Block | $B$ | $\mathrel{\vcentcolon\vcentcolon=}$ | $\begin{array}[t]{lllll}\\{&start&\mathrel{\vcentcolon}&\mathbb{N}\\\ &end&\mathrel{\vcentcolon}&\mathbb{N}\\\ &|slots|&\mathrel{\vcentcolon}&\mathbb{N}\\\ &inputs&\mathrel{\vcentcolon}&\wp({\mathbb{N}+\mathit{Reg}})\\\ &indirects&\mathrel{\vcentcolon}&\wp({\textit{IB}})&\\}\end{array}$
Indirect Block | IB | $\mathrel{\vcentcolon\vcentcolon=}$ | $\begin{array}[t]{lllll}\\{&start&\mathrel{\vcentcolon}&\mathbb{N}\\\ &end&\mathrel{\vcentcolon}&\mathbb{N}\\\ &parent&\mathrel{\vcentcolon}&\textit{Block}&\\}\end{array}$

Figure 26. WebAssembly Structure

###  D.1. Logical Relation

$L$ | $\mathrel{\vcentcolon\vcentcolon=}$ | $\begin{array}[t]{lllll}\\{&\mathit{interface}&\mathrel{\vcentcolon}&\mathbb{N}\rightharpoonup\mathbb{N}\\\ &\mathit{library}&\mathrel{\vcentcolon}&\mathbb{N}\rightharpoonup\mathit{WasmFunction}\\\ &\mathit{code}&\mathrel{\vcentcolon}&\mathit{Code}&\\}\end{array}$
---|---|---

$\begin{array}[]{rcl}\mathrm{World}&\triangleq&\mathbb{N}\times(\mathbb{N}\rightharpoonup\mathbb{N})\times(\mathbb{N}\rightharpoonup\mathbb{N})\\\
\\\ \mathord{\triangleright}&:&\mathrm{World}\rightarrow\mathrm{World}\\\
\mathord{\triangleright}(0,\overline{f_{i}},\overline{f_{l}})&\triangleq&(0,\overline{f_{i}},\overline{f_{l}})\\\
\mathord{\triangleright}(n,\overline{f_{i}},\overline{f_{l}})&\triangleq&(n-1,\overline{f_{i}},\overline{f_{l}})\\\
\\\
(C,\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{2}}}},\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{3}}}}):_{W}&\triangleq&\text{for
}i\in\\{1,2\\}:\\\
&&\begin{array}[t]{l}\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}.\mathit{entry}}=\operatorname{dom}({\pi_{i}(W)})\\\
\forall{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}\in\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}}.\leavevmode\nobreak\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}.\mathit{type}=\pi_{i}(W)({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}.\mathit{entry})\\\
\forall{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}\in\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}}.\leavevmode\nobreak\
(\mathord{\triangleright}W,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}},C|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}.\mathit{instrs}})\in\mathcal{F}\end{array}\end{array}$

$\begin{array}[]{rcl}\mathcal{F}&\triangleq&\left\\{(W,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}},c)\middle|\begin{array}[]{l}\forall\rho\in[\mathit{Frame}],\mathit{ret\mbox{-}addr},A\in[\mathbb{N}],sp,R,M,C,\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}},\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}}}.\\\
\quad\text{let
}\rho^{\prime}=\rho\mathbin{{+}\mspace{-8.0mu}{+}}\\{\mathit{base}\mathbin{\vcentcolon=}sp-|A|,\mathit{ret\mbox{-}addr\mbox{-}loc}\mathbin{\vcentcolon=}sp,\mathit{csr\mbox{-}vals}\mathbin{\vcentcolon=}R(\mathbb{CSR})\\}\\\
\quad|A|={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}}.\mathit{type}\\\
\quad sp>\operatorname{top}(\rho).\mathit{ret\mbox{-}addr\mbox{-}loc}+|A|\\\
\quad[sp\mapsto\mathit{ret\mbox{-}addr},(sp-|A|+i\mapsto
A_{i})_{i\in[0,|A|)}]\leq M\\\
\quad(C,\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}},\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}}}):_{W}\\\
\quad c\leq C\\\
\quad\operatorname{dom}({c})={\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}}.\mathit{instrs}\\\
\quad\forall\ell\in\operatorname{dom}({M})\cap
M_{\mathtt{lib}}.\leavevmode\nobreak\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\langle}}{n},{\mathtt{lib}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rangle}}=M(\ell)\\\
\quad\text{let
}\Psi=\\{pc\mathbin{\vcentcolon=}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}}.\mathit{entry},sp\mathbin{\vcentcolon=}sp,R\mathbin{\vcentcolon=}R,M\mathbin{\vcentcolon=}M,C\mathbin{\vcentcolon=}C\\}\\\
\quad\text{let
}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi}}=\\{\Psi\mathbin{\vcentcolon=}\Psi,\mathit{stack}\mathbin{\vcentcolon=}\rho^{\prime},\mathit{funcs}\mathbin{\vcentcolon=}[{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F^{\prime}}}.\mathit{entry}\mapsto{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F^{\prime}}}]_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F^{\prime}}}\in\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}}\uplus\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}}}}\\}\\\
\Longrightarrow\\\ \quad\forall n^{\prime}\leq W.n.\leavevmode\nobreak\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi}}\mathrel{\accentset{\rho^{\prime}}{\leadsto}^{n^{\prime}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}\Longrightarrow{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}\neq{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathtt{oerror}}}\\\
\quad\text{or }\exists n^{\prime}\leq W.n.\leavevmode\nobreak\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi}}\mathrel{\accentset{\rho^{\prime}}{\leadsto}^{n^{\prime}-1}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}\mathrel{\leadsto}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime\prime}}}\\\
\qquad\text{where
}({{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}}\llparenthesis{\mathtt{ret}}\rrparenthesis\vee{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}}\llparenthesis{\mathtt{gateret}}\rrparenthesis)\wedge{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}.\mathit{stack}=\rho^{\prime}\wedge{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime\prime}}}\neq{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathtt{oerror}}}\end{array}\right\\}\\\
\\\ \mathcal{L}&\triangleq&\left\\{(n,L)\middle|\leavevmode\nobreak\
\begin{array}[]{l}\forall i\in\operatorname{dom}({L.\mathit{library}}).\\\
\quad\text{let }\textit{WF}=L.\mathit{library}(i)\\\ \quad\text{let
}W=(n,L.\mathit{interface},\lambda i\rightarrow
L.\mathit{library}(i).|args|)\\\ \quad\text{let
}\mathit{instrs}=\biguplus_{B\in\textit{WF}.\mathit{blocks}}[B.start,B.end]\\\
\quad\text{let
}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}}=\\{\mathit{instrs}\mathbin{\vcentcolon=}\mathit{instrs},\mathit{entry}\mathbin{\vcentcolon=}\textit{WF}.\mathit{entry}.\mathit{start},\mathit{type}\mathbin{\vcentcolon=}\textit{WF}.|args|\\}\\\
\quad(W,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}},L.\mathit{code}|_{\mathit{instrs}})\in\mathcal{F}\end{array}\right\\}\end{array}$

Figure 27. Function and Library Relations

###### Lemma 0 (FTLR for functions).

Let $W\in\mathrm{World}$ and $c$ be the code for a compiled WebAssembly
function WF such that WF expects application functions in the interface with
locations and types $\pi_{2}(W)$ and in the library with locations and types
$\pi_{3}(W)$. Further let
$\mathit{instrs}=\biguplus_{B\in\textit{WF}.\mathit{blocks}}[B.start,B.end]$
and
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}}=\\{\mathit{instrs}\mathbin{\vcentcolon=}\mathit{instrs},\mathit{entry}\mathbin{\vcentcolon=}\textit{WF}.\mathit{entry}.\mathit{start},\mathit{type}\mathbin{\vcentcolon=}\textit{WF}.|args|\\}$.
Then
$(W,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}},c)\in\mathcal{F}$.

###### Proof.

We first unroll the assumptions of
$(W,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F}},c)\in\mathcal{F}$
reusing the variable names defined there. We will maintain that any steps do
not step to
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathtt{oerror}}}$
so WOLOG we will continually assume $n^{\prime}\leq W.n$ such that
$n^{\prime}$ greater than the number of steps we have taken, otherwise the
case
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi}}\mathrel{\accentset{\rho^{\prime}}{\leadsto}^{n^{\prime}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}\Longrightarrow{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}\neq{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathtt{oerror}}}$
holds.

By the structure of a compiled WebAssembly function and assumption we have
that the stack and stack pointer represent $\textit{WF}.|args|$ arguments. The
abstract interpretation ensures that if we write to $H_{\mathtt{lib}}$ then
that value has label $\mathtt{lib}$ so the checks pass. We are further assured
that we do not read or below the stack frame. The structure of a compiled
WebAssembly block then lets us proceed until we reach one of  (1) a function
call to a library function $\textit{WF}^{\prime}$ such that
$\textit{WF}^{\prime}.\mathit{entry}.\mathit{start}\in\overline{F_{l}.\mathit{entry}}$,
(2) an application function
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F^{\prime}}}$
such that
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F^{\prime}}}.\mathit{entry}\in\overline{F_{i}.\mathit{entry}}$,
(3) or the end of the block.

  1. (1)

The abstract interpretation ensures that we have initialized the arguments
$\textit{WF}^{\prime}.|args|=\pi_{3}(W)(\textit{WF}^{\prime}.\mathit{entry}.\mathit{start})$
or failed a dynamic type check and terminated (thus stepping to a terminal
state that is not an
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathtt{oerror}}}$).
We thus set $\rho_{2}=\rho^{\prime}$ and see that we have constructed
$\rho_{2}^{\prime}=\rho_{2}\mathbin{{+}\mspace{-8.0mu}{+}}\\{\mathit{base}\mathbin{\vcentcolon=}sp-\textit{WF}^{\prime}.|args|,\mathit{ret\mbox{-}addr\mbox{-}loc}\mathbin{\vcentcolon=}sp,\mathit{csr\mbox{-}vals}\mathbin{\vcentcolon=}R(\mathbb{CSR})\\}$.
We further set $\mathit{ret\mbox{-}addr}=pc+1$,
$A=\textit{WF}^{\prime}.|args|$, $sp=sp$, $R=R$, $M=M$, $C=C$,
$\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}}=\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{i}}}}$,
and
$\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}}}=\overline{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}}}$.
By the abstract interpreation we have that all of the remaining checks in
$\mathcal{F}$ pass and that the instantiated
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi}}$ is
equal to our current state. We therefore instantiate
$(\mathord{\triangleright}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}},C|_{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{F_{l}}}.\mathit{instrs}})\in\mathcal{F}$.
If this uses the remaining steps then we are done. Otherwise we get that we
return to $pc+1$ with all values restored and no new $\mathtt{app}$ values
written to the library memory, and our walk through the block may continue.

  2. (2)

Identical to the case for (1).

  3. (3)

The end of a block is followed by a direct jump to another block $B^{\prime}$,
an indirect block $\mathit{IB}$, or we are at an exit block. In the case of
another block $B^{\prime}$ we have by the structure of compiled WebAssembly
code that we have instantiated $B^{\prime}.inputs$. We thus jump to the block
and follow the same proof structure as detailed here. The same is true of an
intermediate block $\mathit{IB}$ except with the extra steps of setting up the
inputs jumping to another block $B^{\prime\prime}$. Lastly if we have reached
the end of an exit block then we have not touched the pushed return address or
callee-save registers and the stack pointer is in the expected location. We
thus execute $\mathtt{ret}$ or $\mathtt{gateret}$ and pass the overlay monitor
checks.

∎

###### Lemma 0 (FTLR for libraries).

For any number of steps $n\in\mathbb{N}$ and compiled WebAssembly library $L$,
$(n,L)\in\mathcal{L}$.

###### Proof.

By unrolling the definition of $\mathcal{L}$ and Lemma 1. ∎

###### Theorem 3 (Adequacy of $\mathcal{L}$).

For any number of steps $n\in\mathbb{N}$, library $L$ such that
$(n,L)\in\mathcal{L}$, program
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi_{0}}}\in\mathit{Program}$
using $L$, and $n^{\prime}\leq n$, if
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi_{0}}}\mathrel{\mathrel{\leadsto}^{n^{\prime}}}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}$
then
${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\Phi^{\prime}}}\neq{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathtt{oerror}}}$.

###### Proof.

Straightforward: by assumption for steps in the application, and by assumption
about application code properly calling the library code and the unrolling of
$\mathcal{L}$ and $\mathcal{F}$. ∎

##  Appendix E Additional Image Benchmarks

Figure 28.  Performance of the WasmLucet heavyweight transitions included in
the Lucet runtime on the image benchmarks in Section 7. Performance when
rendering (a) a simple image with one color, (b) a stock image and (c) a
complex image with random pixels. The performance is the overhead compared to
WasmZero.

Figure 29.  Performance of an ideal isolation scheme (no enforcement overhead)
with heavy trampolines when rendering images. Wasm compilers whose enforcement
overhead is lower than this can outperform even an ideal isolation scheme that
uses heavy weight transitions.


# The ergodic Mean Field Game system for a type of state constraint condition

Mariya Sardarli

##  1 Introduction

The object of this paper is to establish well-posedness (existence and
uniqueness) results for a type of state constraint ergodic Mean Field Game
(MFG) system

$\displaystyle\begin{cases}-\Delta u+H(Du)+\rho=F(x;m)&\text{ in }\Omega,\\\
\Delta m+div(mD_{p}H(Du))=0&\text{ in }\Omega,\\\ m\geq
0,\quad\int_{\Omega}m=1,\end{cases}$ (1.1)

supplemented with the state constraint-type infinite Dirichlet boundary
condition

$\lim\limits_{d(x)\to 0}u(x)=\infty,$ (1.2)

where $d(x):=d(x,\partial\Omega)$ is the distance to the boundary. Here
$\Omega\subset\mathbb{R}^{n}$ is an open bounded subset of $\mathbb{R}^{n}$,
$H:\mathbb{R}^{n}\times\overline{\Omega}\to\mathbb{R}$ is the Hamiltonian
associated with the cost function of an individual agent, $\rho$ is the
ergodic constant, $m$ is the distribution of the agents, and $F:\Omega\times
L^{1}(\Omega)\to\mathbb{R}$ is the interaction term. The Hamilton-Jacobi-
Bellman (HJB) equation is understood in the viscosity sense and the
Kolmogorov-Fokker-Planck (KFP) equation, in the distributional sense.

The MFG $(\ref{eq:proto})$ with the boundary condition $(\ref{intro:bds})$ can
be interpreted as a state constraint-type problem. Indeed, the infinite
boundary condition prevents the underlying stochastic trajectories from
reaching the boundary, forcing them to stay in the domain.

The theory of MFGs was introduced by Lasry and Lions [8] and, in a particular
setting, by Caines, Huang, and Malhamé [4], to describe the interactions of a
large number of small and indistinguishable rational agents. In the absence of
common noise, the behavior of the agents leads to a forward-backward coupled
system of PDEs, consisting of a backward HJB equation describing the
individual agent’s value function, and a forward KFP equation for the
distribution of the law (density) of the population.

The forward-backward system with periodic boundary conditions and either local
or non-local coupling, as well its ergodic stationary counterpart, was first
studied in [8]. Summaries of results may also be found in Cardaliguet [2],
Lions [9], and Gomes, Pimentel, and Voskanyan [3]. Fewer results exist,
however, in the case of either Dirichlet, state constraint or Neumann boundary
conditions.

We next state the main result of this paper for superlinear power-like
Hamiltonians, that is,

$H(p,x)=|p|^{q}\text{ with }\quad q>1,$ (1.3)

and suitable assumptions on the coupling $F$ (see $(F1)$, $(F2)$ and $(F3)$ in
Section 2.2). We emphasize that the particular form of the Hamiltonian is by
no means essential to the analysis that follows. It simply provides a
sufficiently general model problem that results in the necessary asymptotics
of $u$ and $m$ near $\partial\Omega$. In particular, for $1<q\leq 2$,
Hamiltonians $H(p)$ for which

$\delta^{q/(q-1)}H(\delta^{-1/(q-1)}p)-|p|^{q}\to 0,$ (1.4)

locally uniformly in $p$ as $\delta\to 0$ are permissible. We state the result
below, deferring the precise meaning of a solution to the system until Section
3.

######  Theorem.

Let $1<q\leq 2$ and assume either $(F1)$ or $(F2)$.
Then, for all $r>1$, there exists a solution $(u,\rho,m)\in
W^{2,r}_{loc}(\Omega)\times\mathbb{R}\times W^{1,r}_{loc}(\Omega)$ to the
system

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=F(x;m)&\text{ in
}\Omega,\\\ \lim\limits_{d(x)\to 0}u(x)=\infty,\\\ \Delta
m+div(mq|Du|^{q-2}Du)=0&\text{ in }\Omega,\\\ m\geq
0,\quad\int_{\Omega}m=1.\end{cases}$ (1.5)

Moreover, if $F$ satisfies $(F3)$, then the solution is unique.

As will be explained below, the conditions $(F1)$ and $(F2)$ describe non-
local and local couplings respectively, while $(F3)$ is the usual monotonicity
condition.

###  1.1 Background

Heuristically, HJB equations with convex (in the momentum) Hamiltonian $H$ are
associated with the stochastic control problem governed by the dynamics

$dX_{t}=\alpha_{t}dt+\sqrt{2}dB_{t},$

where $(\alpha_{t})_{t\geq 0}$ is a non-anticipating process. In feedback
form, the optimal policy is $-D_{p}H(Du)$.

In settings in which the trajectories of the agents reach the boundary but do
not leave the domain or, as is the case we consider here, do not reach the
boundary at all, the solutions of the HJB equation are referred to as state
constraint solutions.

If the drift $-D_{p}H(Du)$ is bounded, then for all $t>0$,
$P(X_{t}\in\Omega^{c})>0$. It follows that for trajectories to remain inside
$\Omega$, the drift must become unbounded near the boundary, pushing the
agents back into the domain. For Hamiltonians of the form (1.3), or more
generally $(\ref{eq:gen})$, state constraint solutions exist when the equation
is paired with an infinite Neumann condition (reflection cost) or infinite
Dirichlet condition (exit cost), the “correct” choice of boundary condition
depending on whether the Hamiltonian is sub- or super-quadratic. Indeed, when
$q>2$, solutions are bounded but the normal component of the drift blows up at
the boundary. In the sub-quadratic case, the solution itself also blows up.

In the sub-quadratic case, the trajectories never reach the boundary. Hence,
the population density is “almost zero” near $\partial\Omega$. As the behavior
of $m$ at the boundary is coupled with the blowup of $Du$ on $\partial\Omega$,
no boundary condition is required on $m$ in (1.5).

A type of MFG state-constraint problem has been studied by Porretta and
Ricciardi in [10]. The authors impose structural assumptions on their
Hamiltonian so that, for all choices of the control, an agent governed by
these dynamics does not exit the domain $\Omega$. In this instance, no
explicit boundary conditions on the value function are necessary to achieve
well posedness. However, the assumptions of [10] do not permit coercive
Hamiltonians, and, in particular, do not apply to the ones considered in this
paper.

###  1.2 Infinite Dirichlet boundary conditions

In the context of HJB equations, infinite boundary conditions with power-like
Hamiltonians were studied by Lasry and Lions in [7], who considered the HJB
equation

$-\Delta u+|Du|^{q}+\lambda u=g\text{ in }\Omega,$ (1.6)

with $1<q\leq 2$ and $g\in L^{\infty}(\Omega)$, as well as the ergodic problem

$-\Delta u+|Du|^{q}+\rho=g\text{ in }\Omega,$

both coupled with the infinite Dirichlet boundary condition
$(\ref{intro:bds})$.

It was shown in [7] that the boundary value problem

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=g\text{ in }\Omega,\\\
\lim\limits_{d(x)\to 0}u(x)=\infty,\end{cases}$ (1.7)

has a unique viscosity solution $(u,\rho)$, where uniqueness for $u$ is
understood to be modulo an additive constant. Such a solution has unbounded
drift in the direction of the boundary and the agents remain in the domain.

Due to the coupled nature of the MFG system, the existence and well posedness
of the corresponding KFP equation depends in an essential way on the
asymptotics of $u$ and $Du$ near the boundary.

In general, if $b\in L^{1}_{loc}(\Omega)$, the distributional solution $m$ of

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega,\quad\int_{\Omega}m=1,\quad
m\geq 0,$ (1.8)

need not be unique unless a boundary condition is specified.

On the other hand, if

$\displaystyle\lim\limits_{d(x)\to 0}b(x)d(x)=C\nu,$ (1.9)

where $C>0$ and $\nu$ is the outward unit normal to the boundary at the point
closest to $x$, Huang, Ji, Liu and Yi showed in [5] that solutions are unique,
despite the absence of boundary conditions.

To show $(\ref{eq:13})$ for $b=D_{p}H(Du)$ and apply this result, we need to
make precise the asymptotics of $u$ and $Du$ near the boundary.

In [7], the authors proved that if $(u,\rho)$ is a solution of
$(\ref{eq:hjb})$, the asymptotics are precisely of this type. Namely, there
exists a constant $C>0$ such that

$\displaystyle\lim\limits_{d(x)\to
0}\dfrac{Du(x)\cdot\nu(x)}{d^{-1/(q-1)}}=C.$

In this paper, we refine the result and show that the rate of convergence of
the above limit is controlled by the $L^{\infty}-$norm of $g$. This becomes
important when proving stability results related to $(\ref{eq:mainintro})$.

###  1.3 The methodology

As usual, the existence of a solution to the MFG system $(\ref{eq:mainintro})$
is proven via a fixed point argument. To apply such an argument, it is crucial
to obtain bounds on $m$ up to $\partial\Omega$. Toward that end, for
$\delta>0$, we define the subdomain $\Omega_{\delta}\subset\Omega$ by

$\Omega_{\delta}:=\\{x\in\Omega:d(x,\partial\Omega)>\delta\\}.$

As a first step, we show that if $b$ satisfies $(\ref{eq:13})$ with constant
$C$, then there exists $\gamma=\gamma(C)>0$, $\delta=\delta(C)>0$, and
$L=L(C,||b||_{L^{\infty}(\Omega_{\delta})})>0$ such that any solution $m$ of
$(\ref{eq:mbla})$ satisfies

$\int_{\Omega}d^{-\gamma}m\leq L.$ (1.10)

This estimate is needed in both the local and non-local cases.

The space in which the fixed point argument is carried out differs in the non-
local and local coupling settings. In the former, the fixed point is obtained
in the space

$\mathcal{W}_{\gamma}(\Omega):=\\{m\in
L^{1}(\Omega):\int_{\Omega}d^{-\gamma}m<\infty\\},$ (1.11)

where $\gamma$ is as in $(\ref{eq:gam|})$ and, here, depends only on $q$. The
map $T:\mathcal{W}_{\gamma}(\Omega)\to\mathcal{W}_{\gamma}(\Omega)$ formed by
composing the solution maps of $(\ref{eq:hjb})$ and $(\ref{eq:mbla}$)
satisfies the conditions of Schaefer’s fixed point theorem. In particular, the
continuity of $T$ is straightforward due to the regularizing effect of the
coupling.

By contrast, the continuity of the solution map is far from obvious in the
local case, where $F(x;m)=f(m(x))$ for a continuous and bounded function $f$.
Convergence of a sequence $m_{n}$ in $\mathcal{W}_{\gamma}(\Omega)$ does not
yield local uniform convergence of $F(x;m_{n})$ as it does in the non-local
case. Thus the fixed point argument must be performed in a different space.

To deal with this issue, we introduce, for $\delta>0$, the approximating
coupled ergodic system

$\displaystyle\begin{cases}-\Delta
u_{\delta}+|Du_{\delta}|^{q}+\rho_{\delta}=f(\tilde{m}_{\delta}(x)),&\text{ in
}\Omega,\\\ \lim\limits_{d(x)\to 0}u_{\delta}(x)=\infty,\\\ \Delta
m_{\delta}+div\left(m_{\delta}q|Du_{\delta}|^{q-2}Du_{\delta}\right)=0&\text{
in }\Omega_{\delta},\\\
\left(Dm_{\delta}+m_{\delta}q|Du_{\delta}|^{q-2}Du_{\delta}\right)\cdot\nu=0&\text{
on }\partial\Omega_{\delta},\\\ \int_{\Omega}m_{\delta}=1,\end{cases}$ (1.12)

where $(u_{\delta},\rho_{\delta},m_{\delta})\in W^{1,r}\times\mathbb{R}\times
C^{0,\alpha}(\Omega_{\delta})$ and $\tilde{m}_{\delta}$ is a suitable
extension of ${m_{\delta}}$ to all of $\Omega$ satisfying

$||\tilde{m}_{\delta}||_{C^{0,\alpha}(\Omega)}=||m_{\delta}||_{C^{0,\alpha}(\Omega_{\delta})}.$

We note that (1.12) is not the standard MFG system with Neumann boundary
conditions, in which the HJB and KFP equations are set in the same domain and
both have Neumann boundary conditions. Instead, the HJB equation is set in the
entire domain $\Omega$, allowing us to to take advantage of the known
asymptotics of $u$ and $Du$ near the boundary. On the other hand, the Neumann
conditions on the KFP equation allow us to exploit in regularity of
$m_{\delta}$ up to $\partial\Omega_{\delta}$.

As we discuss below, for $b\in L^{\infty}(\Omega)$, the KFP equation

$\displaystyle\begin{cases}\Delta m_{\delta}+div(m_{\delta}b)=0&\text{ in
}\Omega_{\delta},\\\ (Dm_{\delta}+m_{\delta}b)\cdot\nu=0&\text{ on
}\partial\Omega_{\delta},\end{cases}$

has a a unique (modulo a multiplicative constant) distributional solution,
which is positive and Hölder continuous up to the boundary with Hölder
constant depending on the $L^{\infty}-$norm of the drift. If $u$ is the
solution of a HJB equation like $(\ref{eq:hjb})$, $Du$ is bounded on
$\Omega_{\delta}$ and this result applies to the KFP equation in
$(\ref{eq:deltap})$. Moreover, as shown in [7], the local bounds on the
solutions $u_{\delta}$ of the HJB equation are uniform in the
$L^{\infty}-$norm of $f$, which is independent of $m_{\delta}$. Therefore, as
we show below, it is possible to carry out a fixed point argument in
$C^{0,\alpha}(\Omega_{\delta})$.

We are then able to pass from the solution
$(u_{\delta},\rho_{\delta},m_{\delta})$ of the approximating system to a
solution $(u,\rho,m)$ of the original system on $\Omega$ by the stability of
the HJB equation and the local uniform bounds on $m_{\delta}$, which in turn
follow from local uniform bounds on $Du_{\delta}$.

Lastly, under the usual monotonicity assumption we establish uniqueness in
both the local and non-local case. The key observation in the argument is
that, for solutions $(\rho,u,m)$ of (1.5), $m$ decays near the boundary as in
(1.10).

###  1.4 Organization of the paper

The paper is organized as follows. The assumptions on the domain $\Omega$ and
the coupling $F$ are stated in Section 2. Section 3 introduces the notions of
weak solutions of the KFP and HJB equations and proves (local) regularity and
stability results. Section 4 treats the non-local coupling case. In Section 5,
we show the approximating problem (1.12) has a solution and pass to the limit
to obtain a solution to $(\ref{eq:mainintro})$ on the entire domain. Section 6
establishes the uniqueness of the system and quantifies the sense in which $m$
vanishes at the boundary. The Appendix contains two technical lemmata used in
Sections 3 and 5.

##  2 Assumptions and definitions

###  2.1 The domain

Throughout this paper it is assumed that

$\Omega\text{ is a bounded open subset of }\mathbb{R}^{n}\text{ with
}C^{2}\text{-boundary}.$

In particular, the domain satisfies the uniform interior ball condition.
Namely, there exists a $\delta_{0}>0$ such that for every
$x\in\partial\Omega$, the ball $B_{\delta_{0}}(x-\delta_{0}\nu(x))$ is
contained in $\Omega$.

Recall the definition of the subdomains

$\Omega_{\delta}:=\\{x\in\Omega:d(x,\partial\Omega)>\delta\\}.$

Due to the $C^{2}$-regularity of $\Omega$, there exists $\epsilon_{0}\in(0,1)$
such that $d(x,\partial\Omega)\in
C^{2}(\Omega\backslash\Omega_{\epsilon_{0}})$ and for all
$x\in\Omega\backslash\Omega_{\epsilon_{0}}$ there exists a unique point
$\overline{x}\in\partial\Omega$ such that
$d(x,\partial\Omega)=|x-\overline{x}|$. Moreover, on
$\Omega\backslash\Omega_{\epsilon_{0}}$,

$Dd(x,\partial\Omega)=Dd(\overline{x},\partial\Omega)=-\nu(\overline{x}).$

Lastly, we define $d$ to be a $C^{2}(\overline{\Omega})$ extension of
$d(x,\partial\Omega)$ which is equal to $d(x,\partial\Omega)$ on
$\Omega\backslash\Omega_{\epsilon_{0}}$. We additionally require that
$d(x)\leq 1$ in $\Omega$ and $d(x)\geq M_{0}$ in $\Omega_{\epsilon_{0}}$ for
some positive constant $M_{0}$.

###  2.2 The coupling

Our analysis permits a coupling $F:\Omega\times L^{1}(\Omega)\to\mathbb{R}$
which is either non-local, on the one hand, or local, continuous, and bounded
on the other. Namely, we consider couplings $F$ satisfying one of the
following:

  1. (F1)

The map $m\mapsto F(\cdot\ ;m)$ sends bounded sets in $L^{1}(\Omega)$ to
bounded sets in $L^{\infty}(\Omega)$, and is continuous from $L^{1}(\Omega)$
into $L^{\infty}(\Omega)$.

  2. (F2)

$F(x;m)=f(m(x))$ for $f$ a continuous, bounded function.

A coupling satisfying assumption $(F1)$ has an $L^{\infty}(\Omega)$-bound
depending only on $\int_{\Omega}m$. A coupling satisfying $(F2)$ inherits
regularity from the regularity of $m$ and is bounded by
$||f||_{L^{\infty}(\mathbb{R})}$.

When proving uniqueness we will require the additional monotonicity assumption

  1. (F3)

$\int_{\Omega}(F(x;m_{1})-F(x;m_{2}))(m_{1}(x)-m_{2}(x))dx\geq 0$, with
equality only if $m_{1}=m_{2}$ a.e.

###  2.3 The space of measures for the non-local setting

In the introduction, we defined a space $\mathcal{W}_{\gamma}(\Omega)$ which
quantifies the boundary behavior of solutions to KFP equations with an
unbounded drift term. We repeat the definition here for completeness.

######  Definition 2.1.

For $\gamma>0$, define a norm $||\cdot||_{\gamma,\Omega}$ by

$\displaystyle||m||_{\gamma,\Omega}=\int_{\Omega}d^{-\gamma}|m|,$

and recall that $\mathcal{W}_{\gamma}(\Omega)$ is the Banach space induced by
this norm, as in $(\ref{wgamma})$.

###  2.4 The extension of measures defined on subdomains

In the local coupling setting discussed in Section 5 we will consider measures
defined on the subdomain $\Omega_{\delta}$. Since $F$ is defined on
$\Omega\times L^{1}(\Omega)$, it will be necessary to define a suitable
extension of measures on $\Omega_{\delta}$ to $\Omega$ that preserves their
Hölder regularity.

For $\mu\in C^{0,\alpha}(\Omega_{\delta})$ and $x\in\Omega$ we define
$\tilde{\mu}$ by

$\displaystyle\tilde{\mu}(x):=\inf_{y\in\Omega_{\delta}}\\{\mu(y)+||\mu||_{C^{0,\alpha}(\Omega_{\delta})}|x-y|^{\alpha}\\}.$
(2.1)

It is straightforward to check that $\tilde{\mu}(x)=\mu(x)$ on
$\Omega_{\delta}$ and that $\tilde{\mu}\in C^{0,\alpha}(\Omega)$ with the same
Hölder modulus as $\mu$.

In the sections that follow, we abuse notation slightly and define the
coupling $F:\Omega\times C^{0,\alpha}(\Omega_{\delta})\to\mathbb{R}$ by

$\displaystyle F(x;\mu):=F(x;\tilde{\mu}).$

##  3 Preliminaries

In this section, we recall some existence and regularity results for the HJB
equation and the KFP equation. At the end we prove stability results that will
be used in Section 5.

###  3.1 The Kolmogorov Fokker-Planck equation

Throughout this paper we will need to consider the KFP equation in subdomains
of $\Omega$. To that end, we study the following KFP boundary value problem
for a general open bounded domain $V$ and $B\in L^{\infty}(V)$.

$\displaystyle\begin{cases}\Delta\mu+div(\mu B)=0\ &\text{ in }V,\\\ (D\mu+\mu
B)\cdot\nu=0&\text{ on }\partial V,\\\ \int_{V}\mu=1,\,\mu>0\,\text{ in
}V.\end{cases}$ (3.1)

Later we will apply these results when $V=\Omega_{\delta}$ and
$B=D_{p}(H(Du))$.

######  Definition 3.1.

Given $B\in L^{\infty}(V)$, we say that $\mu\in W^{1,2}(V)$ is a Neumann weak
solution of $(\ref{mdelta})$ if $\mu$ is a positive probability measure such
that, for all $\phi\in W^{1,2}(V)$,

$\displaystyle\int_{V}[D\phi(x)\cdot D\mu(x)+B(x)\cdot D\phi(x)\mu(x)]\ dx=0.$

The next lemma gathers existence and uniqueness results for Neumann weak
solutions of $(\ref{mdelta})$ as well as $W_{loc}^{1,r}(V)$ estimates.

######  Lemma 3.1.

For $B\in L^{\infty}(V)$, $(\ref{mdelta})$ admits a unique Neumann weak
solution $\mu\in W^{1,2}(V)$. In addition, $\mu\in W^{1,r}(V)$ for all $r>1$
and, for compact sets $K\subset K^{\prime}\subset V$, $||\mu||_{W^{1,r}(K)}$
is bounded uniformly by a constant that depends only on $K,K^{\prime}$ and
$||B||_{L^{\infty}(K^{\prime})}$.

###### Proof.

The existence, positivity, and uniqueness are proven in [6]. To obtain the
local uniform $W^{1,r}$ estimate, we recall that Theorem 1.1 of [1] yields a
constant $C_{1}=C_{1}(K,||B||_{L^{\infty}(K^{\prime})})>0$ such that

$||\mu||_{W^{1,r}(K)}\leq C_{1}||\mu||_{L^{\infty}(K)}.$ (3.2)

Then Harnack inequality and the normalization condition $\int_{V}\mu=1$ give
some $C_{2}=C_{2}(K,K^{\prime})>0$, such that

$\sup_{K}\mu\leq C_{2}\inf_{K}\mu\leq C_{2}\frac{1}{|K|}.$ (3.3)

Combining $(\ref{eq:r})$ and $(\ref{eq:r2})$ completes the proof. ∎

If the drift in the KFP equation blows-up at the boundary, it is necessary to
modify the notion of weak solution as follows.

######  Definition 3.2.

Given $b\in L^{1}_{loc}(\Omega)$ and $r>1$, we say $\mu\in
W^{1,r}_{loc}(\Omega)$ is a _weak solution_ of

$\displaystyle\Delta\mu+div(\mu b)=0\text{ in }\Omega,$ (3.4)

if $\mu$ is a non-negative function such that, for all $\phi\in
C^{2}_{c}(\Omega)$,

$\displaystyle\int_{\Omega}[\Delta\phi(x)-b(x)\cdot D\phi(x)]\mu(x)\ dx=0.$

Moreover, $\mu$ is a _proper weak solution_ if $\int_{\Omega}\mu=1$.

######  Remark.

The class of test functions in Definition 3.2 can be extended to $C^{2}$
functions that are constant in a neighborhood of $\partial\Omega$.

The following lemma shows that weak solutions of $(\ref{eq:m})$ may arise as
the limit of Neumann weak solutions. The rest of the section provides
conditions under which the limit is a proper weak solution.

######  Lemma 3.2.

Let $m_{\delta}\in W^{1,r}(\Omega_{\delta})$ be a Neumann weak solution of
$(\ref{mdelta})$ for velocity fields $b_{\delta}\in
L^{\infty}(\Omega_{\delta})$. Assume that, as $\delta\to 0$, $b_{\delta}\to b$
locally uniformly. Then, along subsequences, $m_{\delta}\to m\in
W^{1,r}_{loc}(\Omega)$ and $m$ is a weak solution of

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega.$

###### Proof.

Recalling that $||m_{\delta}||_{W^{1,r}(K)}$ is uniformly bounded for each
compact $K\subset\Omega$, it is possible to extract a subsequence such that
$m_{\delta}$ converges locally uniformly to a non-negative
$W^{1,r}_{loc}(\Omega)$ function $m$. To see that $m$ satisfies the desired
equation, consider a test function $\phi\in C^{2}_{c}(\Omega)$. Since
$b_{\delta}$ converges uniformly to $b$ in the support of $\phi$, $\Delta\phi-
b_{\delta}\cdot D\phi$ converges uniformly to $\Delta\phi-b\cdot D\phi$. As
$m_{\delta}\to m$ uniformly in the support of $\phi$, letting $\delta\to 0$ in
the weak formulation $\int_{\Omega_{\delta}}(\Delta\phi-b_{\delta}\cdot
D\phi)m_{\delta}$ leads to the desired result. ∎

It follows from Fatou’s lemma that the limit $m$ in the above lemma is
integrable on $\Omega$ and $\int_{\Omega}m\leq 1$. To ensure that $m$ is a
proper weak solution it is necessary to impose additional structural
conditions on the vector fields $b_{\delta}$. In [1], Bogachev and Rökner give
conditions on the drifts under which the sequence of weak solutions
$m_{\delta}$ is uniformly tight, which in turn yields $L^{1}(\Omega)$
convergence of $m_{\delta}$ to $m$. The version of the theorem relevant to our
setting appears below.

######  Lemma 3.3 (Lemma 1.1 in [1]).

Let $U_{k}$ by a sequence of nested domains and $\mu_{k}$ the corresponding
sequence of Neumann weak solutions to $\Delta\mu_{k}+div(\mu_{k}b_{k})=0$ on
$U_{k}$. Suppose that $b_{k}\in L^{1}_{loc}(U_{k})$ and that, for every $R>0$,
there exists $\alpha_{R}>n$ such that

$\displaystyle\kappa_{R}:=\sup_{k}\int_{U_{R}}|b_{k}(x)|^{\alpha_{R}}dx<\infty.$

If, in addition, there exists $V\in C^{2}(U)$ such that, for some
$c_{k}\to\infty$, $U_{k}=\\{V<c_{k}\\}$, $\\{V=c_{k}\\}$ has Lebesgue measure
zero,

$\displaystyle\lim_{d(x,\partial U)\to 0}V(x)=\infty,$ (3.5)

and

$\displaystyle\lim_{d(x,\partial U)\to 0}\sup_{k}[\Delta V-DV\cdot
b_{k}]=-\infty,$ (3.6)

then the sequence $\mu_{k}$ is uniformly tight.

A function $V$ satisfying the above conditions is referred to as a Lyapanov
function. The connection between Lyapanov functions and existence and
uniqueness of weak solutions of $(\ref{eq:m})$ has been extensively studied in
[1], as well as in [5].

It follows from the Lemma 3.3 that, if there exists a $V$ satisfying
$(\ref{v1})$, and

$\displaystyle\lim_{d(x,\partial U)\to 0}\Delta V-DV\cdot b=-\infty,$ (3.7)

then, setting $b_{k}=b\chi_{U_{k}}$, we obtain a sequence of measures $m_{k}$
that converge to a proper weak solution $m$ of

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega.$ (3.8)

It is proven in [5] that such a solution is unique.

In the present setting, these results will be applied by considering $b(x)$
satisfying, for some $C>1$ and $\epsilon\in(0,C-1)$,

$\displaystyle\lim_{d(x)\to 0}(b(x)\cdot Dd(x))d(x)=-C,$ (3.9)

and $V(x)=d(x)^{-C+1+\epsilon}$, which satisfies $(\ref{v1})$ and
$(\ref{vb2})$. This is stated precisely in the next result, the proof of which
appears in the Appendix.

######  Lemma 3.4.

Given $b:\Omega\to\mathbb{R}^{d}$ satisfying $(\ref{eq:asym})$ with $C>1$,
there exists a unique proper weak solution $m\in W^{1,r}_{loc}(\Omega)$ of

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega,$

and, for every $\epsilon\in(0,C-1)$, there exists a
$\delta=\delta(C,\epsilon)$, such that

$\displaystyle\int_{\Omega}d^{-C-1+\epsilon}m\leq\hat{C}=\hat{C}(C,||b||_{L^{\infty}(\Omega_{\delta})}).$

######  Remark.

The bound on $\int_{\Omega}d^{-C-1+\epsilon}m$ makes precise the intuition
that $m$ “approaches 0” near the boundary $\partial\Omega$.

It is proven in [7] that, for solutions $u$ of the HJB equation in
$(\ref{eq:mainintro})$, $b=q|Du|^{q-2}Du$ satisfies precisely
$(\ref{eq:asym})$ for $C=q/(q-1)$. This is discussed further in the next
section.

###  3.2 The Hamilton-Jacobi-Bellman equation

In this section, we fix $q\in(1,2]$ and $g\in L^{\infty}(\Omega)$ and consider
the stationary HJB equation with infinite Dirichlet conditions

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=g\text{ in }\Omega,\\\
\lim\limits_{d(x)\to 0}u(x)=\infty.\end{cases}$ (3.10)

The estimates detailed here will be used to prove the HJB stability results in
Theorem 3.8.

We begin by stating the definition of solutions for concreteness.

######  Definition 3.3.

For $1<q\leq 2$ and $g\in L^{\infty}(\Omega)$, $(u,\rho)\in
W^{2,r}_{loc}(\Omega)\times\mathbb{R}$ is an _explosive solution_ of
$(\ref{udef})$ if $u$ is a viscosity solution in $\Omega$ and

$\displaystyle\lim_{d(x)\to 0^{+}}u(x)=\infty.$

The existence and uniqueness of solutions to such HJB equations was proven in
[7], along with several local estimates. These results are collected in the
next two theorems. The second theorem is a straightforward consequence of the
bounds established in Theorem IV.I of [7], with a slight modification to fit
our ergodic framework.

######  Theorem 3.5.

(Theorem VI.I in [7])) Let $1<q\leq 2$ and $g\in L^{\infty}(\Omega)$. The
equation $(\ref{udef})$ admits an explosive solution $(u,\rho)\in
W^{2,r}_{loc}(\Omega)\times\mathbb{R}$ for all $1<r<\infty$.

######  Theorem 3.6.

Let $1<q<2$, $g\in L^{\infty}(\Omega)$ and $(u,\rho)\in
W^{2,r}_{loc}(\Omega)\times\mathbb{R}$ be an explosive solution of
$(\ref{eq:hj})$ with $u(x_{0})=0$, for some fixed $x_{0}\in\Omega$ . There
exist positive constants $C_{1},C_{2},C_{3}$, depending only on $\Omega$ and
$||g||_{L^{\infty}(\Omega)}$, such that,

  1. (i)

$|\rho|\leq C_{1}$,

  2. (ii)

$|Du(x)|\leq C_{2}d(x)^{-1/(q-1)}$, and

  3. (iii)

$|u(x)|\leq C_{3}d(x)^{(q-2)/(q-1)}$.

In addition, for any $K\subset K^{\prime}$ compact subsets of $\Omega$, there
exists a constant $C_{4}=C_{4}(d(K,K^{\prime}),n,r)>0$ such that

  1. (iv)

$||u||_{W^{2,r}(K)}\leq
C_{4}(||u||_{L^{\infty}(K^{\prime})}+||g||_{{L^{q}}(K^{\prime})})$.

If $q=2$, the same estimates hold but $(iii)$ must be replaced by

  1. (iii)’

$|u(x)|\leq C_{3}\log(d(x))$.

###### Sketch of Proof of Theorem 3.6.

The first result follows from the construction of $\rho$ in Theorem VI.I of
[7] which shows that $\rho$ arises as the local uniform limit of $\lambda
u_{\lambda}$ as $\lambda\to 0$, where $u_{\lambda}$ is the unique viscosity
solution of

$\displaystyle\begin{cases}-\Delta u_{\lambda}+|Du_{\lambda}|^{q}+\lambda
u_{\lambda}&=g\text{ in }\Omega,\\\ \lim\limits_{d(x)\to
0}u_{\lambda}(x)=\infty.\end{cases}$

If $1<q<2$, for arbitrary $\epsilon>0$, $u_{\lambda}$ has as sub- and super-
solutions $(C_{q}\mp\epsilon)d^{(q-2)/(q-1)}\mp C_{\epsilon}/\lambda$, where
$C_{q}=(q-1)^{(q-2)/(q-1)}(2-q)^{-1}$ and $C_{\epsilon}$ is a function of
$||g||_{L^{\infty}(\Omega)}$. Similarly, for $q=2$, $u_{\lambda}$ has as sub-
and super-solutions $-(1\mp\epsilon)\log(d)\mp C_{\epsilon}/\lambda$, where
$C_{\epsilon}$ is a function of $||g||_{L^{\infty}(\Omega)}$. Sending
$\lambda\to 0$ yields the desired bound on $\rho$.

The second result $(ii)$ is proven in Theorem IV.I of [7] and $(iii)$ is an
immediate consequence of $(ii)$ as $u(x_{0})$ is fixed. The last interior
estimate follows from a bootstrap argument. ∎

In order to study the solution of the KFP equation near the boundary, it will
be necessary to know the precise asymptotics of $Du$. In [7], Lasry and Lions,
and later Porretta and Véron in [11], established the following results.

######  Theorem 3.7.

(Theorem I.1 and II.3 in [7]; Theorem 2.3 in [11]) Let $1<q<2$, $g\in
L^{\infty}(\Omega)$ and $(u,\rho)\in W^{2,r}_{loc}(\Omega)\times\mathbb{R}$ be
a solution of $(\ref{eq:hj})$. Then,

  1. (i)

$\lim\limits_{d(x)\to
0}\dfrac{u(x)}{d^{(q-2)/(q-1)}}=(q-1)^{(q-2)/(q-1)}(2-q)^{-1}$,

  2. and
  3. (ii)

$\lim\limits_{d(x)\to
0}\dfrac{Du(x)\cdot\nu(x)}{d^{-1/(q-1)}}=(q-1)^{-1/(q-1)}.$

The second result also holds for $q=2$, while the former must be replaced by

  1. (i)’

$\lim\limits_{d(x)\to 0}\dfrac{u(x)}{-\log(d(x))}=1$.

######  Remark.

Lemma 7.1 in the Appendix is a slightly stronger version of this theorem,
where the convergence is shown to be uniform in $||g||_{L^{\infty}(\Omega)}$.
This stronger version will be necessary in Section 5.

Lastly, we provide a stability result particular to the types of couplings we
are studying.

######  Theorem 3.8.

Assume $(F1)$ and fix $x_{0}\in\Omega$. Let $\mu_{n}$ be a sequence of
probability measures on $\Omega$ which converge in $L^{1}(\Omega)$ to $\mu$,
and $(u_{n},\rho_{n})\in W^{2,r}_{loc}(\Omega)\times\mathbb{R}$ be the unique
explosive solution of

$\displaystyle\begin{cases}-\Delta
u_{n}+|Du_{n}|^{q}+\rho_{n}=F(x;\mu_{n})\text{ in }\Omega,\\\
\lim\limits_{d(x)\to 0}u_{n}(x)=\infty,\end{cases}$

subject to the normalization condition $u_{n}(x_{0})=0$. Then there exists
$(u,\rho)\in W^{2,r}_{loc}(\Omega)\times\mathbb{R}$, such that
$\rho_{n}\to\rho$ and $u_{n}\to u$ locally uniformly as $n\to\infty$.
Moreover, $(u,\rho)$ is the unique explosive solution of

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=F(x;\mu)\text{ in
}\Omega,\\\ \lim\limits_{d(x)\to 0}u(x)=\infty,u(x_{0})=0.\end{cases}$ (3.11)

###### Proof.

It follows from $(F1)$ that $F(x;\mu_{n})$ and $F(x;\mu)$ are bounded in
$L^{\infty}(\Omega)$. Theorem 3.6 yields that $\rho_{n}$ is bounded and hence
converges along a subsequence to a constant $\rho$. The sequence $u_{n}$ is
bounded in $W_{loc}^{2,r}(\Omega)$ so, by Arzelà-Ascoli, we can extract a
subsequence converging locally uniformly to $u\in W^{2,r}_{loc}(\Omega)$ such
that $Du_{n}\to Du$ locally uniformly. By pointwise convergence, $u(x_{0})=0$.

That $u(x)\to\infty$ as $d(x)\to 0$ follows from the fact that the sequence of
equations have a common explosive subsolution. When $1<q<2$, one such
subsolution is of the form $(C_{q}-\epsilon)d(x)^{(q-2)/(q-1)}-C_{\epsilon}$,
where $C_{q}=(q-1)^{(q-2)/(q-1)}(2-q)^{-1}$, $\epsilon>0$, and
$C_{\epsilon}=C_{\epsilon}(||F(x;\mu)||_{L^{\infty}(\Omega;L^{1}(\Omega))},\epsilon,C_{q})$.
When $q=2$ the corresponding subsolution is $-(1-\epsilon)\log(d(x))$.

Lastly, in view of $(F1)$, $F(x;\mu_{n})$ converges uniformly to $F(x;\mu)$.
It follows from classic viscosity theory that $(u,\rho)$ is a viscosity
solution of $(\ref{ustab})$. As the explosive solution satisfying $u(x_{0})=0$
is unique, the limit is indepedent of the subsequence. ∎

###  3.3 Definition and Main Result

We finish this section with the definition of a solution to the system
$(\ref{eq:mainintro})$ and the main existence and uniqueness result stated in
the introduction.

######  Definition 3.4.

A triplet $(u,\rho,m)\in W^{2,r}_{loc}(\Omega)\times\mathbb{R}\times
W^{1,r}_{loc}(\Omega)$ is a _solution_ of $(\ref{eq:mainintro})$ if $(u,\rho)$
is an explosive solution of

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=F(x;m)\text{ in }\Omega,\\\
\lim\limits_{d(x)\to 0}u(x)=\infty,\end{cases}$

and $m$ is a proper weak solution of

$\displaystyle\Delta m+div\left(mq|Du|^{q-2}Du\right)=0\text{ in }\Omega.$

We are now able to state the main result of the paper.

######  Theorem 3.9.

Assume that $1<q\leq 2$ and either $(F1)$ or $(F2)$ holds. Then
$(\ref{eq:mainintro})$ has a solution for all $r>1$. Moreover, if $(F3)$ also
holds, then the solution is unique.

##  4 Fixed point for the non-local case

In this section we will use Schaefer’s fixed point theorem in
$\mathcal{W}_{\gamma}(\Omega)$ to prove the existence of a solution to
$(\ref{eq:mainintro})$ when $(F1)$ holds.

Let $2<\gamma<(2q-1)/(q-1)$. Define $T_{1}:\mathcal{W}_{\gamma}(\Omega)\to
W^{1,r}_{loc}(\Omega)$ to be the map that sends measures $\mu$ to
$q|Du|^{q-2}Du$, where $(u,\rho)$ is any explosive solution of

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=F(x;{\mu})\text{ in
}\Omega,\\\ \lim\limits_{d(x)\to 0}u(x)=\infty.\end{cases}$ (4.1)

It is immediate from Theorem 3.5 that $T_{1}$ is well defined as explosive
solutions are unique up to a constant.

Define
$T_{2}:T_{1}(\mathcal{W}_{\gamma}(\Omega))\to\mathcal{W}_{\gamma}(\Omega)$ to
be the map that sends $b$ in the image of $T_{1}$ to the proper weak solution
$m$ of the associated continuity equation

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega.$ (4.2)

In view of Lemma 3.4 and Theorem 3.7, $(\ref{eq:fpb})$ has a unique proper
weak solution $m$ for $b$ in the image of $T_{1}$.

Finally define $T=T_{2}\circ T_{1}$. In the next result, we apply Schaefer’s
fixed point theorem to $T$, thereby obtaining a solution of the MFG system
$(\ref{eq:mainintro})$.

######  Theorem 4.1.

For $F$ satisfying $(F1)$ and $2<\gamma<(2q-1)/(q-1)$,
$T:\mathcal{W}_{\gamma}(\Omega)\to\mathcal{W}_{\gamma}(\Omega)$ has a fixed
point.

###### Proof.

Recall that to apply Schaefer’s fixed point theorem, we need to show that $T$
is continuous, $T$ maps bounded sets to precompact sets, and if $F\subset
W_{\gamma}$ is the set defined by

$\displaystyle F=\\{m\in\mathcal{W}_{\gamma}(\Omega)|m=\lambda T(m)\text{ for
some }0\leq\lambda\leq 1\\},$ (4.3)

then $F$ is bounded.

We start by showing that $T$ is continuous. Consider a sequence $\mu_{n}$ that
converges to $\mu$ in $\mathcal{W}_{\gamma}(\Omega)$. To simplify the
notation, define $b_{n}:=T_{1}(\mu_{n})$, $b:=T_{1}(\mu)$, and
$m_{n}:=T_{2}(b_{n})$. We first show that $b_{n}\to b$ locally uniformly. We
then show that the sequence $m_{n}$ converges in
$\mathcal{W}_{\gamma}(\Omega)$ to a measure $m$ and prove that
$m=T_{2}(b)=T(\mu)$.

Fix $x_{0}\in\Omega$ and let $(u_{n},\rho_{n})$ be the unique explosive
solution of $(\ref{eq:hjmu})$ that corresponds to $F(x;{\mu}_{n})$ and
satisfies $u_{n}(x_{0})=0$. Note that $T_{1}(\mu_{n})=q|Du_{n}|^{q-2}Du_{n}$
by the remark following $(\ref{eq:hjmu})$. Since $\mu_{n}$ converges to $\mu$
in $L^{1}(\Omega)$, the local uniform convergence of $T_{1}(\mu_{n})$ to
$T_{1}(\mu)$ follows from Theorem 3.8.

In view of the local uniform bounds on $b_{n}$, Theorem 3.1 implies the
sequence of measures $m_{n}$ are uniformly bounded in $W^{1,r}_{loc}(\Omega)$
for all $r>1$ and, hence, locally uniformly bounded in $C^{0,\alpha}(\Omega)$
for $\alpha<1-n/r$. Along subsequences, the measures $m_{n}$ converge locally
uniformly to a measure $m$. Consequently, we can pass to the limit in the weak
formulation of $m_{n}$ and obtain that $m$ is a non-negative measure that
satisfies the KFP equation with drift $b=T_{1}(\mu)$. It remains to show that
$m_{n}$ converges to $m$ in $\mathcal{W}_{\gamma}(\Omega)$.

Lemma 3.4 yields that if $(2q-1)/(q-1)>\gamma^{\prime}>\gamma$, then the
sequence $m_{n}$ is bounded in $\mathcal{W}_{\gamma^{\prime}}(\Omega)$. This
implies that the sequence $d^{-\gamma}m_{n}$ is uniformly tight. Indeed, for
some constant $C$, independent of the choice of $\gamma$ and
$\gamma^{\prime}$,

$\displaystyle C$
$\displaystyle\geq\int_{\Omega}d^{-\gamma^{\prime}}m_{n}\geq\delta^{-\gamma^{\prime}+\gamma}\left(\int_{\Omega\backslash\Omega_{\delta}}d^{-\gamma}m_{n}\right)$

It then follows from the bound

$\displaystyle\int_{\Omega}d^{-\gamma}|m_{n}-m|\leq\int_{\Omega_{\delta}}d^{-\gamma}|m_{n}-m|+\int_{\Omega\backslash\Omega_{\delta}}d^{-\gamma}m_{n}+\int_{\Omega\backslash\Omega_{\delta}}d^{-\gamma}m,$

that $||m_{n}-m||_{\gamma,\Omega}\to 0$ as $n\to\infty$.

Since convergence in $\mathcal{W}_{\gamma}(\Omega)$ implies convergence in
$L^{1}(\Omega)$, this also proves that $\int_{\Omega}m=1$. The uniqueness of
$m$ then yields the convergence of the full sequence $m_{n}=T(\mu_{n})$ to
$T(\mu)$.

Next we show that $T$ maps bounded sets to precompact ones. Indeed let
$\mu_{n}$ be a bounded sequence in $\mathcal{W}_{\gamma}(\Omega)$, and
consider $u_{n}$, $b_{n}$, and $m_{n}$ defined as before. It suffices to show
that there is a convergent subsequence of $m_{n}$. The sequence $\mu_{n}$ is
bounded in $L^{1}(\Omega)$ and $F(x;\mu_{n})$ is bounded in
$L^{\infty}(\Omega;L^{1}(\Omega))$. Moreover, Theorem 3.6 yields that $b_{n}$
is locally uniformly bounded. Finally, Lemma 3.1 implies that $m_{n}$ is
uniformly bounded in $C^{0,\alpha}_{loc}(\Omega)$. Passing to subsequences,
the measures $m_{n}$ converge locally uniformly to a measure $m$. Proceeding
as in the continuity argument, we find that the sequence $d^{-\gamma}m_{n}$ is
uniformly tight and $||m_{n}-m||_{\gamma,\Omega}\to 0$.

It remains to check that the set $F$ defined above is bounded. To see this,
first observe that if $m=\lambda T(m)$, then $\int_{\Omega}m=\lambda\leq 1$.
Hence $||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$ is bounded by assumption
$(F1)$. Next we apply Theorem 3.6 to find that $q|Du|^{q-2}Du$ is bounded on
$\Omega_{\delta}$ by a constant
$C(\delta,||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))})$. Lemma 3.4 then
implies that $||m||_{\gamma,\Omega}$ is bounded. ∎

##  5 Fixed point argument for the local setting

In this section we will show existence of solutions to $(\ref{eq:mainintro})$
under assumption $(F2)$ by considering the system $(\ref{eq:deltap})$,
introduced in Section 1, which approximates $(\ref{eq:mainintro})$. First, we
will use Schaefer’s fixed point theorem to show the system $(\ref{eq:deltap})$
has a solution. Then, using the uniform tightness results of Lemma 3.3 and the
local uniform bounds of Theorem 3.6, we will show that solutions of
$(\ref{eq:deltap})$ converge locally uniformly to solutions of
$(\ref{eq:mainintro})$.

###  5.1 The approximate problem

We begin by making precise the notion of solution to the approximating system
$(\ref{eq:deltap})$. We then show the existence of a solution using a fixed
point theorem argument similar to that used in the non-local setting.

To utilize the asymptotic behavior of the explosive solutions to HJB equations
at the boundary of $\Omega$ as well as the $W_{loc}^{1,r}$ bounds of Neumann
weak solutions to KFP equations on $\Omega_{\delta}$, the KFP equation of
$(\ref{eq:deltap})$ is solved in the subdomain $\Omega_{\delta}$ and the HJB
equation is solved in $\Omega$.

######  Definition 5.1.

A triplet $(u_{\delta},\rho_{\delta},m_{\delta})\in
W^{2,r}_{loc}(\Omega)\times\mathbb{R}\times W^{1,r}(\Omega_{\delta})$ is a
_solution_ of $(\ref{eq:deltap})$ if $(u_{\delta},\rho_{\delta})$ is an
explosive solution of

$\displaystyle\begin{cases}-\Delta
u_{\delta}+|Du_{\delta}|^{q}+\rho_{\delta}=F(x;{m}_{\delta})\text{ in
}\Omega,\\\ \lim\limits_{d(x)\to 0}u_{\delta}(x)=\infty,\end{cases}$

and $m_{\delta}$ is a Neumann weak solution of

$\displaystyle\Delta
m_{\delta}+div\left(m_{\delta}q|Du_{\delta}|^{q-2}Du_{\delta}\right)=0\text{
in }\Omega_{\delta}.$

When $F$ satisfies $(F2)$, existence of a solution to this system follows from
Schaefer’s fixed point theorem.

Indeed, let $T_{1}:C^{0,\alpha}(\Omega_{\delta})\to W^{1,r}_{loc}(\Omega)$ be
the map that sends measures $\mu$ to $q|Du|^{q-2}Du$, where $(u,\rho)$ is the
explosive solution of

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=F(x;{\mu})\text{ in
}\Omega\\\ \lim\limits_{d(x)\to 0}u(x)=\infty,\end{cases}$

which is well defined for all $\alpha>0$ and $1<r<\infty$. It follows from
Theorem 3.5 that the system has a solution $u\in W^{2,r}_{loc}(\Omega)$ for
every $r>1$, and $Du$ is unique.

Let $T_{2}:W^{1,r}_{loc}(\Omega)\to C^{0,\alpha}(\Omega_{\delta})$ be the map
that sends $b$ to the Neumann weak solution $m_{\delta}$ of

$\displaystyle\begin{cases}\Delta
m_{\delta}+div(m_{\delta}b_{\delta})=0&\text{ in }\Omega_{\delta}\\\
(Dm_{\delta}+m_{\delta}b_{\delta})\cdot\nu=0&\text{ on
}\partial\Omega_{\delta},\end{cases}$

where $b_{\delta}$ is the restriction of $b$ to the domain $\Omega_{\delta}$,
which is also well defined. In view of Theorem 3.5, there is a unique Neumann
weak solution $m_{\delta}$. Moreover, $m_{\delta}$ is in
$W^{1,s}(\Omega_{\delta})$ for all $1<s<\infty$ and, by Sobelov embedding, is
in $C^{0,\alpha}(\Omega_{\delta})$ for all $0<\alpha<1$. Composing the two, we
define $T=T_{2}\circ T_{1}$.

######  Theorem 5.1.

For $F$ satisfying $(F2)$ and $\alpha>0$, $T:C^{0,\alpha}(\Omega_{\delta})\to
C^{0,\alpha}(\Omega_{\delta})$ has a fixed point.

###### Proof.

We apply Schaefer’s fixed point theorem as in Theorem 4.1. As we will be
working with an HJB equation set in $\Omega$ and a KFP equation set in
$\Omega_{\delta}$, we will be exploiting that the method for extending
measures from $\Omega_{\delta}$ to $\Omega$ defined in $(\ref{tilde})$ is
$C^{0,\alpha}$-norm preserving.

To show the continuity of $T$, we consider a sequence $\mu_{n}$ which
converges to $\mu$ in $C^{0,\alpha}(\Omega_{\delta})$, and we start by showing
that $T_{1}(\mu_{n})$ converges to $T_{1}(\mu)$ locally uniformly. To simplify
notation, we let $b_{n}:=T_{1}(\mu_{n})$, $b:=T_{1}(\mu)$, and
$m_{n}:=T_{2}(b_{n})$.

Let $(u_{n},\rho_{n})$ be the unique explosive solution of $(\ref{eq:hj})$
that corresponds to $F(x;{\mu}_{n})$ and satisfies $u_{n}(x_{0})=0$, where
$x_{0}\in\Omega$ is fixed. As the sequence $\mu_{n}$ converges to $\mu$ in
$C^{0,\alpha}(\Omega_{\delta})$, $\mu_{n}$ converges uniformly to $\mu$ in
$\Omega_{\delta}$, and as a result $\tilde{\mu}_{n}$ converges uniformly to
$\tilde{\mu}$ in $\Omega$, where $\tilde{\mu}_{n}$ and $\tilde{\mu}$ are
defined as in $(\ref{tilde})$.

Due to the stability properties of viscosity solutions, to prove continuity of
$T_{1}$ under assumption $(F2)$, it is sufficient to show that if $x_{n}\to x$
in $\Omega$, then $f(\tilde{\mu}_{n}(x_{n}))\to f(\tilde{\mu}(x))$. However,
this follows immediately from the continuity of $f$ and the equicontinuity of
$\tilde{\mu}_{n}$.

Next we show that $T_{2}$ is continuous. Since
$||b_{n}||_{L_{loc}^{\infty}(\Omega)}$ is bounded in $n$, $m_{n}$ is uniformly
bounded in $W^{1,r}(\Omega_{\delta})$ and, hence, bounded in
$C^{0,\beta}(\Omega_{\delta})$ for $\alpha<\beta<1-n/r$. In view of the
compact embedding of $C^{0,\beta}(\Omega_{\delta})$ in
$C^{0,\alpha}(\Omega_{\delta})$, there is a subsequence of $m_{n}$ that
converges in $C^{0,\alpha}(\Omega_{\delta})$ to some $m\in
C^{0,\alpha}(\Omega_{\delta})$. Moreover, $Dm_{n}$ converges weakly to $Dm$.
Consequently, we can pass to the limit in the weak formulation of $m_{n}$ and
obtain that $m\in W^{1,r}(\Omega_{\delta})$ is a positive probability measure
that satisfies the KFP equation with drift $b=T_{1}(\mu)$. The uniqueness of
$m$ yields convergence of $T(\mu_{n})$ to $T(\mu)$ along the full sequence.

Next, to show that $T$ is precompact, we will use the bounds in Theorem 3.6,
Lemma 3.1, and the Sobelov embedding theorem.

For $\mu\in C^{0,\alpha}(\Omega_{\delta})$ and $b=T_{1}(\mu)$, Theorem 3.6
yields $|b(x)|\leq Cd(x)^{-1}$, where $C$ depends on $\Omega$ and
$||F(x;{\mu})||_{L^{\infty}(\Omega;L^{1}(\Omega))}$. Note that by assumption
$(F2)$, $||b||_{L^{\infty}(\Omega_{\delta})}$ is bounded by a constant
depending on $||f||_{{L^{\infty}}(\mathbb{R})}$.

Lemma 3.1 yields that $||T(\mu)||_{W^{1,r}(\Omega_{\delta})}$ is bounded by a
constant depending on $||f||_{L^{\infty}(\mathbb{R})}$. Since
$W^{1,r}(\Omega_{\delta})$ embeds compactly in $C^{0,\alpha}(\Omega_{\delta})$
for $\alpha<1-n/r$, choosing appropriately large $r$, $T$ sends bounded sets
of $C^{0,\alpha}(\Omega_{\delta})$ to precompact ones.

It remains to check that if $F$ is the set defined by

$\displaystyle F=\\{m\in C^{0,\alpha}(\Omega_{\delta})|m=\lambda T(m)\text{
for some }0\leq\lambda\leq 1\\},$

then $F$ is bounded. This follows from Theorem 3.1 as $q|Du|^{q-2}Du$ is
bounded in $L^{\infty}(\Omega_{\delta})$.

∎

###  5.2 Limit of the approximate problem

In this section we prove that in the local setting, the solutions
$(u_{\delta},\rho,m_{\delta})$ of $(\ref{eq:deltap})$ converge locally
uniformly to solutions of $(\ref{eq:mainintro})$. We begin by showing that,
along subsequences, $u_{\delta}$ has a local uniform limit $u$ using the
Arzelà-Ascoli Theorem. We then construct a Lyapanov function and use Lemma 3.2
and Lemma 3.3 to show that the measures $m_{\delta}$ converge locally
uniformly to a proper weak solution $m$ of a continuity equation. Lastly we
show that $u$ is the explosive solution corresponding to $f(m(x))$.

######  Theorem 5.2.

Given $\delta>0$ and $F$ satisfying $(F2)$, let $x_{0}\in\Omega$, and
$(u_{\delta},\rho_{\delta},m_{\delta})\in
W_{loc}^{2,r}(\Omega)\times\mathbb{R}\times W^{1,r}(\Omega_{\delta})$ be a
solution of $(\ref{eq:deltap})$ satisfying $u_{\delta}(x_{0})=0$. Along
subsequences, $\rho_{\delta}$ converges to a constant $\rho$ and $u_{\delta}$
converges in $C^{1}_{loc}(\Omega)$ to $u\in W_{loc}^{2,r}(\Omega)$.

###### Proof.

It follows from Theorem 3.6, and the uniform in $\delta$ bounds on
$||F(x;m_{\delta})||_{L^{\infty}(\Omega;L^{1}(\Omega))}$, that $\rho_{\delta}$
and the $W^{1,r}(\Omega_{\delta})$ bounds of $u_{\delta}$ are bounded
uniformly in $\delta$. The desired convergence along subsequences follows by
the Arzelà-Ascoli Theorem. ∎

For now we make no claims about the equation $u$ solves. We must first make
sense of the limit of the measures $m_{\delta}$.

######  Theorem 5.3.

Given $\delta>0$ and $F$ satisfying $(F2)$, let
$(u_{\delta},\rho_{\delta},m_{\delta})\in
W_{loc}^{2,r}(\Omega)\times\mathbb{R}\times W^{1,r}(\Omega_{\delta})$ be a
solution of $(\ref{eq:deltap})$. Along subsequences, $m_{\delta}$ converges
locally uniformly to $m\in W^{1,r}_{loc}(\Omega)$ which is a proper weak
solution of

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega,$

where $b=q|Du|^{q-2}Du$ and $u$ is the limit function from Theorem 5.2.

###### Proof.

In view of Lemma 3.2 and Lemma 3.3, it suffices to construct a function $V$
satisfying $(\ref{v1})$ and $(\ref{v2})$ for
$b_{\delta}=q|Du_{\delta}|^{q-2}Du_{\delta}$.

Consider $V(x)=d(x)^{-C+1+\epsilon}$, where $C=q/(q-1)$ and $0<\epsilon<C-1$,
which satisfies the desired growth condition $(\ref{v1})$ at the boundary. The
$L^{\infty}(\Omega;L^{1}(\Omega))$-norm of the coupling term
$F(x;{m}_{\delta})$ is uniformly bounded in $\delta$ by
$||f||_{L^{\infty}(\mathbb{R})}$. By Lemma 7.1,
$b_{\delta}(x)\cdot\nu(x)d(x)\to C$ uniformly in $\delta$ as $d(x)\to 0$. It
follows that the limit $\Delta V-b_{\delta}\cdot DV\to-\infty$ at the boundary
is likewise uniform in $\delta$, and $(\ref{v2})$ is satisfied. The uniform
tightness of the measures $m_{\delta}$ ensures that $\int_{\Omega}m=1$ and
$m_{\delta}$ converges to $m$ in $L^{1}(\Omega)$. ∎

We are now ready to state the equation that $u$ solves.

######  Theorem 5.4.

Let $F$ satisfy $(F2)$ and $(u_{\delta},\rho_{\delta},m_{\delta})\in
W_{loc}^{2,r}(\Omega)\times\mathbb{R}\times W^{1,r}(\Omega_{\delta})$ be a
solution of $(\ref{eq:deltap})$. Let $u$, $m$ and $\rho$ be as constructed in
Theorems 5.1 and 5.2. Then $(u,\rho)$ is the explosive solution of

$\displaystyle\begin{cases}-\Delta u+|Du|^{q}+\rho=F(x;m)\text{ in }\Omega,\\\
\lim\limits_{d(x)\to 0}u(x)=\infty.\end{cases}$

###### Proof.

The proof is very similar to the stability argument in Theorem 5.1, and
follows from the local uniform Hölder bounds on $m_{\delta}$. Indeed, consider
a sequence $x_{\delta}\to x$ contained in a compact set $K$, and let
$M=\sup_{\delta}||m_{\delta}||_{C^{0,\alpha}(K)}$. Then,

$\displaystyle|m_{\delta}(x_{\delta})-m(x)|$
$\displaystyle\leq|m_{\delta}(x)-m(x)|+|m_{\delta}(x_{\delta})-m_{\delta}(x)|$
$\displaystyle\leq|m_{\delta}(x)-m(x)|+M|x_{\delta}-x|^{\alpha}.$ (5.1)

By the uniform convergence of $m_{\delta}$ to $m$ on $K$,
$(\ref{eq:expression})$ goes to $0$ as $\delta\to 0$. The continuity of $f$
allows us to pass to the limit in the HJB equation, obtaining that $u$ is a
viscosity solution of

$\displaystyle-\Delta u+|Du|^{q}+\rho=F(x;m)\text{ in }\Omega.$

The argument that $u$ blows up at the boundary is the same as in Theorem 3.8.
∎

##  6 Uniqueness

We are now ready to address the uniqueness claim in Theorem 3.9. Throughout
the discussion we take $1<q<2$ and deal with the case $q=2$ at the end of the
section.

###### Proof of Theorem 3.9.

It only remains to show that uniqueness holds when $F$ satisfies $(F3)$. Here
we follow the classical Lasry-Lions uniqueness argument. Consider two
solutions $(u_{1},\rho_{1},m_{1}),(u_{2},\rho_{2},m_{2})\in
W^{2,r}_{loc}(\Omega)\times\mathbb{R}\times W^{1,r}_{loc}(\Omega)$ of
$(\ref{eq:mainintro})$. Without loss of generality, we may assume that
$u_{1}(x_{0})=u_{2}(x_{0})=0$ for a fixed $x_{0}\in\Omega$.

Letting $\overline{u}:=u_{1}-u_{2}$, $\overline{\rho}:=\rho_{1}-\rho_{2}$, and
$\overline{m}:=m_{1}-m_{2}$, it follows that
$(\overline{u},\overline{\rho},\overline{m})$ satisfies the system

$\displaystyle\begin{cases}-\Delta\overline{u}+|Du_{1}|^{q}-|Du_{2}|^{q}+\overline{\rho}=F(x;m_{1})-F(x;m_{2})&\text{
in }\Omega,\\\ \Delta\overline{m}+div(m_{1}b_{1}-m_{2}b_{2})=0&\text{ in
}\Omega,\end{cases}$

where $b_{1}:=q|Du_{1}|^{q-2}Du_{1}$ and $b_{2}:=q|Du_{2}|^{q-2}Du_{2}$.

Consider a non-negative and compactly supported function $\phi\in
C^{\infty}(\Omega)$; the particular choice of $\phi$ will be made later.

Multiplying the first equation by $\overline{m}\phi$ and integrating by parts,
we obtain

$\displaystyle\int_{\Omega}\left[(D\overline{u})\cdot
D(\overline{m}\phi)+(|Du_{1}|^{q}-|Du_{1}|^{q})\overline{m}\phi+(\overline{\rho})\overline{m}\phi\right]=\int_{\Omega}(F(x;m_{1})-F(x;m_{2}))\overline{m}\phi.$

Similarly, using $\overline{u}\phi$ as a test function in the KFP equation, we
obtain

$\displaystyle\int_{\Omega}\left[-D(\overline{u}\phi)\cdot(D\overline{m})-(m_{1}b_{1}-m_{2}b_{2})\cdot
D(\overline{u}\phi)\right]=0.$

Adding the equations and regrouping gives:

$\displaystyle\int_{\Omega}(F(x;m_{1})-F(x;m_{2}))\overline{m}\phi$
$\displaystyle=\int_{\Omega}\phi\left[(|Du_{1}|^{q}-|Du_{1}|^{q})\overline{m}-(m_{1}b_{1}-m_{2}b_{2})\cdot
D(\overline{u})\right]$
$\displaystyle+\int_{\Omega}(\overline{\rho})\overline{m}\phi$
$\displaystyle-\int_{\Omega}(D\phi)\cdot[(m_{1}b_{1}-m_{2}b_{2})\overline{u}]$
$\displaystyle+\int_{\Omega}(D\phi)\cdot(\overline{m}D\overline{u}-\overline{u}D\overline{m})$
$\displaystyle=\int_{\Omega}\phi m_{1}(|Du_{1}|^{q}-|Du_{2}|^{q}-b_{1}\cdot
D(u_{1}-u_{2}))$ (6.1) $\displaystyle+\int_{\Omega}\phi
m_{2}(|Du_{2}|^{q}-|Du_{1}|^{q}-b_{2}\cdot D(u_{2}-u_{1}))$ (6.2)
$\displaystyle+\int_{\Omega}(\overline{\rho})\overline{m}\phi$
$\displaystyle-\int_{\Omega}(D\phi)\cdot[(m_{1}b_{1}-m_{2}b_{2})\overline{u}+2\overline{m}D\overline{u}].$
$\displaystyle+\int_{\Omega}\overline{m}\Delta\phi\overline{u}$

The left-hand side is positive by the monotonicity assumption $(F3)$, and
$(\ref{con1})$ and $(\ref{con2})$ are negative by the convexity of the
Hamiltonian. We will show that the remaining terms on the right hand side can
be made arbitrarily small for the right choice of $\phi$.

We are now ready to define $\phi$. Let $\psi(s):[0,\infty)\to[0,1]$ be a
smooth function that is $0$ in a neighborhood of $[0,1]$, $1$ in a
neighborhood of $2$, and constant for $s\geq 2$. For
$\delta<\min(1,\epsilon_{0})/2$, let $\phi(x)=\psi(d(x)/\delta)$. There exists
a constant $C_{1}=C_{1}(\Omega,\psi)>0$ such that, in
$\Omega_{\delta}\backslash\Omega_{2\delta}$,

$\displaystyle|D\phi|\leq\dfrac{1}{\delta}||\psi^{\prime}||_{L^{\infty}([0,\infty))}||Dd||_{L^{\infty}(\Omega_{\delta}\backslash\Omega_{2\delta})}\leq
C_{1}\dfrac{1}{\delta}.$ (6.3)

and

$\displaystyle|\Delta\phi|\leq\dfrac{1}{\delta^{2}}||\psi^{\prime\prime}||_{L^{\infty}([0,\infty))}||Dd||_{L^{\infty}(\Omega_{\delta}\backslash\Omega_{2\delta})}^{2}+\dfrac{1}{\delta}||\psi^{\prime}||_{L^{\infty}([0,\infty))}||\Delta
d||_{L^{\infty}(\Omega_{\delta}\backslash\Omega_{2\delta})}\leq
C_{1}\dfrac{1}{\delta^{2}}.$

Together with the local bounds on $|u_{i}|$ and $|Du_{i}|$ from Theorem 3.6,
it follows from $(\ref{eq:dpsi})$ that there exists a constant $C_{2}>0$,
independent of $\delta$, for which

$\displaystyle\left|\int_{\Omega}(D\phi)\cdot[(m_{1}b_{1}-m_{2}b_{2})\overline{u}+2\overline{m}D\overline{u}]\right|$
$\displaystyle\leq
C_{2}\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}\dfrac{1}{\delta}(d^{-1/(q-1)})^{q-1}(m_{1}+m_{2})d^{(q-2)/(q-1)}$
$\displaystyle+C_{2}\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}\dfrac{1}{\delta}d^{-1/(q-1)}|\overline{m}|$
$\displaystyle\leq
2C_{2}\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}d^{-q/(q-1)}(m_{1}+m_{2}+|\overline{m}|).$
(6.4)

Similarly there exists a constant $C_{3}>0$, independent of $\delta$, for
which

$\displaystyle\left|\int_{\Omega}(\Delta\phi)(\overline{u})\overline{m}\right|$
$\displaystyle\leq
C_{3}\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}\dfrac{1}{\delta^{2}}(m_{1}+m_{2})d^{(q-2)/(q-1)}$
(6.5) $\displaystyle\leq
4C_{3}\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}d^{-q/(q-1)}(m_{1}+m_{2}).$
(6.6)

We recall from Lemma 3.4 that, for small $\epsilon$,
$\int_{\Omega}d(x)^{-q/(q-1)-1+\epsilon}m(x)$ is finite. It follows that
$(\ref{b2})$ and $(\ref{b3})$ can be made arbitrarily small for $\delta$ small
enough.

To bound the remaining term, we use that $m_{1}+m_{2}\in L^{1}(\Omega)$ and
$\int_{\Omega}(m_{1}-m_{2})=0$, obtaining

$\displaystyle\left|\int_{\Omega}(\overline{\rho})\overline{m}\phi\right|$
$\displaystyle\leq|\overline{\rho}|\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}|m_{1}-m_{2}|+|\overline{\rho}|\left|\int_{\Omega_{2\delta}}m_{1}-m_{2}\right|$
$\displaystyle\leq|\overline{\rho}|\left(\int_{\Omega\backslash\Omega_{2\delta}}(m_{1}+m_{2})+\left|0-\int_{\Omega\backslash\Omega_{2\delta}}(m_{1}-m_{2})\right|\right)$
$\displaystyle\leq
2|\overline{\rho}|\int_{\Omega\backslash\Omega_{2\delta}}(m_{1}+m_{2}),$

which can also be made arbitrarily small for $\delta$ small enough.

It follows from the monotonicity of $F(x;m)$, that $m_{1}=m_{2}$. That
$u_{1}=u_{2}$ and $\rho_{1}=\rho_{2}$ follows from the uniqueness of solutions
to $(\ref{eq:hj})$. ∎

######  Remark.

To prove uniqueness of solutions to $(\ref{eq:mainintro})$ when $q=2$,
$(\ref{b2})$ should be replaced by an estimate of the form

$\displaystyle
C\int_{\Omega_{\delta}\backslash\Omega_{2\delta}}d^{-1}(-\log(d))(m_{1}+m_{2}),$
(6.7)

where $C>0$ is a constant independent of $\delta$. As $d^{-1}(-\log(d))$ has
growth slower than $d^{-2+\epsilon}$, the rest of the proof then proceeds as
in the $1<q<2$ case.

##  7 Appendix

In this section we prove Lemma 7.1, a stronger version of $(ii)$ in Theorem
3.7, and Lemma 3.4.

######  Lemma 7.1.

Let $m\in W^{1,r}_{loc}(\Omega)$ and consider the unique solution
$(u_{m},\rho_{m})\in W^{2,r}_{loc}(\Omega)\times\mathbb{R}$ of

$\displaystyle\begin{cases}-\Delta u_{m}+|Du_{m}|^{q}+\rho_{m}=F(x;m)\text{ in
}\Omega,\\\ \lim\limits_{d(x)\to 0}u_{m}(x)=\infty.\end{cases}$ (7.1)

Let $b_{m}:=q|Du_{m}|^{q-2}Du_{m}$. Then, uniformly in
$||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$,

$\lim_{d(x)\to 0}(b_{m}(x)\cdot\nu(x))d(x)=\dfrac{q}{q-1}.$

It will be enough to show uniform convergence of
$(Du_{m}(x)\cdot\nu(x))d(x)^{1/(q-1)}$ as $d(x)\to 0$.
We prove this by following the proof of Theorem 2.3 in [11] and tracking how
the convergence depends on $||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$.

In the proof that follows, we argue first when $1<q<2$, and discuss the $q=2$
case at the end of the proof. The same arguments also hold for Hamiltonians
satisfying $(\ref{eq:gen})$ locally uniformly in $p$ as $\delta\to 0$.

###### Proof of Lemma 7.1.

Since $\Omega$ has a uniform $C^{2}$ boundary, by Theorem 3.7 there exists a
$\delta_{0}>0$, depending on $||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$,
such that for every $x\in\partial\Omega$, the ball
$B_{\delta_{0}}(x-\delta_{0}\nu(x))$ is contained in $\Omega$, and, for
$d(x)<\delta_{0}$ ,

$\displaystyle\left|\dfrac{u_{m}(x)}{d(x)^{(q-2)/(q-1)}}-C_{q}\right|\leq\dfrac{C_{q}}{2},$
(7.2)

where $C_{q}=(q-1)^{(q-2)/(q-1)}(2-q)^{-1}$.

We begin by fixing $x_{0}\in\partial\Omega$ and a system of coordinates
$(\eta_{1},\ldots,\eta_{n})$ centered at $x_{0}$ whose $\eta_{1}$-axis is the
inner normal direction. Although we have fixed an $x_{0}$, any bounds and
rates that follow will not depend on $x_{0}\in\partial\Omega$.

Let $O_{\delta}=(\delta,\ldots,0)$ and consider the domain

$\displaystyle D_{\delta}=B(O_{\delta},\delta^{1-\sigma})\cap
B((\delta_{0},0,\ldots,0),\delta_{0}-\delta)\text{ , }\sigma\in(0,1/2).$

Applying the change of variable $\zeta=(\eta-O_{\delta})/\delta$, the domain
$\tilde{D}_{\delta}:=\\{(\eta-O_{\delta})/\delta:\eta\in D_{\delta}\\}$
converges to the half space $\\{\zeta:\zeta_{1}>0\\}$, where $\zeta_{1}$ is
the component of the vector in the $-\nu(x)$ direction. Under this change of
variable we have the bounds

$\displaystyle(\zeta_{1}+1)\delta\leq
d(\eta)\leq(\zeta_{1}+1)\delta+O(\delta^{2-2\sigma}).$ (7.3)

It is now possible to introduce, for $\delta<\delta_{0}$, the blow up function

$\displaystyle
v_{\delta,m}(\zeta)=\dfrac{u_{m}(\delta\zeta+O_{\delta})}{\delta^{(q-2)/(q-1)}},$

with domain $\tilde{D}_{\delta}$, which, in view of $(\ref{d_0})$, is
uniformly bounded in $\delta,||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))},$
and $\zeta$.

Straightforward computations show that $v_{\delta,m}$ describes the blow-up of
$u_{m}$ near the boundary. Indeed, for $\eta_{1}=\delta\zeta_{1}+O_{\delta}$,

$\displaystyle
Dv_{\delta,m}(\zeta_{1})=\dfrac{Du_{m}(\delta\zeta_{1}+O_{\delta})}{C_{q}\delta^{-1/(q-1)}}=\dfrac{Du_{m}(\eta_{1})}{C_{q}d(\eta_{1})^{-1/(q-1)}}(1+\zeta_{1})^{-1/(q-1)}.$

In $\tilde{D}_{\delta}$, $v_{\delta,m}$ solves, in the viscosity sense,

$\displaystyle-\Delta
v_{\delta,m}+\dfrac{\rho_{m}\delta^{q/(q-1)}}{C_{q}}+C_{q}^{q-1}|Dv_{\delta,m}|^{q}=\dfrac{F(\delta\zeta+O_{\delta};m)\delta^{q/(q-1)}}{C_{q}}.$
(7.4)

Moreover, $Dv_{\delta,m}(\zeta)$ satisfies the estimate

$\displaystyle|Dv_{\delta,m}(\zeta)|=\dfrac{|Du_{m}(\delta\zeta+O_{\delta})|}{C_{q}\delta^{-1/(q-1)}}\leq
C_{m}d(\eta)^{-1/(q-1)}\delta^{1/(q-1)}\leq C_{m},$

where $C_{m}=C_{m}(||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))})$ comes from
the locally uniform bound on $Du$ from Theorem 3.6.

Consider a family of $\\{v_{\delta,m}\\}$ with $\delta\to 0$ and measures $m$
such that $||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$ is uniformly
bounded. This is the case, for example, if $(F1)$ or $(F2)$ are satisfied and
the measures $m$ are uniformly bounded in $L^{1}(\Omega)$. It follows from
elliptic regularity that $\\{v_{\delta,m}\\}$ has a convergent subsequence
converging in $C^{1}_{loc}$ to $v$ defined in the half-plane
$\\{\zeta:\zeta_{1}>0\\}$.

Since $\dfrac{\rho_{m}\delta^{q/(q-1)}}{C_{q}}\to 0$ and
$\dfrac{F(\delta\zeta+O_{\delta};m)\delta^{q/(q-1)}}{C_{q}}\to 0$ uniformly,
$v$ is a viscosity solution of

$\displaystyle-\Delta v+C_{q}^{q-1}|Dv|^{q}=0\text{ in
}\\{\zeta:\zeta_{1}>0\\}.$ (7.5)

In addition, $v$ satisfies the boundary conditions

$\displaystyle\lim_{\zeta_{1}\to 0^{+}}v(\zeta)=C_{q}\text{ and
}\lim_{\zeta_{1}\to\infty}v(\zeta)=0.$

The boundary conditions follow from

$\displaystyle
v_{\delta,m}(\zeta)=\dfrac{u_{m}(\eta)}{d(\eta)^{(q-2)/(q-1)}}\dfrac{d(\eta)^{(q-2)/(q-1)}}{\delta^{(q-2)/(q-1)}},$
(7.6)

and the estimates in $(\ref{eta})$. As shown in Theorem 4.1 of [11], the
equation $(\ref{v})$ paired with boundary conditions $(\ref{bounds})$ has a
unique positive solution in the half-plane,
$v(\zeta)=C_{q}(1+\zeta_{1})^{(q-2)/(q-1)}$. From $(\ref{d_0})$, $v$ is indeed
positive. Thus $Dv_{\delta,m}$ converges locally uniformly to
$C_{q}(1+\zeta_{1})^{-1/(q-1)}(2-q)/(q-1)\nu(x_{0})$, and
$(Du_{m}(x)\cdot\nu(x))d(x)^{1/(q-1)}$ converges uniformly in
$||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$, as $d(x)\to 0$.

When $q=2$, rescaling $u_{m}$ as before does not yield an equation of the type
$(\ref{vd})$. To circumvent this, we consider a function of the form
$u_{m}(\eta)+\log(d(\eta))$, which is uniformly bounded in
$||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$ and has a limiting equation of
the desired form.

For the case $q=2$, consider a coordinate system centered at
$x_{0}\in\partial\Omega$, denoted by $O$, with the $\eta_{1}$-axis as the
inner normal direction. By the inner sphere property, there is $\delta_{0}>0$
such that for every $x\in\partial\Omega$, the ball
$B_{\delta_{0}}(x-\delta_{0}\nu(x))$ is contained in $\Omega$. Under this new
coordinate system define

$\displaystyle D_{\delta}=B(O,\delta^{1-\sigma})\cap
B((\delta_{0},0,\ldots,0),\delta_{0})\text{ , }\sigma\in(0,1/2).$

Note that we changed the definition of $D_{\delta}$ here.

Restricting $u_{m}$ to $D_{\delta}$, the function

$\displaystyle v_{\delta,m}(\zeta)=u_{m}(\delta\zeta)+\log(\delta),$

is defined on the domain $\tilde{D}_{\delta}:=\\{\eta/\delta:\eta\in
D_{\delta}\\}$. As before, $\tilde{D}_{\delta}$ converges to the half space
$\\{\zeta:\zeta_{1}>0\\}$, where $\zeta_{1}$ is the component of the vector in
the $-\nu(x)$ direction. Under this change of variable we have the bounds

$\displaystyle\zeta_{1}\delta\leq
d(\eta)\leq\zeta_{1}\delta+O(\delta^{2-2\sigma}).$

In $\tilde{D}_{\delta}$, $v_{\delta,m}$ satisfies, in the viscosity sense,

$\displaystyle-\Delta
v_{\delta,m}+\rho_{m}\delta^{2}+|Dv_{\delta,m}|^{2}=F(\delta\zeta;m)\delta^{2}.$

By Theorem II.3 of [7], $u_{m}(\delta\zeta)+\log(d(\eta))$ is bounded by a
constant $C$ which is uniform in
$||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$. From the bounds $(\ref{eta})$
on $d(\eta)$ it follows that $v_{\delta,m}$ is bounded locally uniformly. From
standard elliptic theory, $v_{\delta}$ is relatively compact in $C^{1}_{loc}$.
Thus, along subsequences $v_{\delta,m}$ converges in $C^{1}_{loc}$ to a
function $v$, defined in the upper-half plane, which is a viscosity solution
of

$\displaystyle-\Delta v+|Dv|^{2}=0.$

Moreover, for constant $C$ as defined above,

$\displaystyle v_{\delta,m}$
$\displaystyle=u_{m}(\delta\zeta)+\log(d(\eta))+\log\left(\dfrac{\delta}{d(\eta)}\right)$
$\displaystyle\geq-C+\log\left(\dfrac{1}{\zeta_{1}+O(\delta^{1-2\sigma})}\right).$

Thus we have the boundary condition

$\displaystyle\lim\limits_{\zeta_{1}\to 0^{+}}v(\zeta)=\infty.$

Solutions of the system

$\displaystyle\begin{cases}-\Delta v+|Dv|^{2}=0\text{ in
}\\{\zeta\in\mathbb{R}^{n}\,|\,\zeta_{1}>0\\},\\\ \lim\limits_{\zeta_{1}\to
0^{+}}v(\zeta)=\infty,\end{cases}$

are of the form $-\log\zeta_{1}+\kappa$ for a constant $\kappa$, which follows
from the change of variable $w=e^{-v}$. Thus $Dv_{\delta,m}$ converges locally
uniformly to $(1/\zeta_{1})\nu(x_{0})$, and $(Du_{m}(x)\cdot\nu(x))d(x)$
converges uniformly in $||F(x;m)||_{L^{\infty}(\Omega;L^{1}(\Omega))}$, as
$d(x)\to 0$.

∎

We now prove Lemma 3.4 from Section 3.1.

######  Lemma 3.4.

Let $b:\Omega\to\mathbb{R}^{d}$ satisfy $(\ref{eq:asym})$ with $C>1$. Then
there exists a unique proper weak solution $m\in W^{1,r}_{loc}(\Omega)$ of

$\displaystyle\Delta m+div(mb)=0\text{ in }\Omega,$ (7.7)

and, for every $\epsilon\in(0,C-1)$, there exists a
$\delta=\delta(C,\epsilon)$, such that

$\displaystyle\int_{\Omega}d^{-C-1+\epsilon}m\leq\hat{C}=\hat{C}(C,||b||_{L^{\infty}(\Omega_{\delta})}).$
(7.8)

###### Proof.

The existence of a proper weak solution $m$ was proven in Section 3.1. To
prove (7.8), we mimick the proof of Lemma 1.1 in [1].

Let $V(x)=d(x)^{-C+1+\epsilon}$ for $0<\epsilon<C-1$. In the neighborhood
$\Omega\backslash\Omega_{\epsilon_{0}}$ of $\partial\Omega$,

$\displaystyle\Delta V-b\cdot DV$
$\displaystyle=(-C+1+\epsilon)d^{-C-1+\epsilon}[(-C+\epsilon)|Dd|^{2}+d\Delta
d-(b\cdot Dd)d]$
$\displaystyle=(-C+1+\epsilon)d^{-C-1+\epsilon}[-C+\epsilon+d\Delta d-(b\cdot
Dd)d].$

Since the last factor approaches $\epsilon$ as $d\to 0$, in a neighborhood of
$\partial\Omega$, we have

$\displaystyle\Delta V-b\cdot DV\lesssim-d^{-C-1+\epsilon}.$ (7.9)

Let $\delta_{0}>0$ be such that, for $x\in\Omega/\Omega_{\delta_{0}}$,

$\displaystyle\Delta V-b\cdot DV\leq-1,$

and define $S:=\int_{\Omega_{\delta_{0}}}|\Delta V-b\cdot DV|m$.

Note that since $m$ is a probability measure on $\Omega$, $S$ is bounded by
the $L^{\infty}$ norm of $|\Delta V-b\cdot DV|$ on $\Omega_{\delta_{0}}$.

We claim that, for all $\delta<\delta_{0}$,

$\int_{\Omega_{\delta}}|\Delta V-b\cdot DV|m\leq 2S.$ (7.10)

Since $S$ is independent of $\delta$, it would then follow that

$\displaystyle\int_{\Omega}|\Delta V-b\cdot DV|m\leq 2S,$

and the growth of $\Delta V-b\cdot DV$ in $(\ref{eq:Vass})$ completes the
proof of the lemma.

To prove $(\ref{eq:V})$, fix $\delta$, $s$, and $r$, such that
$0<\delta_{0}^{-C+1+\epsilon}<s<\delta^{-C+1+\epsilon}<r$. Consider a
nondecreasing cut-off function $\phi\in C^{2}(\mathbb{R})$ such that
$\phi^{\prime\prime}\leq 0$, $\phi(t)=t$ for $0\leq t\leq s$, and $\phi(t)=r$
for $t\geq r$. Since $\phi(V(x))$ is constant in a neighborhood of the
boundary $\partial\Omega$, it follows that it is a permissible test function
in $(\ref{eq:mapp})$. The sign of $\phi^{\prime\prime}$ yields,

$\displaystyle 0$ $\displaystyle=\int_{\Omega}[\Delta(\phi(V))-b\cdot
D(\phi(V)]$
$\displaystyle=\int_{\Omega}[\phi^{\prime\prime}(V)|DV|^{2}+\phi^{\prime}(V)\Delta
V-\phi^{\prime}(V)b\cdot DV]m$
$\displaystyle\leq\int_{\Omega}\phi^{\prime}(V)[\Delta V-b\cdot DV]m.$

The last integrand is zero outside $\\{V\leq r\\}$, equals $(\Delta V-b\cdot
DV)m$ on $\\{V\leq s\\}$ and, is non-positive on $\\{s\leq V\leq r\\}$.
Consequently,

$\displaystyle\int_{\\{V\leq s\\}}(\Delta V-b\cdot DV)m\geq 0.$

Letting $s\to\delta^{-C+1+\epsilon}$ yields

$\displaystyle\int_{\Omega_{\delta}}(\Delta V-b\cdot DV)m\geq 0.$

Since $\Delta V-b\cdot DV\leq 0$ on
$\Omega_{\delta}\backslash\Omega_{\delta_{0}}$, and
$\int_{\Omega_{\delta_{0}}}|\Delta V-b\cdot DV|m=S$, it follows that

$\displaystyle\int_{\Omega_{\delta}}|\Delta V-b\cdot DV|m\leq 2S.$

∎

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