{2,1}_{\tilde{q}}$. Therefore, by Sobolev embeddings one has a bound
of $|Du|^{\gamma}$ in $L^{p}$, where $1/p=1/\tilde{q}-1/(N+2)$, and $p>N+2$.
Hence, linear parabolic regularity allows to control $u$ in
$W^{2,1}_{\min\\{p,q\\}}$ on a smaller set, which in turn yields Lipschitz
estimates. A further application of parabolic regularity gives finally bounds
in $W^{2,1}_{q}$.

Note that this extends Lipschitz estimates to the full range $q>N+2$.
Previously, these were known [4] up to $q>\frac{N+2}{2(\gamma^{\prime}-1)}$
(when $\gamma>3$).

######  Remark 6.2 (On the sign of $f$).

Since $f$ can be unbounded, its sign plays a major role. If one has in mind
that $f$ appears in the running cost of an optimal control problem, then
optimal trajectories are pushed away by positive singular points of $f$, while
they are strongly attracted by negative ones. Heuristically, the former
affects the value function $u$ less than the latter. If one assumes $f\geq 0$,
then it shown in [3] that

$f\in L^{q},\qquad q>1+\frac{N}{\gamma}$

is sufficient for $u$ to be Hölder continuous, even in first-order problems.
Note that if $\gamma>2$, then $1+\frac{N}{\gamma}$ is always smaller than our
threshold $q_{0}$, and it even improves as $\gamma$ increases.

Here, we do not assume any sign condition on $f$. It is not clear whether the
critical integrability $q_{0}$ is related just to the behavior of the negative
part of $f$, but we believe that for some $f\notin L^{q_{0}}$, $u$ is bounded
from below. This is strictly related to exponent $q_{0}^{\prime}$ appearing in
(A.7), which is “natural” by invariance properties of the Fokker-Planck
equation, but its sharpness is not clear, as no counterexamples below
$q_{0}^{\prime}$ are available yet.

Not that if no sign condition is imposed on $f$, then our results hold also
for the so-called “repulsive” case $h<0$. By the methods developed here, it is
not clear if they can be extended to arbitrary (possibly vanishing) $h$.

##  Appendix A Miscellaneous estimates

We first start with some estimates on solutions to Fokker-Planck equations.
Let $\sigma>0$, $b$ a measurable vector field such that $b\in
L^{N+2}(Q_{R,\tau})$ and $\operatorname{div}b\in
L^{\frac{N+2}{2}}(Q_{R,\tau})$. Below, $m$ be a solution of

$\begin{cases}\partial_{t}m-\sigma\Delta m-\operatorname{div}(bm)=0&\text{on
$B_{R}\times(0,\tau)$}\\\ m(x,t)=0&\text{on $\partial B_{R}\times(0,\tau)$}\\\
m|_{t=0}=\delta_{0}\end{cases}$ (A.1)

Such a solution can be constructed by approximation, that is, taking limits of
classical solutions with smooth approximating drifts $b_{k}$ and smooth
initial data $m_{k}(0)$. Due to the singular nature of $m(0)$, $m\in
C([0,\tau];{\mathcal{M}}(\Omega))$ is a distributional solution. Furthermore,
$m$ is on $Q_{R,\tau}$ a strong solution, since classical parabolic Calderón-
Zygmund estimates apply away from $t=0$. In particular, $m\in
W^{2,1}_{\tilde{q}}(B_{R}\times(\tau^{\prime},\tau))$ for any
$0<\tau^{\prime}<\tau$ and $1<\tilde{q}<(N+2)/2$ (see for example [15, Theorem
IV.9.1]). Moreover, $m\in L^{q_{0}}(Q_{R,\tau})$, since the estimate (A.7)
below “sees” only the $L^{1}$-norm of the initial datum. Finally, the trace of
$m(t)$ on $\partial B_{R}$ is well defined for a.e. $t$, and it belongs to
$W^{2-1/\tilde{q}}_{\tilde{q}}(\partial B_{R})$.

We list several properties of $m$. First, $-Dm\cdot\nu\geq 0$ a.e. on
$\partial B_{R}\times(0,\tau)$. Indeed, $m$ is a (limit of) non-negative
solution vanishing on the lateral boundary $\partial B_{R}\times(0,\tau)$,
hence $Dm\cdot\zeta\leq 0$ on $\partial B_{R}\times(0,\tau)$ for any
$\zeta\cdot\nu>0$, and clearly $\nu\cdot\nu>0$. Moreover, for a.e.
$s\in(0,\tau]$,

$\int_{B_{R}}m(x,s)\,\operatorname{d\\!}x\leq\int_{B_{R}}m(x,s)\,\operatorname{d\\!}x-\sigma\iint_{\partial
B_{R}\times(0,s)}Dm\cdot\nu\operatorname{d\\!}x\operatorname{d\\!}t=1.$ (A.2)

Note that $m$ is the density of a killed Brownian particle starting from $x=0$
at time $t=0$, on the domain $B_{R}$). The term $-\sigma\iint Dm\cdot\nu$
represents the probability that the particle has reached $\partial B_{R}$
before $t=\tau$. We will estimate below such probability, as a corollary of
the following lemma.

######  Lemma A.1.

Assume that $|b|m\in L^{1}(Q_{R,\tau})$, and let $\alpha\in(0,1)$. Then, for
some $C$ depending on $\alpha$,

$\int_{B_{R}}|x|^{\alpha}m(x,\tau)\,\operatorname{d\\!}x\leq
C\left(\iint_{B_{R}\times(0,\tau)}|b|m\operatorname{d\\!}x\operatorname{d\\!}t\right)^{\alpha}+C\left(\sigma\tau\right)^{\alpha/2}.$

###### Proof.

Let $\chi\in C^{\infty}([0,+\infty);[0,+\infty))$ be such that $\chi(r)\equiv
1$ for all $r\geq 1$ and $\chi(0)=0$. For $0<y\leq R$, test the equation for
$m$ by $\chi_{y}(x)=\chi(|x|/y)$ to get

$\int_{B_{R}}\chi_{y}(x)m(x,\tau)\,\operatorname{d\\!}x-\sigma\iint_{\partial
B_{R}\times(0,\tau)}\chi_{y}Dm\cdot\nu\operatorname{d\\!}x\operatorname{d\\!}t=\\\
=\sigma\iint_{B_{R}\times(0,\tau)}m\Delta\chi_{y}\operatorname{d\\!}x\operatorname{d\\!}t-\iint_{B_{R}\times(0,\tau)}mb\cdot
D\chi_{y}\operatorname{d\\!}x\operatorname{d\\!}t,$

whence

$\int_{|x|\geq y}m(x,\tau)\,\operatorname{d\\!}x-\sigma\iint_{\partial
B_{R}\times(0,\tau)}Dm\cdot\nu\operatorname{d\\!}x\operatorname{d\\!}t\leq\frac{C\sigma\tau}{y^{2}}+\frac{C}{y}\iint_{B_{R}\times(0,\tau)}|b|m\operatorname{d\\!}x\operatorname{d\\!}t.$
(A.3)

In particular, by (A.2), setting $k_{1}=C\sigma\tau$ and
$k_{2}=C\iint_{B_{R}\times(0,\tau)}|b|m$,

$\int_{|x|\geq
y}m(x,\tau)\,\operatorname{d\\!}x\leq\min\left\\{\frac{k_{1}}{y^{2}}+\frac{k_{2}}{y},1\right\\}.$
(A.4)

Now,

$\int_{B_{R}}|x|^{\alpha}m(x,\tau)\,\operatorname{d\\!}x=\alpha\int_{B_{R}}\int_{0}^{|x|}y^{\alpha-1}m(x,\tau)\,\operatorname{d\\!}y\,\operatorname{d\\!}x=\alpha\int_{0}^{R}y^{\alpha-1}\int_{|x|\geq
y}m(x,\tau)\,\operatorname{d\\!}x\,\operatorname{d\\!}y\\\
\stackrel{{\scriptstyle\eqref{eqzz2}}}{{\leq}}\alpha\int_{0}^{R}y^{\alpha-1}\min\left\\{{k_{1}}{y^{-2}},1\right\\}\,\operatorname{d\\!}y+\alpha\int_{0}^{R}y^{\alpha-1}\min\left\\{k_{2}{y^{-1}},1\right\\}\,\operatorname{d\\!}y\\\
\leq
k_{1}\alpha\int_{k_{1}^{1/2}}^{+\infty}y^{\alpha-3}\,\operatorname{d\\!}y+\alpha\int_{0}^{k_{1}^{1/2}}y^{\alpha-1}\,\operatorname{d\\!}y+k_{2}\alpha\int_{k_{2}}^{+\infty}y^{\alpha-2}\,\operatorname{d\\!}y+\alpha\int_{0}^{k_{2}}y^{\alpha-1}\,\operatorname{d\\!}y\\\
=\alpha\frac{k_{1}^{\alpha/2}}{2-\alpha}+k_{1}^{\alpha/2}+\alpha\frac{k_{2}^{\alpha}}{1-\alpha}+k_{2}^{\alpha}.$

Substituting $k_{1},k_{2}$ gives the assertion. ∎

As a consequence (A.2) and (A.3) (with $y=R$), we have

$-\sigma\iint_{\partial
B_{R}\times(0,\tau)}Dm\cdot\nu\operatorname{d\\!}x\operatorname{d\\!}t\leq
C\frac{\tau^{1/\gamma}}{R}\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t\right)^{1/\gamma^{\prime}}+C\frac{\sigma\tau}{R^{2}}$
(A.5)

and by the previous lemma,

$\int_{B_{R}}|x|^{\alpha}m(x,\tau)\,\operatorname{d\\!}x\leq
C\tau^{\alpha/\gamma}\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t\right)^{\alpha/\gamma^{\prime}}+C\left(\sigma\tau\right)^{\alpha/2}$
(A.6)

######  Lemma A.2.

Suppose that $R^{2}\geq\tau\sigma$. There exists $C$ depending on $q_{0}$ such
that

$\sigma^{\gamma^{\prime}\frac{N+1}{N+2}}\,\|m\|_{L^{q^{\prime}_{0}}(Q_{R,\tau})}\leq
C\left(\iint_{B_{R}\times(0,\tau)}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t+\sigma^{\gamma^{\prime}/2}\tau^{\alpha_{0}/2}\right).$
(A.7)

###### Proof.

We start with the case $\sigma=R=1$, following the same lines as [7,
Proposition 2.4], that is

$\begin{cases}\partial_{t}\tilde{m}-\Delta\tilde{m}-\operatorname{div}(\tilde{b}\tilde{m})=0&\text{on
$B_{1}\times(0,\bar{s})$}\\\ m(z,s)=0&\text{on $\partial
B_{1}\times(0,\bar{s})$}\\\ m(0)=\delta_{0},\end{cases}$ (A.8)

for any $\bar{s}\in(0,1]$. We might assume that $\tilde{m}$ is smooth, the
general case follows by approximation. Let $\zeta$ be the solution of the dual
problem

$\begin{cases}-\partial_{t}\zeta-\Delta\zeta=\tilde{m}^{q_{0}^{\prime}-1}&\text{on
$B_{1}\times(0,\bar{s})$}\\\ \zeta(z,s)=0&\text{on $\partial
B_{1}\times(0,\bar{s})$}\\\ \zeta(z,\bar{s})=0&\text{on $B_{1}$}.\end{cases}$

We now employ $L^{q_{0}}$-parabolic regularity, and embedding theorems of
$W^{2,1}_{q_{0}}(Q_{1,\bar{s}})$ into $W^{1,0}_{q}(Q_{1,\bar{s}})$ and
$C^{\alpha_{0}/2}(0,\bar{s};C^{\alpha_{0}}(B_{1}))$, where
$q^{-1}=q_{0}^{-1}-(N+2)^{-1}$ and
$\alpha_{0}=2-\frac{N+2}{q_{0}}=2-\gamma^{\prime}$ (see [15, Theorem IV.9.1]
and [15, Lemma II.3.3], recall that $\frac{N+2}{2}<q_{0}<N+2$), to get

$\displaystyle|\zeta(0,0)|\leq
C^{\prime}\bar{s}^{\alpha_{0}/2}\|\zeta\|_{W^{2,1}_{q_{0}}(Q_{1,\bar{s}})}\leq$
$\displaystyle
C\bar{s}^{\alpha_{0}/2}\|\tilde{m}^{q^{\prime}_{0}-1}\|_{L^{q_{0}}(Q_{1,\bar{s}})}=C\bar{s}^{\alpha_{0}/2}\|\tilde{m}\|^{q^{\prime}_{0}-1}_{L^{q^{\prime}_{0}}(Q_{1,\bar{s}})},$
(A.9) $\displaystyle\|D\zeta\|_{L^{q}(Q_{1,\bar{s}})}\leq
C^{\prime}\|\zeta\|_{W^{2,1}_{q_{0}}(Q_{1,\bar{s}})}\leq$ $\displaystyle
C\|\tilde{m}^{q^{\prime}_{0}-1}\|_{L^{q_{0}}(Q_{1,\bar{s}})}=C\|\tilde{m}\|^{q^{\prime}_{0}-1}_{L^{q^{\prime}_{0}}(Q_{1,\bar{s}})}.$
(A.10)

Note that $C,C^{\prime}$ do not depend on $\bar{s}\leq 1$. Then, by duality
between $\tilde{m}$ and $\zeta$,

$\iint_{Q_{1,\bar{s}}}\tilde{m}^{q^{\prime}_{0}}=|\zeta(0,0)|-\iint_{Q_{1,\bar{s}}}\tilde{b}\tilde{m}\cdot
D\zeta.$

Hence, by Hölder inequality ($1=\frac{1}{\gamma^{\prime}}+\frac{1}{\gamma
q_{0}^{\prime}}+\frac{1}{q}$),

$\displaystyle\|\tilde{m}\|^{q^{\prime}_{0}}_{L^{q^{\prime}_{0}}(Q_{1,\bar{s}})}$
$\displaystyle\leq
C\|\tilde{b}\tilde{m}^{1/\gamma^{\prime}}\|_{L^{\gamma^{\prime}}(Q_{1,\bar{s}})}\|\tilde{m}^{1/\gamma}\|_{L^{\gamma
q^{\prime}_{0}}(Q_{1,\bar{s}})}\|D\zeta\|_{L^{q}(Q_{1,\bar{s}})}+|\zeta(0,0)|$
$\displaystyle\stackrel{{\scriptstyle\eqref{embedd}}}{{\leq}}C\left(\left(\iint_{Q_{1,\bar{s}}}|\tilde{b}|^{\gamma^{\prime}}\tilde{m}\right)^{1/\gamma^{\prime}}\|\tilde{m}\|^{1/\gamma}_{L^{q^{\prime}_{0}}(Q_{1,\bar{s}})}+\bar{s}^{\alpha_{0}/2}\right)\|\tilde{m}\|^{q^{\prime}_{0}-1}_{L^{q^{\prime}_{0}}(Q_{1,\bar{s}})},$

and the desired inequality (A.7) follows by means of Young’s inequality.

The general case, for $m$ solving (A.1), can be obtained by scaling: setting
$\tilde{m}(z,s)=R^{N}m(Rz,R^{2}\sigma^{-1}s)$,
$\tilde{b}(z,s)=R\sigma^{-1}b(Rz,R^{2}\sigma^{-1}s)$, and $\bar{s}=\tau
R^{-2}\sigma\leq 1$, then (A.8) holds. Therefore, by a straightforward
computation

$\sigma^{1/{q_{0}^{\prime}}}R^{N(1-1/q_{0}^{\prime})-2/q_{0}^{\prime}}\|m\|_{L^{q^{\prime}_{0}}(Q_{R,\tau})}=\|\tilde{m}\|_{L^{q^{\prime}_{0}}(Q_{1,\bar{s}})}\leq
C\left(\iint_{Q_{1,\bar{s}}}|\tilde{b}|^{\gamma^{\prime}}\tilde{m}\operatorname{d\\!}z\operatorname{d\\!}s+\bar{s}^{\alpha_{0}/2}\right)=\\\
=C\left(\frac{R^{\gamma^{\prime}-2}}{\sigma^{\gamma^{\prime}-1}}\iint_{Q_{R,\tau}}|b|^{\gamma^{\prime}}m\operatorname{d\\!}x\operatorname{d\\!}t+\frac{\sigma^{\alpha_{0}/2}}{R^{\alpha_{0}}}\tau^{\alpha_{0}/2}\right),$

which yields the conclusion, since
$N(1-1/q_{0}^{\prime})-2/q_{0}^{\prime}+2-\gamma^{\prime}=0$,
$\alpha_{0}=2-\gamma^{\prime}$ and
$\alpha_{0}/2+\gamma^{\prime}-1=\gamma^{\prime}/2$. ∎

We now state a property of $\zeta\mapsto|\zeta|^{\gamma^{\prime}}$.

######  Lemma A.3.

For any $\gamma^{\prime}\in(1,2)$, there exists $C$ depending on
$\gamma^{\prime}$ such that

$|\zeta+\xi|^{\gamma^{\prime}}-|\zeta|^{\gamma^{\prime}}\leq
C(|\zeta|^{\gamma^{\prime}-1}|\xi|+|\xi|^{\gamma^{\prime}})$ (A.11)

for all $\zeta,\xi\in{\mathbb{R}}^{N}$.

###### Proof.

Since ${\mathbb{R}}^{n}\ni p\mapsto|p|^{\gamma^{\prime}}$ is convex,

$|\zeta+\xi|^{\gamma^{\prime}}-|\zeta|^{\gamma^{\prime}}\leq\gamma^{\prime}|\zeta+\xi|^{\gamma^{\prime}-2}(\zeta+\xi)\cdot\xi.$

If $|\zeta|\leq|\xi|$,

$|\zeta+\xi|^{\gamma^{\prime}}-|\zeta|^{\gamma^{\prime}}\leq\gamma^{\prime}|\zeta+\xi|^{\gamma^{\prime}-1}|\xi|\leq\gamma^{\prime}2^{\gamma^{\prime}-1}|\xi|^{\gamma^{\prime}}.$

Else, $|\zeta|>|\xi|$; since $(0,\infty)\ni t\mapsto t^{\gamma^{\prime}-1}$ is
concave, and $(0,\infty)\ni t\mapsto t^{\gamma^{\prime}-2}$ is decreasing,

$\displaystyle|\zeta+\xi|^{\gamma^{\prime}}-|\zeta|^{\gamma^{\prime}}$
$\displaystyle\leq\gamma^{\prime}(|\zeta+\xi|^{\gamma^{\prime}-1}-|\zeta|^{\gamma^{\prime}-1})|\xi|+\gamma^{\prime}|\zeta|^{\gamma^{\prime}-1}|\xi|$
$\displaystyle\leq\gamma^{\prime}(\gamma^{\prime}-1)|\zeta|^{\gamma^{\prime}-2}(|\zeta+\xi|-|\zeta|)|\xi|+\gamma^{\prime}|\zeta|^{\gamma^{\prime}-1}|\xi|$
$\displaystyle\leq\gamma^{\prime}(\gamma^{\prime}-1)|\zeta|^{\gamma^{\prime}-2}|\xi|^{2}+\gamma^{\prime}|\zeta|^{\gamma^{\prime}-1}|\xi|$
$\displaystyle\leq\gamma^{\prime}(\gamma^{\prime}-1)|\xi|^{\gamma^{\prime}}+\gamma^{\prime}|\zeta|^{\gamma^{\prime}-1}|\xi|.$

∎

We finally present a crucial result on the $W^{2,1}_{q}$ regularity of
solutions to HJ equations enjoying suitable Hölder bounds. This is a parabolic
analogue of [8, Proposition 3.3].

######  Proposition A.4.

Let $R>0$, $g\in L^{q}(Q_{2R})$ and $v\in W^{2,1}_{q}(Q_{2R})$ be such that
$v(0)=0$,

$\|g\|_{L^{q}(Q_{2R})}+[v]_{\alpha;Q_{2R}}\leq
c_{1}\qquad\left|-\partial_{t}v-\Delta v\right|\leq c_{2}|Dv|^{\gamma}+g$

a.e. in $Q_{2R}$, for some $c_{1},c_{2}>0$, and $\alpha+\frac{N+2}{q}=2$.
Then, there exists $K$ depending on $c_{1},c_{2},R,N,q$ such that

$\|\partial_{t}v\|_{L^{q}(Q_{R})}+\|D^{2}v\|_{L^{q}(Q_{R})}\leq K.$

Moreover, the same statement holds if $Q_{R}$ and $Q_{2R}$ are replaced by
cylinders $Q^{\prime}\subset\joinrel\subset Q$ respectively.

###### Proof.

First, since $[v]_{\alpha;Q_{2R}}\leq c_{1}$ and $v(0)=0$,

$\|v\|_{L^{q}(Q_{2R})}\leq c_{1}CR^{\alpha+\frac{N+2}{q}}=c_{1}CR^{2}.$ (A.12)

Let $R\leq\rho\leq 2R$ and $0<\sigma<1$. Arguing as in the proof of [16,
Theorem 7.22], and using Hölder’s inequality and (A.12)

$\begin{split}\|D^{2}v\|_{L^{q}(Q_{\sigma\rho})}&\leq\frac{C}{(1-\sigma)^{2}R^{2}}\left(R^{2}\|c_{2}|Dv|^{\gamma}+g\|_{L^{q}(Q_{\rho})}+\|v\|_{L^{q}(Q_{\rho})}\right)\\\
&\leq\frac{C}{(1-\sigma)^{2}}\left(R^{\gamma-\frac{N+2}{q}(\gamma-1)}\|Dv\|_{p;Q_{\rho}}^{\gamma}+1\right).\end{split}$
(A.13)

where $\frac{1}{p}=\frac{1}{q}-\frac{1}{N+2}$. By the Gagliardo-Nirenberg
interpolation inequality [18], there exists $C$ (independent of $\rho$) such
that for a.e. $t\in(0,\rho^{2})$,

$\|Dv(t)\|_{p;B_{\rho}}\leq
C\left(\|D^{2}v(t)\|_{L^{q}(B_{\rho})}^{a}[v(t)]_{\alpha;B_{\rho}}^{1-a}+[v(t)]_{\alpha;B_{\rho}}\right),\qquad\text{where
}a=1-\frac{q}{N+2}<\frac{1}{\gamma}.$

Since $ap=q$, if we raise the previous inequality to the power $p$ and
integrate in time we obtain

$\|Dv\|^{p}_{p;Q_{\rho}}\leq C(\|D^{2}v\|_{L^{q}(Q_{\rho})}^{q}+R^{2}).$

If $\|D^{2}v\|_{L^{q}(Q_{R})}^{q}\leq R^{2}$, then there is nothing else to
prove, since $\|D^{2}v\|_{q,Q_{R}}$ is bounded, and a control on
$\|\partial_{t}v\|_{q,Q_{R}}$ follows immediately by integrating the
differential inequality for $v$. Otherwise, plugging the resulting inequality
into (A.13) implies

$\|D^{2}v\|_{L^{q}(Q_{\sigma\rho})}\leq\frac{C}{(1-\sigma)^{2}}\left(R^{\gamma-\frac{N+2}{q}(\gamma-1)}\|D^{2}v\|^{a\gamma}_{L^{q}(Q_{\rho})}+1\right).$
(A.14)

As before, if
$R^{\gamma-\frac{N+2}{q}(\gamma-1)}\|D^{2}v\|^{a\gamma}_{L^{q}(Q_{R})}\leq 1$,
then we are done. Otherwise, (A.14) yields, for every $R\leq\rho\leq 2R$ and
$0<\sigma<1$,

$\|D^{2}v\|_{L^{q}(Q_{\sigma\rho})}\leq\frac{E}{(1-\sigma)^{2}}\|D^{2}v\|_{L^{q}(Q_{\rho})}^{a\gamma}\
,$ (A.15)

where $E=CR^{\gamma-\frac{N+2}{q}(\gamma-1)}$, and $C$ is independent of
$\rho,\sigma$. We may then conclude following exactly the same lines as in the
proof of [8, Proposition 3.3]; the statement with $Q^{\prime},Q$ follows by a
standard covering argument. ∎

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Marco Cirant
Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova
Via Trieste 63, 35121 Padova, Italy
<EMAIL_ADDRESS>