# Well-posedness for a transmission problem connecting first and second-order
operators

Héctor A. Chang-Lara  Department of Mathematics, CIMAT, Guanajuato, Mexico
<EMAIL_ADDRESS>
###### Abstract.

We establish the existence and uniqueness of viscosity solutions within a
domain $\Omega\subseteq{\mathbb{R}}^{n}$ for a class of equations governed by
elliptic and eikonal type equations in disjoint regions. Our primary
motivation stems from the Hamilton-Jacobi equation that arises in the context
of a stochastic optimal control problem.

###### Key words and phrases:

Discontinuous dynamics, transmission problems, viscosity solutions, comparison
principle

###### 1991 Mathematics Subject Classification:

35D40, 35B51, 35F21, 49L12, 49L25

##  1\. Introduction

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be an open set, and let
$\Omega_{B}\subseteq\Omega$ also be open, non-empty, with
$\Omega_{E}:=\Omega\setminus\overline{\Omega_{B}}$ being non-empty as well. We
consider a stochastic optimal control problem in which the objective is to
minimize the expected time taken by a particle to travel from an initial
position $x\in\Omega$ to its first exit from $\Omega$. The controller can
determine the direction of the particle at any given moment when the particle
is in the region $\overline{\Omega}_{E}\cap\Omega$, maintaining a constant
speed, assumed to be one without loss of generality. In the complementary
region $\Omega_{B}$, the particle follows a Brownian motion. This dynamics
allow the particle to switch between the regions multiple times before
exiting. See Figure 1.

Figure 1. A possible realization of a path from $x$ to $\partial\Omega$. The
dynamics on $\Omega_{E}$ are chosen by the controller and follow a Brownian
motion in $\Omega_{B}$.

The analysis of the optimal strategy is closely related to the Hamilton-
Jacobi-Bellman equation satisfied by the value function
$u:\overline{\Omega}\to{\mathbb{R}}$, defined as the least expected exit time
when the particle starts at $x$. In the regions $\Omega_{E}$, $\Omega_{B}$,
and $\partial\Omega$, the value function satisfies an eikonal equation, a
Poisson equation, and a boundary condition, respectively

$\displaystyle\begin{cases}|Du|=1\text{ in }\Omega_{E},\\\
\frac{1}{2}(-\Delta)u=1\text{ in }\Omega_{B},\\\ u=0\text{ in
}\partial\Omega\end{cases}$

In order to obtain a unique solution, we anticipate an additional transmission
condition over the interface
$\Gamma:=\partial\Omega_{E}\cap\Omega=\partial\Omega_{B}\cap\Omega$. The
primary goal of this work is to reveal this transmission condition from a
partial differential equations point of view. Moreover, we establish the
existence and uniqueness of viscosity solutions for a broader family of
equations, using the model described above as the principal guiding example.

A discrete implementation of this problem suggests that over the interface
$\Gamma$ either one of the equations must hold. Indeed, we have observed that
numerical solutions converge to a limit whenever the points in the
discretization may include or not points on the interface. When we have an
interface point a natural condition would be to consider the two possible
dynamics, eikonal or random walk, and choose the one leading to the least
expected time. We plan to present a deeper analysis of our findings in a
forthcoming work.

The relaxed notion of the problem where either of two given equations can hold
over the boundary has been previously used in the fields of optimal control
theory and viscosity solutions for quite some time [3, Section V.4] and [16,
Section 7]. It is the natural requirement for achieving stability of viscosity
solutions. Specifically, for the sub-solution equations, we must consider

$\begin{cases}|Du|-1\leq 0\text{ in }\Omega_{E},\\\
\frac{1}{2}(-\Delta)u-1\leq 0\text{ in }\Omega_{B},\\\
\min(|Du|-1,\frac{1}{2}(-\Delta)u-1)\leq 0\text{ on }\Gamma.\end{cases}$

Meanwhile, for the super-solution equations, we reverse the direction of all
inequalities and replace the minimum with a maximum in the expression over the
interface

$\begin{cases}|Du|-1\geq 0\text{ in }\Omega_{E},\\\
\frac{1}{2}(-\Delta)u-1\geq 0\text{ in }\Omega_{B},\\\
\max(|Du|-1,\frac{1}{2}(-\Delta)u-1)\geq 0\text{ on }\Gamma.\end{cases}$

A solution is then a function that simultaneously satisfies both sub-solution
and super-solution conditions. Colloquially, we say that

(1.1) $\displaystyle\begin{cases}|Du|-1=0\text{ in }\Omega_{E},\\\
\frac{1}{2}(-\Delta)u-1=0\text{ in }\Omega_{B},\\\ \text{Either $|Du|-1$ or
$\frac{1}{2}(-\Delta)u-1=0$ on $\Gamma$}.\end{cases}$

Although it might appear that the condition over the interface is too weak to
guarantee the uniqueness of solutions, our aim is to demonstrate that, in the
context of viscosity solutions and flat interfaces, this is surprisingly not
the case. We will do this by deducing a stronger equation for the interface
and establishing a comparison principle.

There are several challenges associated with the comparison principle for this
problem. Firstly, note that the operator governing this equation over $\Omega$
as a whole is discontinuous across $\Gamma$. This lack of translation
invariance renders the comparison principle a non-trivial question. On the
other hand, we may hope to apply the comparison principle for general boundary
conditions, as established by Barles in [4]. However, the eikonal operator on
$\Gamma$ is not monotone in the exterior normal direction to
$\partial\Omega_{B}$, which prevents the direct application of such theory.

###  1.1. Main results

In an effort to enhance the applicability of our results to related problems,
we have extended our hypotheses beyond the eikonal/Poisson equations
illustrated in this introduction. The corresponding hypotheses will be
announced in each section. For the moment and to keep things simple, let us
state our contributions in the context of the example we have already
discussed.

We demonstrate as a consequence of Theorem 3.1 that for any $C^{2}$ regular
interface $\Gamma$, the problem (1.1) is equivalent to the following stronger
equation

(1.2) $\displaystyle\begin{cases}|Du|-1=0\text{ in }\Omega_{E}\cup\Gamma,\\\
\frac{1}{2}(-\Delta)u-1=0\text{ in }\Omega_{B}.\end{cases}$

Colloquially speaking, we could say that the eikonal mechanism dominates the
dynamics at the interface. For Theorem 3.1 we mainly need a coercivity from
the second order operator resulting from uniform ellipticity.

For flat interfaces we establish a comparison principle for the equation (1.2)
in Corollary 4.3. Our general comparison principle is stated in Theorem 4.1
for continuous sub and super-solutions with their corresponding equations
being separated by some gap (see the hypotheses (Sub) and (Sup) at the
beginning of Section 4). This gap can be taken to be just zero whenever the
operators satisfies some further coercivity type assumptions, such as in the
case of convex operators (the case in the introduction), parabolic problems,
and equations arising from geometric discounted costs, see Corollary 4.3 and
Remark 4.2. Theorem 4.1 also relies on further hypotheses on the operators
(see the hypotheses ($H_{-}$) and ($H_{+}$) also at the beginning of Section
4).

In the preliminary section we have included, besides the main definitions, a
few results which are direct sapplications of the classical theory. For
instance, the existence of viscosity solution by Perron’s method stated in
Theorem 2.7, relies on the global regularity for eikonal type and uniformly
elliptic equations.

###  1.2. Related work

One of the main examples of problems that combine disjoint regions with
diffusion and eikonal type regimes, emerges in the theory of singular
stochastic control, as seen in Chapter VIII of the book by Fleming and Soner
[21]. The representative equation for the value function takes the following
form, known as a gradient constrained problem

$\max(H(Du),Lu)=0\text{ in }\Omega.$

In the equation above, $L$ is a second-order elliptic operator, and $H$ is
typically a convex function. A stochastic control problem that motivates this
equation for $Lu=\tfrac{1}{2}(-\Delta)u-1$ and $H(Du)=|Du|-1$ can be described
as follows: We aim to minimize the expected exit time of a particle starting
at $x\in\Omega$, such that at each instant, we can decide to move either with
Brownian motion or with speed one in a chosen direction. In a few words, this
is an extension of the problem in the introduction where the controlled also
can conveniently choose the region $\Omega_{B}$.

As usual, the value function $u$ represents the least expected exit time when
the particle starts from $x$. From a dynamic programming principle, we derive
the Hamilton-Jacobi equation of singular control provided above. This model
has significant applications in Merton’s portfolio problem and spacecraft
control, with multiple references available in [21, Section VIII.7]. Other
motivations for gradient constrained problems, arising from variational
inequalities, can be found in applications of elastoplasticity of materials
[44]. The survey [20] explore the connections with recent developments for the
obstacle problem.

The literature concerning the analysis of solutions for gradient constrained
problems with a convex $H$ is extensive. Notable works include the $C^{1,1}$
optimal regularity of solutions by Evans [18, 19], Wiegner [47], Ishii and
Koike [30], and Soner and Shreve [42, 41]. Brezis and Sibony [10] show the
equivalence with an obstacle problem. Recent works include those by Andersson,
Shahgholian, and Weiss [1], Hynd [23, 24, 25], Hynd and Mawi [26], and Safdari
[37, 38, 39]. In the last few years, attention has also been given to cases
where $H$ is non-convex, motivated by applications in optimal dividend
strategies for multiple insurances [2]. See Safdari [40] and our own
collaboration with Pimentel [15] for the analysis of solutions.

In the previous scenario, the solution is the one that determines the regions
where either the first-order or the second-order operator is active.
Furthermore, the solution must globally satisfy $Lu\leq 0$ and $H(Du)\leq 0$,
which imposes significant rigidity on the function. The problem considered in
this paper fixes the regions and does not assume global relations like in the
gradient constrained problem; as a result, solutions are expected to be more
flexible across this interface. In our case, we no longer expect better
regularity than Lipschitz continuity, this is illustrated by the examples in
the Section 1.3.2 and the Section 2.2.

Another research direction related to our model involves transmission
problems. These typically deal with differential equations over given disjoint
domains connected by some prescribed condition over their common boundaries.
Borsuk’s book [9] provides a detailed exposition of second-order problems.
Some recent regularity results have been established by Caffarelli, Soria-
Carro, and Stinga in [13], and by Soria-Carro and Stinga in [43]. Problems
connecting operators with different (fractional) orders have been studied, for
instance, by Kriventsov in [32], D’Elia, Perego, Bochev, and Littlewood in
[17], and Capana and Rossi in [14]. An intermediate step in our proof of
Theorem 3.1 involves uncovering a transmission-type equation for our problem
(Lemma 3.2, Corollary 3.3, and Lemma 3.4).

The works referenced in the previous paragraph feature some ellipticity
condition on both sides of the interface. Problems governed by hyperbolic or
eikonal-type operators on both sides are also very relevant in applications
and have been studied recently. These works are motivated by models in
deterministic optimal control. The contributions we have in mind include those
by Barles, Briani, and Chasseigne [5, 6], Lions and Souganidis [34, 35],
Imbert and Monneau [27], and more recently Barles, Briani, Chasseigne, and
Imbert [7]. Our weak formulation on the interface is analogous to the so-
called natural junction condition found in [7, Equations (1.3) and (1.4)].

In a recent publication by Imbert and Nguyen [28], the authors consider
junction problems with second order operators on either side of the interface
or junction. The model in this case assumes that the equations degenerate to
first order problem at the junction. As in the models discussed in the
previous paragraph, there is an additional junction condition from where it is
shown that the problem is well posed. Similar to one of our results, an
intermediate step consists the equivalence between a relaxed and a strong
notion of viscosity solutions (know as flux limited solution).

In contrast to the first order problems, the presence of uniformly elliptic
effects up to the interface seems to make our model rigid enough to yield
uniqueness of solutions by itself, without additional conditions on the
interface which in this case may lead to an over-determined ill-posed problem.

###  1.3. Main ideas

In this brief section we aim to emphasize the fundamental points that underlie
the main theorems presented in this paper. We acknowledge that a significant
portion of the article is devoted to discussing technical modifications on
well-established classical notions in the theory of viscosity solutions, these
have been included for the sake of completeness. However, this may obscure the
novel ideas and challenges that we have encountered in our work and will
appear toward the Section 3 and the Section 4. To address this concern, we
would like to provide a brief discussion in this introduction.

The upcoming presentation is informal and assumes a certain level of
familiarity with the concept of viscosity solutions, which will be revisited
in Section 2.

####  1.3.1. Strong equation

To show that the weak equation (1.1) implies the strong equation (1.2), we
demonstrate as an intermediate step that (1.1) can also be characterized with
test functions that may have discontinuous gradients at the interface. In
other words, they belong to the following space of functions (here
$B_{r}(x_{0})\subseteq\Omega$)

$C(B_{r}(x_{0}))\cap C^{2}(B_{r}(x_{0})\cap(\Omega_{E}\cup\Gamma))\cap
C^{2}(B_{r}(x_{0})\cap(\Omega_{B}\cup\Gamma)).$

Whenever the contact occurs at the interface, we find that either the eikonal
equation holds on the Brownian side, or there is a transmission condition
between the normal derivatives.

To be more specific, we show that (1.1) implies the following problem in an
appropriate viscosity formulation

$\displaystyle\begin{cases}|Du|-1=0\text{ in }\Omega_{E},\\\
\frac{1}{2}(-\Delta)u-1=0\text{ in }\Omega_{B},\\\ \text{Either $|Du_{B}|-1$
or $\partial_{\nu}u_{E}-\partial_{\nu}u_{B}=0$ on $\Gamma$}.\end{cases}$

In the equation above, we have denoted by $\nu$ the interior normal of
$\Omega_{B}$ over $\Gamma$ (or the exterior normal of $\Omega_{E}$ over
$\Gamma$). Moreover, $u_{B}$ and $u_{E}$ represent the restrictions of $u$ to
$\Omega_{B}$ and $\Omega_{E}$, respectively. Geometrically,
$\partial_{\nu}u_{E}-\partial_{\nu}u_{B}$ is positive if the graph of $u$
forms a concave angle along $\Gamma$, and negative if the angle is instead
convex.

####  1.3.2. Existence and uniqueness of solutions in one dimension

Let $\Omega=(-1,1)$, $\Omega_{E}=(-1,0)$, $\Omega_{B}=(0,1)$, and
$\Gamma=\\{0\\}$. Our goal is to see that there is at most one viscosity
solution $u\in C(\overline{\Omega})$ for

(1.3) $\displaystyle\begin{cases}|u^{\prime}|-1=0\text{ in }\Omega_{E},\\\
-u^{\prime\prime}=0\text{ in }\Omega_{B},\\\ \text{Either $|u^{\prime}|-1$ or
$-u^{\prime\prime}=0$ on $\Gamma$},\\\ u(-1)=0,\\\ u(1)=\alpha.\end{cases}$

The two equations and the condition at $x=-1$ indicate that for some parameter
$\beta\in[-1,0]$

$u(x)=\begin{cases}1+\beta-|x-\beta|\text{ in }\Omega_{E},\\\ \alpha
x+(1+2\beta)(1-x)\text{ in }\Omega_{B}.\end{cases}$

See Figure 2. We would like to see that there is at most one $\beta$ for which
$u$ is a solution.

$x$$y$(-1,0)($\beta$,$1+\beta$)(0,$1+2\beta$)(1,$\alpha$) Figure 2. Graph of a
one dimensional candidate for the solution of (1.3).

Case 1: $\alpha\geq 0$. If $\beta<0$ then $u$ is not a super-solution because
around zero

$u(x)=u(0)+\begin{cases}-x\text{ if }x<0,\\\ \gamma x\text{ if }x\geq
0\end{cases},\qquad\gamma=\alpha-(1+2\beta)>-1.$

Then it can be touched at $x_{0}=0$ from below by

$\varphi(x):=u(0)+\frac{\min(\gamma,1)-1}{2}x+x^{2}$

which does not satisfy $|\varphi^{\prime}(x_{0})|-1\geq 0$ or
$-\varphi^{\prime\prime}(x_{0})\geq 0$. The only option left is $\beta=0$ in
which case

$u(x)=\begin{cases}1+x\text{ if }x<0,\\\ (\alpha-1)x+1\text{ if }x\geq
0,\end{cases}$

is indeed a viscosity solution. This assertion can be verified in two steps,
depending on whether $\alpha$ is larger than $2$ or lies between $0$ and $2$:
If $\alpha>2$, then $u$ can only be touched from below at $x_{0}=0$. In this
case, the derivative of the test function is at least $1$, satisfying the
super-solution condition. If $\alpha\in[0,2)$, then $u$ can only be touched
from above at $x_{0}=0$. In this case, the derivative of the test function
lies between $-1$ and $1$, satisfying the sub-solution condition. If
$\alpha=2$, then $u=1+x$, and it clearly satisfies both the sub and super-
solution criteria at every point.

Case 2: $\alpha\in[-2,0)$. Now we can check in a similar fashion as before
that $\beta=\alpha/2$ is the only possibility that gives a viscosity solution.
This choice of the parameter $\beta$ is the one that makes $u$ a translation
of the function $-|x|$

$u(x)=1+\beta-|x-\beta|=1+\alpha/2-|x-\alpha/2|.$

Case 3: $\alpha<-2$. In this case there are no solutions for the given
boundary values.

####  1.3.3. Comparison principle in one dimension

Let $\Omega=(-1,1)$, $\Omega_{E}=(-1,0)$, $\Omega_{B}=(0,1)$, and
$\Gamma=\\{0\\}$ as in the previous example. Assume that $v$ is a super-
solution of

$\begin{cases}|v^{\prime}|-1\geq 0\text{ in }\Omega_{E},\\\
-v^{\prime\prime}\geq 0\text{ in }\Omega_{B},\\\
\max(|v^{\prime}|-1,-v^{\prime\prime})\geq 0\text{ on }\Gamma,\end{cases}$

meanwhile $u$ a sub-solution of the following equation, which leaves a gap
$\eta>0$ between both equations

$\begin{cases}|u^{\prime}|-1+\eta\leq 0\text{ in }\Omega_{E},\\\
-u^{\prime\prime}\leq 0\text{ in }\Omega_{B},\\\
\min(|u^{\prime}|-1+\eta,-u^{\prime\prime})\leq 0\text{ on
}\Gamma.\end{cases}$

Assuming that both functions are continuous and moreover smooth outside of
zero, we will see that a contradiction arises by assuming that the graph of
$v$ touches the graph of $u$ from above at the origin.

To the left of zero we find that $u^{\prime}_{E}(0)\in[-1+\eta,1-\eta]$. Since
$v_{E}$ is a super-solution touching $u_{E}$ at zero, we must have that
$v_{E}^{\prime}(0)\leq-1$.

Let us now examine the behavior of the functions to the right of zero. If
$u^{\prime}_{B}(0)<-1+\eta$, the graph of $u$ forms a concave angle at zero,
with an upper supporting line of slope less than $-1+\eta$. In this case, we
can construct a concave paraboloid $\varphi$ that touches $u$ from above at
the origin, and further impose $\varphi^{\prime}(0)<-1+\eta$. However, this
contradicts the sub-solution condition at the interface, indicating that
$u^{\prime}_{B}(0)\geq-1+\eta$ must hold instead. Since $v_{B}$ touches
$u_{B}$ at zero, we conclude that also $v_{B}^{\prime}(0)\geq-1+\eta$. Notice
that this step indicates that the eikonal equation gets somehow transmitted
towards the Brownian side at the interface.

Putting the two conclusions on $v$ together, we notice that the graph of $v$
forms a convex angle at zero, with a lower supporting plane of slope
$-1+\eta/2$. This contradicts $v$ being a super-solution, as it can be touched
by a convex paraboloid $\varphi$ with
$\varphi^{\prime}(0)=-1+\eta/2\in(-1,1)$.

The case with no gap, $\eta=0$, can be recovered by an approximation argument.
See the comments after Remark 4.2 and Corollary 4.3.

####  1.3.4. Challenges and strategies

The previous and rather simple reasoning in one dimension works because we are
assuming some regularity on the solutions. In higher dimensions, the solutions
are more flexible because of the variations in the directions parallel to the
interface, therefore the comparison principle for viscosity solutions becomes
delicate.

Our strategy involves the use of inf/sup regularizations and boundary
estimates. We perform inf/sup convolutions along the directions parallel to
the flat interface in order to assume that the sub-solution $u$ is semi-convex
along the interface, and the super-solution $v$ is semi-concave also along the
interface. If we assume by contradiction that $v$ touches $u$ from above at
some point $x_{0}\in\Gamma$, the semi-convexity assumption implies that the
functions $C^{1,1}$ at $x_{0}$ along the interface. In other words, they are
squeezed between two paraboloids at $x_{0}$ and along the interface.

Lemma 4.6 and its Corollary 4.7 show that if the boundary data for the
solution of a uniformly elliptic equation is trapped between two paraboloids,
then the solution is differentiable at such point. The main observation is
that the normal derivative is well defined. The argument for this proof is due
to Caffarelli [31, Lemma 4.31] and [22, Theorem 9.31]. On the other hand, we
can see Lemma 4.12 and Lemma 4.13 as analogous boundary regularity results for
solutions of first order equations.

While we can present a fairly general result, our approach does not appear to
yield a comparison principle for non-flat interfaces and general non-
translation invariant operators.

###  1.4. Further questions

One of the main reasons that justify the analysis of this problem relies on a
verification theorem for some particular class of games. In [15] we proposed
and analyzed a Hamilton Jacobi equation for the following one: a particle
starts at some position $x\in\Omega$ and it is driven by two players with
opposite goals, the first one wants to maximize the expected exit time from
$\Omega$, meanwhile the second wants to minimize it. The first player chooses
$\Omega_{B}\subseteq\Omega$ and the second player fixes a unit vector field
$v:\Omega_{E}=\Omega\setminus\overline{\Omega_{B}}\to\partial B_{1}$ such that
the dynamics are given by the following SDE ($B_{t}$ is a Brownian motion in
${\mathbb{R}}^{n}$)

$d\gamma=\mathbbm{1}_{\Omega_{E}}(\gamma)v(\gamma)dt+\mathbbm{1}_{\Omega_{B}}(\gamma)dB_{t},\qquad\gamma(0)=x.$

In a casual manner, we can describe the dynamic of $\gamma$ as being
influenced by two players as follows: the first one determines who drives,
however when this player takes the wheel it does so in some random fashion and
without any preference for any given direction (a drunkard’s walk). Meanwhile
the second (and sober) player, whenever it gets the opportunity, aims to
escape $\Omega$ by driving at maximum speed in some given set of directions.

The value function for this game is the least expected exit time and should
satisfy the gradient constrained problem

$\begin{cases}\min(|Du|-1,\tfrac{1}{2}(-\Delta)u-1)=0\text{ in }\Omega,\\\
u=0\text{ on }\partial\Omega.\end{cases}$

The corresponding verification theorem remains an open problem. One possible
way to address this question would be to fix the set $\Omega_{B}$ and solve a
corresponding optimal control problem (for the second player) and then
consider the maximum among all the values. If we let $u_{\Omega_{B}}$ to be
the solution of (1.1) with zero boundary data, we should then expect that the
solution $u$ of the problem above is actually given by the envelope

$u=\max_{\Omega_{B}\subseteq\Omega}u_{\Omega_{B}}.$

One of the reasons why we are not able to answer this is because we do not
even know if $u_{\Omega_{B}}$ is well defined for a general
$\Omega_{B}\subseteq\Omega$ (or at least a dense set in some suitable
topology).

On the other hand, if both players seek to minimize the expected exit time
under the same rules, then we are actually considering an optimal control
problem where the value function should satisfy the following gradient
constrained problem

$\begin{cases}\max(|Du|-1,\tfrac{1}{2}(-\Delta)u-1)=0\in\Omega,\\\ u=0\text{
on }\partial\Omega.\end{cases}$

This is the simplest model for the problem usually known as gradient
constrained that was addressed in the previous section [18, 19, 47, 30, 42,
41, 10, 1, 23, 24, 25, 26, 37, 38, 39]. Once again, it would not be surprising
that in this case $u=\min_{\Omega_{B}\subseteq\Omega}u_{\Omega_{B}}$. As far
as we know, this question has not been addressed before in the literature
either.

In either case, the well-posedness for the problems that determine
$u_{\Omega_{B}}$ not only fulfills a theoretical inquiry. They can also be
used to estimate how far from optimal is a given strategy, as they are bounds
for the corresponding value function of interest.

###  1.5. Organization of the paper

In Section 2, we provide precise definitions of viscosity solutions for
equations with discontinuous dynamics. We revisit some classical results, such
as stability and Perron’s method. The existence of solutions, stated in
Theorem 2.7, is a rather straightforward consequence of known results in the
classical theory and does not require the comparison principle.

Section 3 is devoted to show that the equations (1.1) and (1.2) are
equivalent. This result can be interpreted as the canonical transmission
condition at the interface. Theorem 3.1 is stated to a more general class of
problems.

The main result of this work, the comparison principle, is proven in Section 4
for flat interfaces. As already announced, we require of inf/sup
regularizations and some boundary regularity estimates proven in Section 4.2.

###  1.6. Acknowledgments

I would like to thank Ryan Hynd, Arturo Arellano, and Edgard Pimentel for
their helpful feedback on this project. The author was supported by CONACyT-
MEXICO grant A1-S-48577.

##  2\. Preliminaries

###  2.1. Notation

For a real number or a real-value function $\alpha$, we denote its positive
and negative part as $\alpha_{\pm}:=\max(\pm\alpha,0)$.

Given a point $x\in{\mathbb{R}}^{n}$ we use
$x_{1},x_{2},\ldots,x_{n}\in{\mathbb{R}}$ to denote its coordinates. The
notation $x^{\prime}\in{\mathbb{R}}^{n-1}$ may be used to denote the first
$(n-1)$ coordinates of $x$, i.e. $x^{\prime}=(x_{1},x_{2},\ldots,x_{n-1})$ and
$x=(x^{\prime},x_{n})$. Occasionally, we may also use
$x^{\prime}\in{\mathbb{R}}^{n}$ as a point which is actually in
$\\{x_{n}=0\\}$. Sometimes $x_{0}$ may denote a fixed point in
${\mathbb{R}}^{n}$, or perhaps $\\{x_{k}\\}$ may be a sequence of points in
${\mathbb{R}}^{n}$. In such cases, the coordinates will be denoted by
$(x_{k})_{1},(x_{k})_{2},\ldots,(x_{k})_{n}$ and
$x_{k}^{\prime}=((x_{k})_{1},(x_{k})_{2},\ldots,(x_{k})_{n-1})$.

The open ball in ${\mathbb{R}}^{n}$ of radius $r>0$ and center
$x_{0}\in{\mathbb{R}}^{n}$ is denoted by $B_{r}(x_{0})$ and we may omit the
center when $x_{0}=0$. Whenever we talk about a ball in dimension $(n-1)$, we
denote it by $B^{n-1}_{r}(x_{0}^{\prime})$ or $B^{n-1}_{r}$.

We use ${\mathbb{R}}^{n\times n}_{\rm sym}$ to denote the space of $n\times n$
symmetric matrices. For $M_{1},M_{2}\in{\mathbb{R}}^{n\times n}_{\rm sym}$ we
say that $M_{1}\leq M_{2}$ iff $\xi\cdot M_{1}\xi\leq\xi\cdot M_{2}\xi$ for
any $\xi\in{\mathbb{R}}^{n}\setminus\\{0\\}$. The remaining inequalities
($\geq$, $<$, and $>$) are understood in a similar way.

We use the notation $L^{\infty}$, $C$, $C^{k}$, and $C^{k,\alpha}$ to
respectively denote the spaces of bounded functions, continuous functions,
$k^{th}$-order continuously differentiable functions, and $k^{th}$-order
continuously differentiable functions with derivatives of order $k$ being
$\alpha$-Hölder continuous ($\alpha\in(0,1]$).

Given a set $\Omega\subseteq{\mathbb{R}}^{n}$, a relatively open subset
$\Gamma\subseteq\partial\Omega$ is said to be uniformly Lipschitz regular iff
for every $x_{0}\in\Gamma$ there exists some change of variables
$\Phi:B_{1}\to U$ where $U=\Phi(B_{1})$ is a neighborhood of $x_{0}=\Phi(0)$,
and $\Phi$ is a bi-Lipchitz map, with a Lipschitz norm independent of $x_{0}$,
such that

$\displaystyle\Phi(B_{1}\cap\\{x_{n}<0\\})=U\cap({\mathbb{R}}^{n}\setminus\overline{\Omega}),$
$\displaystyle\Phi(B_{1}\cap\\{x_{n}>0\\})=U\cap\Omega,$
$\displaystyle\Phi(B_{1}\cap\\{x_{n}=0\\})=U\cap\Gamma.$

We will also consider the scenario where the maps are assumed to be
$C^{2}$-regular diffeomorphism, in this case we say that $\Gamma$ is
$C^{2}$-regular.

A second-order operator over $\Omega\subseteq{\mathbb{R}}^{n}$ is defined in
terms of a function $H=H(M,p,z,x)\in C({\mathbb{R}}^{n\times n}_{\rm
sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega)$. For $u$ second-
order differentiable at $x_{0}\in\Omega$ we compute

$Hu(x_{0}):=H(D^{2}u(x_{0}),Du(x_{0}),u(x_{0}),x_{0}).$

We may occasionally also refer to the function $H$ as the operator.

In the present article, we will say that an operator $H$ is degenerate
elliptic if for any pair
$(M_{1},p,z_{1},x),(M_{2},p,z_{2},x)\in{\mathbb{R}}^{n\times n}_{\rm
sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega$

$M_{1}\geq M_{2},\quad z_{1}\leq z_{2}\qquad\Rightarrow\qquad
H(M_{1},p,z_{1},x)\leq H(M_{2},p,z_{2},x).$

Our convention makes $H(M,z)=-\operatorname{tr}(AM)+\lambda z$ degenerate
elliptic for any $A\in{\mathbb{R}}^{n\times n}_{sym}$ with $A\geq 0$, and
$\lambda\geq 0$. The monotonicity with respect to $z$ is usually called
properness in the literature, meanwhile the monotonicity with respect to the
Hessian is the one referred to as degenerate ellipticity.

If $H=H(M,p,z)$ is independent of $x\in\Omega$ we say that the operator is
translation invariant.

We say the operator is convex if for every $x\in\Omega$ the function
$H(\cdot,\cdot,\cdot,x)$ is convex. It is instead quasi-convex if for every
$x\in\Omega$ and $\lambda\in{\mathbb{R}}$, the sub-level sets
$\\{H(\cdot,\cdot,\cdot,x)\leq\lambda\\}$ are convex.

If $H=H(p,z,x)$ is independent of the matrix variable we say that it is a
first-order operator and

$Hu(x_{0}):=H(Du(x_{0}),u(x_{0}),x_{0})$

can be evaluated for $u$ first-order differentiable at $x_{0}$.

If a first-order operator satisfies that for every $z,\lambda\in{\mathbb{R}}$,
the set $\bigcup_{x\in\Omega}\\{H(\cdot,z,x)\leq\lambda\\}$ is bounded, then
we say that it has bounded sub-level sets. Clearly the bound on the sets
depends on $z$; however, as we will be considering bounded solutions we can
also make the sub-level sets uniformly bounded in $z$. To be precise, let us
consider that $|z|\leq M$, then
$\bigcup_{(z,x)\in[-M,M]\times\Omega}\\{H(\cdot,z,x)\leq\lambda\\}$ is bounded
by the continuity of $H$. This property is usually a consequence of some sort
of coercivity or super-linearity assumption on $H$.

A second-order operator is uniformly elliptic if it is controlled by a family
of elliptic linear operators with uniformly bounded coefficients (from above
and away from zero). In order to do this it is convenient to introduce as well
the extremal Pucci operators. These are defined with respect of some interval
$[\lambda,\Lambda]\subseteq(0,\infty)$ such that
${\mathcal{M}}^{\pm}_{\lambda,\Lambda}:{\mathbb{R}}^{n\times n}_{\rm
sym}\to{\mathbb{R}}$ is given by

$\displaystyle{\mathcal{M}}^{+}_{\lambda,\Lambda}(M)$
$\displaystyle:=\sup\\{-\operatorname{tr}(AM)\ |\ A\in{\mathbb{R}}^{n\times
n}_{\rm sym},\,\lambda I\leq A\leq\Lambda
I\\}=-\sum_{e\in\operatorname{eig}(M)}(\lambda e_{+}-\Lambda e_{-}),$
$\displaystyle{\mathcal{M}}^{-}_{\lambda,\Lambda}(M)$
$\displaystyle:=\inf\\{-\operatorname{tr}(AM)\ |\ A\in{\mathbb{R}}^{n\times
n}_{\rm sym},\,\lambda I\leq A\leq\Lambda
I\\}=-\sum_{e\in\operatorname{eig}(M)}(\Lambda e_{+}-\lambda e_{-}).$

We say that $H$ gives a uniformly elliptic operator with respect to the
interval $[\lambda,\Lambda]\subseteq(0,\infty)$ iff for every
$(M_{1},p_{1},z_{1},x),(M_{2},p_{2},z_{2},x)\in{\mathbb{R}}^{n\times n}_{\rm
sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega$, $M=M_{2}-M_{1}$,
$p=p_{2}-p_{1}$, and $z=z_{2}-z_{1}$, we have that

$\displaystyle{\mathcal{M}}^{-}_{\lambda,\Lambda}(M)-\Lambda|p|-\Lambda
z_{-}\leq H(M_{2},p_{2},z_{2},x)-H(M_{1},p_{1},z_{1},x)$
$\displaystyle\leq{\mathcal{M}}^{+}_{\lambda,\Lambda}(M)+\Lambda|p|+\Lambda
z_{+}.$

###  2.2. Viscosity solutions

Consider for $j\in\\{1,\ldots,k\\}$, degenerate elliptic operators $H_{j}$
defined over some subset $\Omega_{j}\subseteq\Omega$. In the following
definition we give a notion of sub and super-solutions of the problem

(2.4) $\displaystyle\begin{cases}H_{1}u=0\text{ in }\Omega_{1},\\\ \vdots\\\
H_{k}u=0\text{ in }\Omega_{k}.\end{cases}$

We say that a function $\varphi:\Omega_{\varphi}\to{\mathbb{R}}$ touches
another function $u:\Omega_{u}\to{\mathbb{R}}$ from above at some
$x_{0}\in\Omega_{\varphi}\cap\Omega_{u}$ iff

$u\leq\varphi\text{ in $\Omega_{\varphi}\cap\Omega_{u}$, with equality at
$x_{0}$.}$

If additionally,

$u<\varphi\text{ in $\Omega_{\varphi}\cap\Omega_{u}\setminus\\{x_{0}\\}$,}$

we say that the contact is strict. Contact from below is defined similarly.

######  Definition 2.1.

A function $u\in C(\Omega)$ is a viscosity sub-solution of (2.4) iff for every
$\varphi\in C^{2}(B_{r}(x_{0}))$ that touches $u$ from above at
$x_{0}\in\Omega$, it holds that

$H_{j}\varphi(x_{0})\leq 0\text{ if }x_{0}\in\Omega_{j}.$

A function $v\in C(\Omega)$ is a viscosity super-solution of (2.4) iff for
every $\varphi\in C^{2}(B_{r}(x_{0}))$ that touches $v$ from below at
$x_{0}\in\Omega$, it holds that

$H_{j}\varphi(x_{0})\geq 0\text{ if }x_{0}\in\Omega_{j}.$

Finally, $u\in C(\Omega)$ is a solution of (2.4) iff it is both a sub and
super-solution of the respective problem.

For sub-solutions, we could also say that the function satisfies the following
inequalities in the viscosity sense

$\begin{cases}H_{1}u\leq 0\text{ in }\Omega_{1},\\\ \vdots\\\ H_{k}u\leq
0\text{ in }\Omega_{k}.\end{cases}$

A similar notation is also used for super-solutions. Finally, if we ask for
strict contact in the definitions above we obtain the exact same notion of
solutions, a useful trick in a few proofs.

In the next definition we consider
$\Omega_{-},\Gamma,\Omega_{+}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$, such
that $\Gamma=\Omega\setminus(\Omega_{-}\cup\Omega_{+})$. The two main
operators $H_{-}$ and $H_{+}$ are defined over $\Omega_{-}\cup\Gamma$ and
$\Omega_{+}\cup\Gamma$ respectively. Our goal is to define the viscosity
solutions for the following problem

(2.5) $\displaystyle\begin{cases}H_{-}u=0\text{ in }\Omega_{-},\\\
H_{+}u=0\text{ in }\Omega_{+},\\\ \text{Either $H_{-}u$ or $H_{+}u=0$ on
$\Gamma$}.\end{cases}$

In this case the operators used for sub and super solutions are different over
the set $\Gamma$.

######  Definition 2.2.

A function $u\in C(\Omega)$ is a viscosity sub-solution of (2.5) iff it is a
viscosity sub-solution of

$\begin{cases}H_{-}u\leq 0\text{ in }\Omega_{-},\\\ H_{+}u\leq 0\text{ in
}\Omega_{+},\\\ \min(H_{-}u,H_{+}u)\leq 0\text{ on }\Gamma.\end{cases}$

A function $u\in C(\Omega)$ is a viscosity super-solution of (2.5) iff it is a
viscosity super-solution of

$\begin{cases}H_{-}u\geq 0\text{ in }\Omega_{-},\\\ H_{+}u\geq 0\text{ in
}\Omega_{+},\\\ \max(H_{-}u,H_{+}u)\geq 0\text{ on }\Gamma.\end{cases}$

Finally, $u\in C(\Omega)$ is a solution of (2.5) iff it is both a sub and
super-solution of the respective problem.

We leave as an exercise to check that the examples discussed in Section 1.3.2
are viscosity solutions. Let us give a further example of a viscosity solution
in an annular domain $\Omega=B_{R}\setminus B_{r}\subseteq{\mathbb{R}}^{n}$
with $n\geq 2$ and $0<r<R$, taken from [15]. The operators under consideration
will be as in the introduction $H_{-}=|p|-1$ and
$H_{+}=\tfrac{1}{2}(-\Delta)-1$.

Let $\rho\in(r,R)$, $\Omega_{-}=B_{R}\setminus B_{\rho}$,
$\Omega_{+}=B_{\rho}\setminus B_{r}$ and

$u(x):=\begin{cases}R-|x|\text{ in }\overline{\Omega_{-}},\\\
A+B\Phi(|x|)-|x|^{2}/n\text{ in
}\overline{\Omega}\setminus\overline{\Omega_{-}}.\end{cases}$

The function $\Phi$ is a multiple of the fundamental solution for the
Laplacian

$\Phi(s)=\begin{cases}-\ln s\text{ if }n=2,\\\ s^{2-n}\text{ if }n\geq
2.\end{cases}$

The constants $\rho$, $A$, and $B$ are chosen such that $u$ is continuous in
$\Omega$, and attains the boundary value $u=0$ on $\partial\Omega$, and
$|Du|\leq 1$ on both sides of $\Gamma$. The specific equations one should
consider are the following and have infinite solutions

$\displaystyle A+B\Phi(r)-r^{2}/n=0,$ $\displaystyle
A+B\Phi(\rho)-\rho^{2}/n=R-\rho,$ $\displaystyle
B\Phi^{\prime}(\rho)-2\rho/n\leq 1.$

See Figure 3.

Under these requirements it is rather easy to check that $u$ is a viscosity
solution of (1.1) which is Lipschitz continuous but not $C^{1}$. One may
wonder if solutions will be semi-concave, as for the eikonal equation.
However, the one-dimensional examples in Section 1.3.2 show that this is not
necessarily the case.

Figure 3. (a) Graph of a viscosity solution of (1.1) in an annular domain. (b)
Lateral view.

####  2.2.1. Stability and Perron’s method

The flexibility of the problem (2.5) over $\Gamma$ has the benefit of
providing the following stability property.

######  Lemma 2.1.

Let $\Omega_{-},\Omega_{+}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be all
open sets, and let $\Gamma:=\Omega\setminus(\Omega_{-}\cup\Omega_{+})$. Let
$H_{\pm}$ be degenerate elliptic operators over $\Omega_{\pm}\cup\Gamma$
respectively, and let $\\{u_{k}\\}\subseteq C(\Omega)$ be a sequence of
viscosity sub-solutions for (2.5) that converges uniformly to some $u\in
C(\Omega)$. Then $u$ is also a viscosity sub-solution for (2.5).

###### Proof.

Let $\varphi\in C^{2}(B_{r}(x_{0}))$ be a test function that strictly touches
$u$ from above at $x_{0}\in\Omega$. Up to a vertical translation and a sub-
sequence, $\varphi+m_{k}$ touches $u_{k}$ from above at $x_{k}\to x_{0}$, and
$m_{k}\to 0$, due to the uniform convergence.

If $x_{0}\in\Omega_{+}$ then we can assume without loss of generality that
$\\{x_{k}\\}\subseteq\Omega_{+}$ as well. This means that
$H_{+}(\varphi+m_{k})(x_{k})\leq 0$ and the desired inequality for
$H_{+}\varphi(x_{0})$ follows by continuity. The same reasoning applies if
$x_{0}\in\Omega_{-}$ instead.

If $x_{0}\in\Gamma$, then either $H_{-}(\varphi+m_{k})(x_{k})\leq 0$ or
$H_{+}(\varphi+m_{k})(x_{k})\leq 0$ must be true an infinite number of times.
In either case we conclude once again by continuity that
$\min(H_{-}\varphi(x_{0}),H_{+}\varphi(x_{0}))\leq 0$. ∎

The following consequence of the stability given by the previous lemma is the
first step towards the construction of the viscosity solution of the boundary
value problem by Perron’s method.

######  Corollary 2.2.

Let $\Omega_{\pm}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be all open sets,
and let $\Gamma:=\Omega\setminus(\Omega_{-}\cup\Omega_{+})$. Let $H_{\pm}$ be
degenerate elliptic operators over $\Omega_{\pm}\cup\Gamma$ respectively. Let
$\mathcal{S}$ be a set of viscosity sub-solutions for the problem (2.5) which
is equicontinuous and bounded. Then the upper envelope
$u_{\mathcal{S}}(x):=\sup_{u\in\mathcal{S}}u(x)$ is also a sub-solution of
(2.5).

In the case where $\mathcal{S}$ is finite, the equicontinuity and boundedness
hypotheses are superfluous and the result is known as the lattice property:
The maximum of two sub-solutions is a sub-solution.

The following existence result does not require the comparison principle to
hold. However, it assumes as in the previous lemma that the upper envelope of
the family of sub-solutions of the Dirichlet problem turns out to be
continuous up to the boundary.

######  Lemma 2.3 (Perron’s Solution).

Let $\Omega_{\pm}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be all open sets
such that $\partial\Omega\neq\emptyset$, and let
$\Gamma:=\Omega\setminus(\Omega_{-}\cup\Omega_{+})$. Let $H_{\pm}$ be
degenerate elliptic operators over $\Omega_{\pm}\cup\Gamma$ respectively.
Given $g\in C(\partial\Omega)$, define the set $\mathcal{S}_{g}$ such that

$\mathcal{S}_{g}:=\\{u\in C(\overline{\Omega})\ |\ \text{$u$ is a viscosity
sub-solution of \eqref{eq:main3} and $u=g$ in $\partial\Omega$}\\}.$

If there exists an equicontinuous and bounded subset
$\mathcal{S}\subseteq\mathcal{S}_{g}$ such that

$u_{g}(x):=\sup_{u\in\mathcal{S}_{g}}u(x)=\sup_{u\in\mathcal{S}}u(x),$

then $u_{g}$ is a viscosity solution of (2.5) on $\Omega$ taking the boundary
value $g$ continuously on $\partial\Omega$.

Notice that if $\Omega$ is bounded, as we will set in the next section, the
boundedness condition in $\mathcal{S}$ follows from the equicontinuity
assumption.

###### Proof.

By Corollary 2.2 we know that $u_{g}$ is indeed a sub-solution. In this proof
we will just check that $u_{g}$ is also a super-solution.

Assume by contradiction that $\varphi\in C^{2}(\overline{B_{r}(x_{0})})$
strictly touches $u_{g}$ from below at some $x_{0}\in\Gamma$ over
$B_{r}(x_{0})\subseteq\Omega$, but nonetheless
$\max(H_{-}\varphi,H_{+}\varphi)<0$ over $B_{r}(x_{0})$. Although we focus on
the case where the contact occurs over the interface, the following argument
can also be adapted to the case where $x_{0}$ belongs to either $\Omega_{-}$
or $\Omega_{+}$.

Letting $\delta\in(0,\max_{\partial B_{r}(x_{0})}(u_{g}-\varphi))$, so that we
still have $\max(H_{-}(\varphi+\delta),H_{+}(\varphi+\delta))<0$ over
$B_{r}(x_{0})$, we construct

$v:=\begin{cases}\max(u_{g},\varphi+\delta)\text{ in }B_{r}(x_{0}),\\\
u_{g}\text{ in }\overline{\Omega}\setminus B_{r}(x_{0}).\end{cases}$

This is a function in $\mathcal{S}_{g}$ that contradicts the maximality of
$u_{g}$. ∎

######  Remark 2.4.

One can easily extend the notions of viscosity solutions to semi-continuous
functions. In this case there are a similar stability properties under half-
relaxed limits or gamma-convergence. These will not be used in our work as the
regularity of the operators will allow us to assume that the solutions are
always continuous.

####  2.2.2. Compactness for sub-solutions

Just as sub-solutions of the eikonal equation are Lipschitz continuous, we can
show that viscosity sub-solutions of first-order, degenerate elliptic
operators have a modulus of continuity that depends on the zero sub-level set
of the operator, which we assume to be bounded.

######  Lemma 2.5.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be an open bounded set with uniformly
Lipschitz regular boundary. Let $H\in
C({\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega)$ be first-order, degenerate
elliptic operator with bounded sub-level sets. Then any bounded sub-solution
$u\in C(\Omega)$ of $Hu\leq 0$ in $B_{r}(x_{0})$ is Lipschitz continuous.

###### Proof.

Follows because $u$ is also a sub-solution of the eikonal equation $|Du|-R\leq
0$ in $\Omega$ for some radius $R>0$ sufficiently large such that
$\bigcup_{(z,x)\in[-M,M]\times\Omega}\\{H(\cdot,z,x)\leq 0\\}\subseteq B_{R}$
with $M:=\|u\|_{L^{\infty}(\Omega)}$. This observation allows to show first
that $[u]_{C^{0,1}(\Omega)}\leq R$ if $\Omega$ is convex. In general, the
Lipschitz assumption on $\partial\Omega$ suffices to extend the Lipschitz
regularity to the whole domain by a covering argument. A detailed discussion
can be found in [3, Section IV.3.1]. ∎

In the following lemma, we consider sub-solutions of (2.5) with continuous
boundary data on $\Omega$. We then perform a replacement over $\Omega_{+}$
using the solution of the problem $H_{+}u=0$ in $\Omega_{+}$. A sufficient
condition for this construction to be well-defined is that $H_{+}$ is
uniformly elliptic and $\Omega_{+}$ is a bounded set with uniformly Lipschitz
regular boundary (also an exterior cone condition would be enough). This
ensures the existence of barriers controlling the behavior at the boundary, as
shown in [36]. The result is a continuous function on $\overline{\Omega}$ with
a modulus of continuity depending on the problem’s data.

######  Lemma 2.6.

Let $\Omega_{+}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be open bounded sets,
$\Omega_{-}=\Omega\setminus\overline{\Omega_{+}}$, and
$\Gamma=\Omega\setminus(\Omega_{+}\cup\Omega_{-})=\partial\Omega_{\pm}\cap\Omega$.
Assume that $\Omega_{\pm}$ have uniformly Lipschitz regular boundaries.
Consider $H_{\pm}$ degenerate elliptic operators defined over
$\Omega_{\pm}\cup\Gamma$ respectively, such that $H_{-}$ is a first-order
operator with bounded sub-level sets, while $H_{+}$ is a uniformly elliptic
second-order operator. Let $g\in C(\partial\Omega)$ be the boundary data of
the problem.

Given $u\in C(\overline{\Omega})$, a bounded sub-solution of (2.5) such that
$u=g$ over $\partial\Omega$, consider $\bar{u}\in C(\overline{\Omega})$ to be
the solution of the Dirichlet boundary value problem

$\begin{cases}H_{+}\bar{u}=0\text{ in }\Omega_{+},\\\ \bar{u}=u\text{ on
}\Omega\setminus\Omega_{+}.\end{cases}$

Then $\bar{u}$ is well defined and its modulus of continuity depends only on
$\Omega$, $\Omega_{\pm}$, $H_{\pm}$, and $g$. Moreover, $\bar{u}$ is a sub-
solution of (2.5).

We sketch the main arguments in the proof which combines some of the
fundamental constructions and estimates for uniformly elliptic equations
developed by several authors.

###### Proof.

By Lemma 2.5 we get that $u$ is Lipschitz on $\Omega\setminus\Omega_{+}$. From
this and the existence of barriers provided in [36] we obtain the existence of
viscosity solution by Perron’s method [16, Section 4] and [29]. In other
words, $\bar{u}$ is well defined.

The uniform modulus of continuity for the solution is a consequence of the
interior Hölder estimates for uniformly elliptic equations [11, 46], and the
existence of barriers with uniform modulus, once again found in the explicit
construction given in [36]. The procedure for combining the modulus of
continuity at the boundary with the interior regularity estimate is standard.
See for example, the end of Chapter 4 in [12].

To see that $\bar{u}$ is a sub-solution, we just have to consider the
situation in which a test function touches $\bar{u}$ from above over the
interface, being the other two cases immediate by construction. In this
scenario, the test function also touches the original sub-solution $u$, and
the desired inequality follows promptly. ∎

By combining the previous result with Perron’s method stated in Lemma 2.3 we
recover the existence of solutions to the Dirichlet problem.

######  Theorem 2.7.

Let $\Omega_{+}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be open bounded sets,
$\Omega_{-}=\Omega\setminus\overline{\Omega_{+}}$, and
$\Gamma=\Omega\setminus(\Omega_{+}\cup\Omega_{-})=\partial\Omega_{\pm}\cap\Omega$.
Assume that $\Omega_{\pm}$ have uniformly Lipschitz regular boundaries.
Consider $H_{\pm}$ as degenerate elliptic operators defined over
$\Omega_{\pm}\cup\Gamma$ respectively, such that $H_{-}$ is a first-order
operator with bounded sub-level sets, while $H_{+}$ is a uniformly elliptic
second-order operator.

Given $g\in C(\overline{\Omega})$ being a sub-solution of (2.5), there exists
a viscosity solution of (2.5) in $\Omega$ taking the boundary value $g$ on
$\partial\Omega$.

For instance, the function $g=0$ serves as a sub-solution for the original
problem discussed in the introduction.

######  Corollary 2.8.

Let $\Omega_{+}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be open bounded sets,
$\Omega_{-}=\Omega\setminus\overline{\Omega_{+}}$, and
$\Gamma=\Omega\setminus(\Omega_{+}\cup\Omega_{-})=\partial\Omega_{\pm}\cap\Omega$.
Assume that $\Omega_{\pm}$ have uniformly Lipschitz regular boundaries. Then
there exists a viscosity solution $u\in C(\overline{\Omega})$ of (1.1) with
$u=0$ on $\partial\Omega$.

In the Section 1.3.2 we discussed the one dimensional problem over
$\Omega=(-1,1)$, with $\Omega_{+}=(0,1)$. Recall that in this case, when we
set $g(-1)=0$ and $g(1)<-2$ there are no solutions to the boundary value
problem. In a few words, any pair of sub-solutions in $\Omega_{\pm}$, with the
given boundary values, and coinciding at the interface $\Gamma=\\{0\\}$, will
not satisfy the sub-solution property at zero.

From this point the main goal of the article is to establish uniqueness of
solutions for the Dirichlet boundary value problem.

##  3\. Strong equation

In this section we find a stronger characterization for the problem (2.5) with
a $C^{2}$-regular interface $\Gamma$, and $H_{+}$ uniformly elliptic. Namely,
the solutions of (2.5) are exactly the solutions of

(3.6) $\displaystyle\begin{cases}H_{-}u=0\text{ in }\Omega_{-}\cup\Gamma,\\\
H_{+}u=0\text{ in }\Omega_{+}.\end{cases}$

######  Theorem 3.1.

Let $\Omega_{+}\subseteq\Omega\subseteq{\mathbb{R}}^{n}$ be open sets,
$\Omega_{-}=\Omega\setminus\overline{\Omega_{+}}$, and
$\Gamma=\Omega\setminus(\Omega_{+}\cup\Omega_{-})=\partial\Omega_{\pm}\cap\Omega$
a $C^{2}$-regular interface. Let $H_{\pm}$ be degenerate elliptic operators
over $\Omega_{\pm}\cup\Gamma$, with $H_{+}$ uniformly elliptic. Then $u\in
C(\Omega)$ is a viscosity sub-solution (super-solution or solution) of the
relaxed problem (2.5) if and only if it is a viscosity sub-solution (super-
solution or solution) of the strong problem (3.6).

After a local $C^{2}$-regular change of variables we can assume without loss
of generality that $\Omega=B_{1}$, $\Omega_{\pm}=B_{1}\cap\\{\pm x_{n}>0\\}$
and $\Gamma=B_{1}\cap\\{x_{n}=0\\}$. This is an assumption adopted in the rest
of the section.

As it will be shown in our proof, we only need a coercivity hypothesis on
$H_{+}$ with respect to the Hessian variable. Namely that for any
$(M,p,z,x)\in{\mathbb{R}}^{n\times n}_{\rm
sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega$ there exists some
$\alpha>0$ such that111For $v,w\in{\mathbb{R}}^{n}$, we denote $v\otimes
w\in{\mathbb{R}}^{n\times n}$ such that $(v\otimes w)_{ij}=v_{i}w_{j}$.

$H_{+}(M-\alpha(e_{n}\otimes e_{n}),p,z,x)>0.$

This follows from a uniform ellipticity hypothesis because of the inequality

$H_{+}(M-\alpha(e_{n}\otimes
e_{n}),p,z,x)-H_{+}(M,p,z,x)\geq{\mathcal{M}}^{-}_{\lambda,\Lambda}(-\alpha(e_{n}\otimes
e_{n}))=\alpha\lambda.$

In the following lemmas we denote by $\varphi_{\pm}$ the restrictions of a
given function $\varphi:\Omega\subseteq{\mathbb{R}}^{n}\to{\mathbb{R}}$ to
$\Omega_{\pm}\cup\Gamma$ respectively. The hypothesis $\varphi\in
C(\Omega)\cap C^{2}(\Omega_{-}\cup\Gamma)\cap C^{2}(\Omega_{+}\cup\Gamma)$
means that $\varphi$ is continuous across $\Gamma$, and each one of the
restrictions $\varphi_{\pm}$ are $C^{2}$-regular up to $\Gamma$.

######  Lemma 3.2.

Let $\Omega=B_{1}$, $\Omega_{\pm}=B_{1}\cap\\{\pm x_{n}>0\\}$, and
$\Gamma=B_{1}\cap\\{x_{n}=0\\}$. Let $H_{\pm}$ be degenerate elliptic
operators over $\Omega_{\pm}\cup\Gamma$, with $H_{+}$ uniformly elliptic.
Then, for any $\varphi\in C(\Omega)\cap C^{2}(\Omega_{-}\cup\Gamma)\cap
C^{2}(\Omega_{+}\cup\Gamma)$ such that

$\min(H_{-}\varphi_{+}(0),\partial_{n}\varphi_{-}(0)-\partial_{n}\varphi_{+}(0))>0,$

there exists a test function $\psi\in C^{2}(B_{\rho})$, with $\rho\in(0,1)$,
that touches $\varphi$ from above at the origin and satisfies

$\min(H_{-}\psi(0),H_{+}\psi(0))>0.$

We would like to emphasize the geometric interpretation of the inequality
$\partial_{n}\varphi_{-}(0)-\partial_{n}\varphi_{+}(0)>0$. Specifically, this
condition implies that the graph of $\varphi$ forms a concave angle around the
origin and along $\Gamma$.

###### Proof.

Given $\varepsilon\in(0,1)$, and $\alpha>0$, consider the functions

$\displaystyle P_{+}(x)$ $\displaystyle:=\varphi(0)+D\varphi_{+}(0)\cdot
x+\tfrac{1}{2}x\cdot D^{2}\varphi_{+}(0)x+\tfrac{\varepsilon}{2}|x|^{2},$
$\displaystyle P_{-}(x)$ $\displaystyle:=\varphi(0)+D\varphi_{-}(0)\cdot
x+\frac{1}{2}x\cdot D^{2}\varphi_{-}(0)x+\tfrac{\varepsilon}{2}|x|^{2},$
$\displaystyle\psi(x)$ $\displaystyle:=P_{+}(x)+\varepsilon x_{n}-\alpha
x_{n}^{2}.$

By having that $\varphi_{+}$ and $\varphi_{-}$ are second-order
differentiables at the origin we have that there exists some
$\rho_{0}\in(0,1)$ sufficiently small such that

$\begin{cases}\varphi_{+}\leq P_{+}\text{ in }B_{\rho_{0}}\cap\\{x_{n}\geq
0\\},\\\ \varphi_{-}\leq P_{-}\text{ in }B_{\rho_{0}}\cap\\{x_{n}\leq
0\\}.\end{cases}$

For $\varepsilon\in(0,1)$ sufficiently small we get by continuity and the
degenerate ellipticity that

$H_{-}\psi(0)>0\qquad\text{ and
}\qquad\partial_{n}\varphi_{-}(0)-\partial_{n}\varphi_{+}(0)-2\varepsilon>0.$

Take now $\alpha$ sufficiently large such that, thanks to the uniform
ellipticity, we can enforce $H_{+}\psi(0)>0$. Hence the proposed $\psi$
already satisfies the desired inequality,

$\min(H_{-}\psi(0),H_{+}\psi(0))>0.$

To finish the proof we need to show that $\psi$ touches $\varphi$ from above
at the origin over some sufficiently small ball $B_{\rho}$. This follows if
for some $\rho\in(0,\rho_{0})$ it holds that

$\begin{cases}P_{+}\leq\psi\text{ in }B_{\rho}\cap\\{x_{n}>0\\},\\\
P_{-}\leq\psi\text{ in }B_{\rho}\cap\\{x_{n}<0\\}.\end{cases}$

The inequality over $B_{\rho}\cap\\{x_{n}>0\\}$ is equivalent to checking that
$\varepsilon x_{n}\geq\alpha x_{n}^{2}$. This is clearly the case once we take
$\rho\leq\varepsilon/\alpha$.

To prove the inequality over $B_{\rho}\cap\\{x_{n}<0\\}$, we need to show that
$\varepsilon x_{n}\geq P_{-}-P_{+}+\alpha x_{n}^{2}$ holds over the same
domain. Here, we should keep in mind that $P_{-}-P_{+}=x_{n}R$, where $R(x)$
is defined as follows

$R(x):=(\partial_{n}\varphi_{-}(0)-\partial_{n}\varphi_{+}(0))+\sum_{j=1}^{n-1}(\partial_{nj}\varphi_{-}(0)-\partial_{nj}\varphi_{+}(0))x_{j}+\frac{1}{2}(\partial_{nn}\varphi_{-}(0)-\partial_{nn}\varphi_{+}(0))x_{n}.$

We use that
$\partial_{n}\varphi_{-}(0)-\partial_{n}\varphi_{+}(0)>2\varepsilon$ and bound
the quadratic terms by $C\rho|x_{n}|$, then $P_{-}-P_{+}+\alpha x_{n}^{2}\leq
2\varepsilon x_{n}+C\rho|x_{n}|$. This quantity gets bounded by $\varepsilon
x_{n}$ if we choose $\rho\leq\varepsilon/C$. ∎

As a consequence of the previous lemma we see that the transmission condition
only takes into consideration the uniform ellipticity of $H_{+}$ and ignores
the rest of its structure. The following corollary gives an intermediate
formulation, still in the viscosity sense spirit, between the problems (2.5)
and (3.6). We skip the proof as it follows directly from the definitions.

######  Corollary 3.3.

Let $\Omega=B_{1}$, $\Omega_{\pm}=B_{1}\cap\\{\pm x_{n}>0\\}$, and
$\Gamma=B_{1}\cap\\{x_{n}=0\\}$. Let $H_{\pm}$ be degenerate elliptic
operators over $\Omega_{\pm}\cup\Gamma$, with $H_{+}$ uniformly elliptic. Let
$u\in C(\Omega)$ be a viscosity sub-solution of (2.5). Then for every

$\varphi\in C(B_{r})\cap C^{2}(B_{r}\cap(\Omega_{-}\cup\Gamma))\cap
C^{2}(B_{r}\cap(\Omega_{+}\cup\Gamma))$

that touches $u$ from above at $0\,(\in\Gamma)$ it holds that

$\min(H_{-}\varphi_{+}(0),\partial_{n}\varphi_{-}(0)-\partial_{n}\varphi_{+}(0))\leq
0.$

This following lemma makes the final connection with the strong problem (3.6).

######  Lemma 3.4.

Let $\Omega=B_{1}$, $\Omega_{\pm}=B_{1}\cap\\{\pm x_{n}>0\\}$, and
$\Gamma=B_{1}\cap\\{x_{n}=0\\}$. Let $H_{\pm}$ be degenerate elliptic
operators over $\Omega_{\pm}\cup\Gamma$, with $H_{+}$ uniformly elliptic. Let
$u\in C(\Omega)$ be such that for every

$\varphi\in C(B_{r}(x_{0}))\cap
C^{2}(B_{r}(x_{0})\cap(\Omega_{-}\cup\Gamma))\cap
C^{2}(B_{r}(x_{0})\cap(\Omega_{+}\cup\Gamma))$

that touches $u$ from above at $x_{0}\in\Omega$ it holds that

$\begin{cases}H_{-}\varphi(x_{0})\leq 0\text{ if }x_{0}\in\Omega_{-},\\\
H_{+}\varphi(x_{0})\leq 0\text{ if }x_{0}\in\Omega_{+},\\\
\min(H_{-}\varphi_{+}(x_{0}),\partial_{n}\varphi_{-}(x_{0})-\partial_{n}\varphi_{+}(x_{0}))\leq
0\text{ if }x_{0}\in\Gamma.\end{cases}$

Then $u$ is a viscosity sub-solution of (3.6).

###### Proof.

Let $\varphi\in C^{2}(B_{r}(x_{0}))$ touching $u$ from above at
$x_{0}\in\Omega$. The verification of the sub-solution condition reduces to
check the case $x_{0}\in\Gamma$. Assume by contradiction that $H_{-}\varphi>0$
over $B_{\rho_{1}}(x_{0})\cap(\Omega_{-}\cup\Gamma)$ for some small
$\rho_{1}\in(0,r)$.

Let

$\psi(x):=\varphi(x)+\varepsilon\eta(|x^{\prime}-x_{0}|/\rho_{1})x_{n}(x_{n}+\rho_{1}/2)$

where $\eta\in C^{\infty}_{0}((-1/2,1/2))$ is even and non-negative, with
$\eta=1$ in $(-1/4,1/4)$. The value of $\varepsilon>0$ is chosen sufficiently
small such that $H_{-}\psi>0$ over
$B_{\rho_{1}}(x_{0})\cap(\Omega_{-}\cup\Gamma)$.

By construction $\psi\geq\varphi$ in
$\partial(B_{\rho_{1}}(x_{0})\cap\Omega_{-})=(\partial
B_{\rho_{1}}(x_{0})\cap\\{x_{n}<0\\})\cup(B_{\rho_{1}}(x_{0})\cap\\{x_{n}=0\\})$,
and $\psi(x_{0})=\varphi(x_{0})$. So, either $\psi$ continues touching $u$
form above at $x_{0}$, or there is some $c>0$ such that $\psi+c$ touches $u$
from above at some point $x^{*}\in B_{\rho_{1}}(x_{0})\cap\Omega_{-}$.
However, this last alternative would contradict the main hypothesis on $u$,
because $\psi+c$ is a test function for which $H_{-}(\psi+c)(x^{*})\geq
H_{-}\psi(x^{*})>0$ given that $x^{*}\in B_{\rho_{1}}(x_{0})\cap\Omega_{-}$.

Hence we must have that $\psi$ continues touching $u$ form above at $x_{0}$.
We construct in this way the following test function over
$B_{\rho_{1}/4}(x_{0})$

$\phi(x):=\begin{cases}\psi(x)\text{ if }x\in
B_{\rho_{1}/4}(x_{0})\cap\\{x_{n}\leq 0\\},\\\ \varphi(x)\text{ if }x\in
B_{\rho_{1}/4}(x_{0})\cap\\{x_{n}>0\\}.\end{cases}$

Given that

$\partial_{n}\phi_{-}(x_{0})=\partial_{n}\varphi(x_{0})+\varepsilon\rho_{1}/2>\partial_{n}\varphi(x_{0})=\partial_{n}\phi_{+}(x_{0}),$

we must have from the main hypothesis on $u$ that $H_{-}\phi_{+}(x_{0})\leq
0$. However this goes against our initial assumption given that
$H_{-}\phi_{+}(x_{0})=H_{-}\varphi(x_{0})>0$. ∎

The previous lemmas also have analogues for super-solutions with similar
proofs. As a consequence we get the proof of Theorem 3.1 in the case of a flat
interface. The general scenario with $C^{2}$ interfaces follows by a standard
change of variables argument around each point of the interface where the
equation is tested. The $C^{2}$ regularity of the change of variables suffices
to get operators with the same hypothesis as in the flat case.

##  4\. Comparison Principle

In this section we consider the open and bounded sets
$\Omega\subseteq{\mathbb{R}}^{n}$, $\Omega_{\pm}=\Omega\cap\\{\pm x_{n}>0\\}$,
and the flat interface $\Gamma=\Omega\cap\\{x_{n}=0\\}$. These are the
hypotheses on the operators with respect to some fixed uniform ellipticity
parameters $[\lambda,\Lambda]\subseteq(0,\infty)$:

  * ($H_{-}$)

Let $H_{-}\in
C^{0,1}_{loc}({\mathbb{R}}^{n}\times{\mathbb{R}}\times(\Omega_{-}\cup\Gamma))$
be a first-order quasi-convex operator with bounded sub-level sets.

  * ($H_{+}$)

Let $H_{+}\in C({\mathbb{R}}^{n\times n}_{\rm
sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times(\Omega_{+}\cup\Gamma))$ be
degenerate elliptic of the form

$H_{+}(M,p,z,x)=F(M,p,z)+f(z,x),$

where $F\in C({\mathbb{R}}^{n\times n}_{\rm
sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}})$ is translation invariant and
uniformly elliptic with respect to the interval $[\lambda,\Lambda]$; and $f\in
C({\mathbb{R}}\times(\Omega_{+}\cup\Gamma))$ is increasing in the $z$
variable.

Our hypotheses on the operators include a large class of examples, in
particular we can treat the Poisson and eikonal equations which appear in the
introduction. However, our method of proof can not say much about diffusions
that may depend on the position (i.e. $H_{+}(M,x)=-\operatorname{tr}(A(x)M)$),
not even in the case of smooth coefficients.

Notice as well that the hypotheses on each operator guarantee that each one
separately enjoys a comparison principle [16, 12].

Finally, we consider the comparison principle for two equations separated by
some gap $\eta\geq 0$. For $u,-v\in C(\overline{\Omega})\cap
C^{0,1}(\Omega_{-}\cup\Gamma)$, we have the following assumptions:

  * (Sub)

The function $u$ is a viscosity sub-solution of

$\begin{cases}H_{-}u+\eta\leq 0\text{ in }\Omega_{-}\cup\Gamma,\\\
H_{+}u+\eta\leq 0\text{ in }\Omega_{+}.\end{cases}$

  * (Sup)

The function $v$ is a viscosity super-solution of

$\begin{cases}H_{-}v\geq 0\text{ in }\Omega_{-}\cup\Gamma,\\\ H_{+}v\geq
0\text{ in }\Omega_{+}.\end{cases}$

######  Theorem 4.1.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ open and bounded,
$\Omega_{\pm}=\Omega\cap\\{\pm x_{n}>0\\}$, and
$\Gamma=\Omega\cap\\{x_{n}=0\\}$. Let $H_{\pm}$ be degenerate elliptic
operators that satisfy ($H_{-}$) and ($H_{+}$) respectively. Let $u,v\in
C(\overline{\Omega})\cap C^{0,1}(\Omega_{-})$ satisfy (Sub) and (Sup)
respectively with $\eta>0$. Then

$u\leq v\text{ on }\partial\Omega\qquad\Rightarrow\qquad u\leq v\text{ in
}\Omega.$

Under some additional conditions it is possible to remove the gap between the
equations. For instance:

  * (Mz)

For some $\lambda>0$,

$h\geq 0\qquad\Rightarrow\qquad H_{\pm}(p,z+h,x)\geq H_{\pm}(p,z,x)+\lambda
h.$

  * (Mp)

For some $\lambda>0$, and $e\in\partial B_{1}$

$h\geq 0\qquad\Rightarrow\qquad H_{\pm}(p+he,z,x)\geq H_{\pm}(p,z,x)+\lambda
h.$

  * (C)

$H_{\pm}$ are convex operators independent of $z$, and there exist $\beta\in
C^{2}(\overline{\Omega})$ and $\bar{\eta}>0$ such that

$\begin{cases}H_{-}\beta+\bar{\eta}\leq 0\text{ in }\Omega_{-}\cup\Gamma,\\\
H_{+}\beta+\bar{\eta}\leq 0\text{ in }\Omega_{+}.\end{cases}$

######  Remark 4.2.

The first case is usually applied to operators of the form
$H_{-}(p,z,x)=\lambda z+H(p,x)$ appearing in optimal control problem with
geometric discount. The second case can be applied to show a comparison
principle in time evolving problems, being $e$ the time direction. Finally,
the last case applies to our model problem with the eikonal operator
$H_{-}(p)=|p|-1$ and $H_{+}(M)=\tfrac{1}{2}(-\operatorname{tr}M)-1$ by taking
$\beta=0$ and $\bar{\eta}=1$.

######  Corollary 4.3.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ open and bounded,
$\Omega_{\pm}=\Omega\cap\\{\pm x_{n}>0\\}$, and
$\Gamma=\Omega\cap\\{x_{n}=0\\}$. Let $H_{\pm}$ be degenerate elliptic
operators that satisfy ($H_{-}$) and ($H_{+}$) respectively and at least of
the hypotheses (Mz), (Mp), or (C). Let $u,v\in C(\overline{\Omega})\cap
C^{0,1}(\Omega_{-})$ satisfy (Sub) and (Sup) respectively with $\eta=0$. Then

$u\leq v\text{ on }\partial\Omega\qquad\Rightarrow\qquad u\leq v\text{ in
}\Omega.$

In any of the three cases the idea is to approximate the sub-solution with a
limiting sequence that satisfies the hypothesis of Theorem 4.1 with a positive
gap. Let us illustrate the last one, hypothesis (C), pertinent to the optimal
control problem discussed in the introduction.

Given that $H_{\pm}$ are independent of $z$ we may translate the barrier and
assume that $\beta\leq 0$ on $\partial\Omega$. Assume now that $u,v\in
C(\overline{\Omega})\cap C^{0,1}(\Omega_{-})$ satisfy ($H_{-}$), ($H_{+}$)
with $\eta=0$, and $u\leq v$ on $\partial\Omega$. Then $u_{t}=(1-t)u+t\beta$
and $v_{t}=v$ satisfy ($H_{-}$) and ($H_{+}$) with $\eta=t\bar{\eta}$
($t\in(0,1)$), and still $u_{t}\leq v$ on $\partial\Omega$. Hence $u_{t}\leq
v_{t}$ in $\Omega$ and the desired conclusion for $u$ and $v$ gets recovered
in the limit as $t\to 0^{+}$.

######  Corollary 4.4.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ open and bounded,
$\Omega_{\pm}=\Omega\cap\\{\pm x_{n}>0\\}$, and
$\Gamma=\Omega\cap\\{x_{n}=0\\}$. Assume that $\Omega_{\pm}$ have uniformly
Lipschitz regular boundaries. Let $H_{\pm}$ be degenerate elliptic operators
that satisfy ($H_{-}$) and ($H_{+}$) respectively and at least of the
hypotheses (Mz), (Mp), or (C).

Given $g\in C(\overline{\Omega})$ being a sub-solution of (2.5), there exists
a unique viscosity solution of (2.5) in $\Omega$ taking the boundary value $g$
on $\partial\Omega$.

The proof of Theorem 4.1 goes by contradiction, here is a summary of the steps
involved: Assume that $u\leq v$ over the boundary $\partial\Omega$,
nevertheless $\\{u>v\\}$ is non-empty. We use the hypothesis on the operators
and the regularity of the solutions to replace the solutions by some inf/sup
regularizations (along directions parallel to the interface) preserving all
the other hypothesis and gaining semi-convexity along the interface.

By translating $v$ upwards (denoted the same) we can then assume that $v$
touches $u$ from above at some point $x_{0}\in\Gamma$ (being the other cases
clearly not possible by the classical comparison principles). At this point we
also replace $u$ and $v$ over $\Omega_{+}$ with exact solutions of the
uniformly elliptic equation. Finally, we observe that over $\Gamma$ the
solutions are trapped between two paraboloids, from this control and the
equations on both sides, we can extract further regularity that finally allow
us to reproduce the argument in Section 1.3.3 for the one dimensional case.
The boundary estimates are discussed in Section 4.2.

###  4.1. Inf/sup convolutions

Here is a quick review of the inf/sup convolutions and its properties. The
main difference is that we only perform the regularization in directions
parallel to $\\{x_{n}=0\\}={\mathbb{R}}^{n-1}$.

Given $\Omega\subseteq{\mathbb{R}}^{n}$ and $r>0$ we denote

$\Omega^{r}:=\\{x\in\Omega\ |\ \overline{B_{r}(x)}\subseteq\Omega\\}.$

######  Definition 4.1.

Let $\Omega\subseteq{\mathbb{R}}^{n}$, $u\in C(\Omega)\cap
L^{\infty}(\Omega)$, $\varepsilon\in(0,1)$, $M=\|u\|_{L^{\infty}(\Omega)}$,
and $r=2\sqrt{M\varepsilon}$.

We define sup-convolution $u^{\varepsilon}:\Omega^{r}\to{\mathbb{R}}$ such
that

$\displaystyle u^{\varepsilon}(x)$
$\displaystyle:=\sup\\{u(y^{\prime},x_{n})-\tfrac{1}{2\varepsilon}|y^{\prime}-x^{\prime}|^{2}\
|\ (y^{\prime},x_{n})\in\Omega\\}.$

We define the inf-convolution $u_{\varepsilon}:\Omega^{r}\to{\mathbb{R}}$ such
that

$\displaystyle u_{\varepsilon}(x)$
$\displaystyle:=\inf\\{u(y^{\prime},x_{n})+\tfrac{1}{2\varepsilon}|y^{\prime}-x^{\prime}|^{2}\
|\ (y^{\prime},x_{n})\in\Omega\\}.$

By choosing $x\in\Omega^{r}$, with $r=2\sqrt{M\varepsilon}$ and
$M=\|u\|_{L^{\infty}(\Omega)}$, we guarantee that the supremum and infimum
above are actually maximum and minimum attained at some point over the disc
$\\{(y^{\prime},x_{n})\ |\ |y^{\prime}-x^{\prime}|\leq r\\}$.

The proof of the following lemma is a standard adaptation of the ideas in [12,
Chapter 5]. As usual there is a analog version for super-solutions and their
corresponding inf-convolution.

######  Lemma 4.5.

Let $\Omega\subseteq{\mathbb{R}}^{n}$, $u\in C(\Omega)\cap
L^{\infty}(\Omega)$, $\varepsilon\in(0,1)$, $M=\|u\|_{L^{\infty}(\Omega)}$,
and $r=2\sqrt{M\varepsilon}$. In the last two properties we will also denote
$\Omega^{r}_{\pm}=\Omega^{r}\cap\\{\pm x_{n}>0\\}$, and
$\Gamma^{r}=\Omega^{r}\cap\\{x_{n}=0\\}$. The following properties hold:

  * •

Semi-convexity: For each $h\in{\mathbb{R}}$ such that
$\Omega^{r}\cap\\{x_{n}=h\\}\neq\emptyset$, the function
$x^{\prime}\in\\{x^{\prime}\in{\mathbb{R}}^{n-1}\ |\
(x^{\prime},h)\in\Omega^{r}\\}\mapsto
u^{\varepsilon}(x^{\prime},h)+\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}$ is
convex.

  * •

Approximation: The functions $u^{\varepsilon}$ decrease to $u$ locally
uniformly as $\varepsilon\searrow 0$.

  * •

Lipschitz solutions for first-order Lipschitz equations: Given $H_{-}\in
C^{0,1}_{loc}({\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega_{-}\cap\Gamma)$
and $u\in C^{0,1}(\Omega_{-}\cap\Gamma)$ there exists a constant $C$ such that
if $u$ is a viscosity sub-solution of the problem

$H_{-}u\leq 0\text{ in }\Omega_{-}\cap\Gamma$

we also get that $u^{\varepsilon}$ is a sub-solution of

$H_{-}u-C\varepsilon\leq 0\text{ in }\Omega_{-}^{r}\cap\Gamma^{r}$

  * •

Solutions for translation invariant equations: Let $H_{+}\in
C({\mathbb{R}}^{n\times
n}_{sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times\Omega_{+})$ such that

$H_{+}(M,p,z,x)=F(M,p,z)+f(z,x)$

with $F\in C({\mathbb{R}}^{n\times
n}_{sym}\times{\mathbb{R}}^{n}\times{\mathbb{R}})$ translation invariant and
$f\in C({\mathbb{R}}\times\Omega_{+})$, both degenerate elliptic. Given $u\in
C(\Omega_{+}\cap\Gamma)$, there exists a modulus of continuity $\omega$ such
that if $u$ is a viscosity sub-solution of the problem

$H_{+}u\leq 0\text{ in }\Omega_{+},$

we also get that $u^{\varepsilon}$ is a sub-solution of

$H_{+}u-\omega(\varepsilon)\leq 0\text{ in }\Omega_{+}^{r}.$

###  4.2. Regularity estimates

The regularity results in this section use as hypotheses the semi-convexity
that results from the inf/sup-convolutions.

####  4.2.1. Uniformly elliptic regime

The following lemma is closely related to the $C^{1,\alpha}$ boundary
regularity estimate due to Krylov [33]. Its proof is part of an argument due
to Caffarelli [31, Lemma 4.31] and [22, proof of Theorem 9.31] using the
Harnack inequality.

######  Lemma 4.6.

Let $[\lambda,\Lambda]\subseteq(0,\infty)$, $\varepsilon\in(0,1)$,
$p^{\prime}\in{\mathbb{R}}^{n-1}$ with $|p^{\prime}|\leq\Lambda$, and let
$u\in C(\overline{B_{1}\cap\\{x_{n}>0\\}})$ satisfy

$\begin{cases}{\mathcal{M}}^{+}_{\lambda,\Lambda}(D^{2}u)+\Lambda|Du|+\Lambda
u_{+}+\Lambda\geq 0\text{ in }B_{1}\cap\\{x_{n}>0\\},\\\
{\mathcal{M}}^{-}_{\lambda,\Lambda}(D^{2}u)-\Lambda|Du|-\Lambda
u_{-}-\Lambda\leq 0\text{ in }B_{1}\cap\\{x_{n}>0\\},\\\
u\geq-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot x^{\prime}\text{
on }B_{1}\cap\\{x_{n}=0\\}.\end{cases}$

Then one of the following alternatives must be true:

  1. (1)

For every $p_{n}\in{\mathbb{R}}$ there exists a sufficiently small radius
$r\in(0,1)$ such that

$u\geq-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}+p_{n}x_{n}\text{ in }B_{r}\cap\\{x_{n}\geq 0\\}.$

  2. (2)

There exists $p=(p^{\prime},p_{n})\in{\mathbb{R}}^{n}$ such that the following
holds: For every $\eta>0$ there is a radius $r\in(0,1)$ such that

$\begin{cases}u-(-\frac{1}{2\varepsilon}|x^{\prime}|^{2}+p\cdot x)\geq-\eta
x_{n}\text{ in }B_{r}\cap\\{x_{n}\geq 0\\},\\\
u-(-\frac{1}{2\varepsilon}|x^{\prime}|^{2}+p\cdot x)\leq\eta x_{n}\text{ in
}B_{r}\cap\\{x_{n}>\eta|x|\\}.\end{cases}$

###### Proof.

Let $\phi\in C(\overline{B_{1}\cap\\{x_{n}>0\\}})$ be defined by the boundary
value problem

$\begin{cases}{\mathcal{M}}^{+}_{\lambda,\Lambda}(D^{2}\phi)+\Lambda|D\phi|+\Lambda\phi_{+}=-\Lambda\text{
in }B_{1}\cap\\{x_{n}>0\\},\\\
\phi=-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot x^{\prime}\text{
on }B_{1}\cap\\{x_{n}=0\\},\\\ \phi=\min_{\partial
B_{1}\cap\\{x_{n}>0\\}}u\text{ on }\partial
B_{1}\cap\\{x_{n}>0\\}.\end{cases}$

Global estimates for convex operators imply that this problem has a classical
solution with $\phi\in C^{1}(B_{1/2}\cap\\{x_{n}\geq 0\\})$, [33].

This barrier shows that for $r\in(0,1/2)$ the following construction is well
defined as a finite number

$\rho(r):=\sup\\{\rho\in{\mathbb{R}}\ |\
u(x)\geq-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}+\rho x_{n}\text{ for all }x\in B_{r}\cap\\{x_{n}>0\\}\\}.$

The first alternative in the theorem holds if $\lim_{r\to
0^{+}}\rho(r)=+\infty$. So let us assume that $p_{n}:=\lim_{r\to
0^{+}}\rho(r)<+\infty$.

By construction we automatically have the lower bound and all we need to show
is the upper bound in non-tangential domains of the form
$B_{r}\cap\\{x_{n}>\eta|x|\\}$. Assume by contradiction that there is some
$\eta\in(0,1/2)$ and a sequence $\\{y_{k}\\}$ converging to the origin with
$(y_{k})_{n}>\eta|y_{k}|$ such that

(4.7) $\displaystyle
u(y_{k})>-\tfrac{1}{2\varepsilon}|y_{k}^{\prime}|^{2}+p\cdot
y_{k}+\eta(y_{k})_{n}.$

Let $r_{k}:=|y_{k}|\in(0,1/4)$ and consider the following construction defined
in $B_{2}\cap\\{x_{n}>0\\}$

$v_{k}(x):=r_{k}^{-1}u_{k}(r_{k}x)-(-\tfrac{r_{k}}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}+\rho(2r_{k})x_{n}).$

In other words, $v_{k}$ is a Lipschitz rescaling of the non-negative
difference between $u_{k}$ and the approximating polynomial over
$B_{2r_{k}}\cap\\{x_{n}>0\\}$

$u(x)=-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}+\rho(2r_{k})x_{n}+r_{k}v_{k}(r_{k}^{-1}x).$

We will use that the function $v_{k}$ also satisfies in the viscosity sense

$\begin{cases}{\mathcal{M}}^{+}_{\lambda,\Lambda}(D^{2}v)+\Lambda
r_{k}|Dv|+\Lambda r_{k}^{2}v_{+}+Cr_{k}\geq 0\text{ in
}B_{2}\cap\\{x_{n}>0\\},\\\
{\mathcal{M}}^{-}_{\lambda,\Lambda}(D^{2}v)-\Lambda r_{k}|Dv|-\Lambda
r_{k}^{2}v_{-}-Cr_{k}\leq 0\text{ in }B_{2}\cap\\{x_{n}>0\\},\end{cases}$

for some constant $C$ independent of $k$. The verification of this equation in
the viscosity sense is straightforward from the equation for $u$.

Given (4.7), we get that for $v_{k}$ the following must hold

$\sup_{K}v_{k}\geq\eta^{2},$

where $K:=\partial B_{1}\cap\\{x_{n}\geq\eta|x|\\}$ is a compact subset of
$B_{2}\cap\\{x_{n}>0\\}$.

Let $\mu:=c_{1}\lambda/\Lambda$ for some $c_{1}>0$ small to be fixed as a
universal constant. Thanks to the Harnack inequality we get that

$\inf_{B_{1/2}^{n-1}\times\\{x_{n}=\mu\\}}v_{k}\geq c-Cr_{k},$

where the constant $c=c(\mu)>0$ depends on $\mu$. By taking $k$ sufficiently
large we get that $\inf_{B_{1/2}^{n-1}\times\\{x_{n}=\mu\\}}v_{k}\geq
c_{0}>0$, for $c_{0}=c(\mu)/2$ independent of $k$. The goal is to propagate
this lower bound towards $\\{x_{n}=0\\}$ with the use of a lower barrier for
$w_{k}(x):=v_{k}(x)/x_{n}$. This quotient satisfies the following degenerate
elliptic equation in the viscosity sense222For $v,w\in{\mathbb{R}}^{n}$, we
denote $v\odot w:=\tfrac{1}{2}(v\otimes w+w\otimes v)\in{\mathbb{R}}^{n\times
n}_{\rm sym}$

${\mathcal{M}}^{+}_{\lambda,\Lambda}(x_{n}D^{2}w+2e_{n}\odot Dw)+\Lambda
r_{k}|x_{n}Dw+we_{n}|+\Lambda r_{k}^{2}x_{n}w+Cr_{k}\geq 0\text{ in
}B_{1}^{n-1}\times[0,1).$

Let

$\phi(x):=\mu+x_{n}-8\mu|x^{\prime}|^{2}.$

Then $\phi\leq 0$ on $(\partial
B_{1/2}^{n-1}\times(0,\mu])\cup(B_{1/2}^{n-1}\times\\{x_{n}=0\\})$ and

$\displaystyle{\mathcal{M}}^{+}_{\lambda,\Lambda}(x_{n}D^{2}\phi+2e_{n}\odot
D\phi)(x)$ $\displaystyle=\sup_{\lambda I\leq A\leq\Lambda
I}\left(-x_{n}\sum_{i,j=1}^{n}a_{ij}\partial_{ij}\phi(x)-2\sum_{i=1}^{n}a_{in}\partial_{i}\phi(x)\right),$
$\displaystyle=\sup_{\lambda I\leq A\leq\Lambda
I}\left(16\mu\sum_{i=1}^{n-1}(a_{ii}x_{n}+2a_{in}x_{i})-2a_{nn}\right),$
$\displaystyle\leq C\mu\Lambda-2\lambda,$ $\displaystyle\leq-\lambda.$

The last inequality can be achieved by taking $c_{1}=\mu\Lambda/\lambda$
sufficiently small.

The remaining terms can be considered lower order perturbations

$r_{k}|x_{n}D\phi(x)+\phi(x)e_{n}|+r^{2}_{k}x_{n}\phi(x)\leq
C\mu\leq\lambda/(2\Lambda).$

Once again we require $c_{1}$ to be perhaps even smaller than before for the
last inequality to hold.

Hence, for $k\gg 1$

${\mathcal{M}}^{+}_{\lambda,\Lambda}(x_{n}D^{2}\phi+2e_{n}\odot D\phi)+\Lambda
r_{k}|x_{n}D\phi+\phi e_{n}|+\Lambda r^{2}_{k}x_{n}\phi\leq-\lambda/2\leq-
Cr_{k}\text{ in }B_{1/2}^{n-1}\times(0,\mu].$

By comparison, $w_{k}\geq c_{0}\phi/(2\mu^{2})$ in
$B_{1/2}^{n-1}\times(0,\mu]$.

We finally conclude by noticing that, for $\mu$ sufficiently small, $\phi$ is
bounded away from zero in the cylinder $B_{\mu}^{n-1}\times(0,\mu]$, hence for
some $\theta>0$ independent of $k$

$\inf_{B_{\mu}^{n-1}\times(0,\mu]}v_{k}/x_{n}\geq\theta.$

Finally, when we transfer this information back to $u$ we get that

$u(x)\geq-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}+(\rho(2r_{k})+\theta)x_{n}\text{ in }B^{n-1}_{\mu
r_{k}}\times[0,\mu r_{k}).$

This means that $\rho(\mu r_{k})-\rho(2r_{k})\geq\theta>0$, which contradicts
the convergence of $\rho(r)$ to a finite limit as $r\to 0^{+}$. ∎

In the following corollary we assume that the trace of the solution over the
boundary gets trapped between two paraboloids. The conclusion is a bit
stronger than differenciability.

######  Corollary 4.7.

Let $[\lambda,\Lambda]\subseteq(0,\infty)$, $\varepsilon\in(0,1)$,
$p^{\prime}\in{\mathbb{R}}^{n-1}$ with $|p^{\prime}|\leq\Lambda$, and let
$u\in C(\overline{B_{1}\cap\\{x_{n}>0\\}})$ satisfy

$\begin{cases}{\mathcal{M}}^{+}_{\lambda,\Lambda}(D^{2}u)+\Lambda|Du|+\Lambda
u_{+}\geq-\Lambda\text{ in }B_{1}\cap\\{x_{n}>0\\},\\\
{\mathcal{M}}^{-}_{\lambda,\Lambda}(D^{2}u)-\Lambda|Du|-\Lambda
u_{-}\leq\Lambda\text{ in }B_{1}\cap\\{x_{n}>0\\},\\\
-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot x^{\prime}\leq
u\leq\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot x^{\prime}\text{
on }B_{1}\cap\\{x_{n}=0\\}.\end{cases}$

Then $u$ is differentiable at the origin with $Du(0)=(p^{\prime},p_{n})$ and

$\liminf_{\\{x_{n}>0\\}\ni x\to
0}\frac{u(x)-(-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+Du(0)\cdot
x)}{x_{n}}=\limsup_{\\{x_{n}>0\\}\ni x\to
0}\frac{u(x)-(\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+Du(0)\cdot
x)}{x_{n}}=0.$

Notice that we could also state the limits in the conclusion by saying that

$\displaystyle u(x)\geq Du(0)\cdot
x-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+o(x_{n}),$ $\displaystyle u(x)\leq
Du(0)\cdot x+\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+o(x_{n}).$

This is a finer estimate than the differentiablity of $u$ at zero given by
$u(x)=Du(0)\cdot x+o(|x|)$.

####  4.2.2. Eikonal regime

In this section we present some regularity estimates for first order equations
with quasi-convex operators $H\in C({\mathbb{R}}^{n})$. We measure this in
terms of the modulus of continuity given by the one-homogeneous functions

$\phi^{\pm}(x):=\pm\max_{p\in\\{H\leq 0\\}}(\pm p\cdot x).$

The first few results (up to Corollary 4.11) are known in the literature,
however the hypotheses seem to be a bit different. In [8] and also [45,
Section 2] the operators are assumed to be convex. We decided to include
complete proofs in the quasi-convex setting for the results we will be needing
ahead.

######  Lemma 4.8.

Let $H\in C({\mathbb{R}}^{n})$ be a first-order operator with bounded sub-
level sets and let $u\in C(B_{r})$ be a viscosity sub-solution of $Hu\leq 0$
in $B_{r}$. Then $u\leq u(0)+\phi^{+}$ in $B_{r}$.

###### Proof.

Given $R\gg 1$, consider

$\phi_{R}(x):=\max_{p\in Z_{R}}p\cdot x\qquad\text{ where }\qquad
Z_{R}:=\bigcap_{\overline{B_{R}(p)}\supseteq\\{H\leq
0\\}}\overline{B_{R}(p)}.$

Notice then that $\phi_{R}\searrow\phi^{+}$ as $R\to\infty$. Indeed, the
inequality $\phi_{R}\geq\phi^{+}$ and the monotonicity of the sequence follow
because $Z_{R}$ decreases to $\operatorname{conv}(\\{H\leq
0\\})\supseteq\\{H\leq 0\\}$, the convex hull of $\\{H\leq 0\\}$. Let us fix
$x^{*}\in{\mathbb{R}}^{n}$ and let $p_{R}\in Z_{R}$ such that
$\phi_{R}(x^{*})=p_{R}\cdot x^{*}$. By compactness we get that for some
sequence $R_{i}\to\infty$ the points $p_{R_{i}}\to
p^{*}\in\operatorname{conv}(\\{H\leq 0\\})$. Given that $p^{*}=\lambda
p_{1}+(1-\lambda)p_{2}$ for some $p_{1},p_{2}\in\\{H\leq 0\\}$ and
$\lambda\in[0,1]$, we get that

$\lim_{R\to\infty}\phi_{R}(x^{*})\geq\phi^{+}(x^{*})=\lambda\phi^{+}(x^{*})+(1-\lambda)\phi^{+}(x^{*})\geq(\lambda
p_{1}+(1-\lambda)p_{2})\cdot x^{*}=\lim_{R\to\infty}\phi_{R}(x^{*}).$

The advantage of this construction is that $Z_{R}$ is strictly convex and then
$\phi_{R}$ is differentiable outside the origin. By noticing that $\phi_{R}$
is also $1$-homogeneous we get the following identity for any
$x\in{\mathbb{R}}^{n}\setminus\\{0\\}$,

$\phi_{R}(x)=D\phi_{R}(x)\cdot x.$

To check that $Z_{R}$ is strictly convex we consider $p_{1}\neq p_{2}\in
Z_{R}$, $p_{1/2}:=(p_{1}+p_{2})/2$, and show that for
$r=R-\sqrt{R^{2}-|p_{2}-p_{1}|^{2}/4}$ it happens that
$B_{r}(p_{1/2})\subseteq Z_{R}$. For any
$\overline{B_{R}(p_{0})}\supseteq\\{p_{1},p_{2}\\}$ we clearly have that
$B_{r}(p_{1/2})\subseteq\overline{B_{R}(p_{0})}$. As this happens for any
$\overline{B_{R}(p_{0})}\supseteq\\{H\leq 0\\}$ we immediately conclude that
$B_{r}(p_{1/2})\subseteq Z_{R}$.

To see that the convex function $\phi_{R}$ is differentiable for any
$x^{*}\neq 0$ we notice that $\phi_{R}(x^{*})=p^{*}\cdot x^{*}$ for a unique
$p^{*}\in Z_{R}$. Otherwise, $\phi_{R}(x^{*})=p_{1}\cdot x^{*}=p_{2}\cdot
x^{*}=p_{1/2}\cdot x^{*}$ for $p_{1}\neq p_{2}\in Z_{R}$ and
$p_{1/2}=(p_{1}+p_{2})/2$. Then some $p\in B_{r}(p_{1/2})\subseteq Z_{R}$
would improve the maximum of $p\cdot x^{*}$, contradicting the definition of
$\phi_{R}(x^{*})$.

To finally show that $u(x)\leq u(0)+\phi_{R}(x)$ for $R\gg 1$, assume by
contradiction that

$m:=\sup_{x\in B_{r}}(u(x)-u(0)-\phi_{R}(x))>0.$

Hence, for some $\eta>0$ the following function must touch $u$ from above at
some $x^{*}\in B_{r}\setminus\\{0\\}$

$\psi(x):=u(0)+\tfrac{m}{2}+\phi_{R}(x)+\tfrac{\eta}{r-|x|}.$

At this point we must notice that for any $t>0$,
$H(D\phi_{R}(x^{*})+tx^{*})>0$. Indeed, if instead
$H(D\phi_{R}(x^{*})+tx^{*})\leq 0$ we get the contradiction

$\phi_{R}(x^{*})=\max_{p\in Z_{R}}p\cdot
x^{*}\geq(D\phi_{R}(x^{*})+tx^{*})\cdot x^{*}>D\phi_{R}(x^{*})\cdot
x^{*}=\phi_{R}(x^{*}).$

This observation implies that $H(D\psi(x^{*}))$ is strictly positive, which is
inconsistent with $u$ being a sub-solution of the equation. ∎

By invoking a covering argument and reversing the roles of the points in the
previous lemma we obtain the following corollary.

######  Corollary 4.9.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be open convex set. Let $H\in
C({\mathbb{R}}^{n})$ be a first-order operator with bounded sub-level sets and
let $u\in C(\Omega)$ be a sub-solution of $Hu\leq 0$ in $\Omega$. Then for any
$x,y\in B_{r}(x_{0})\subseteq B_{2r}(x_{0})\subseteq\Omega$

$\phi^{-}(x-y)\leq u(x)-u(y)\leq\phi^{+}(x-y).$

By Corollary 2.2, we deduce that $\phi^{+}$ is a sub-solution of $H\phi\leq 0$
in ${\mathbb{R}}^{n}$. Additionally, thanks to the following lemma, we can
conclude that $\phi^{-}$ is as well a sub-solution of $H\phi\leq 0$ in
${\mathbb{R}}^{n}$ if $\\{H\leq 0\\}$ is convex (or if we assume that $H$ is
quasi-convex). It is worth recalling that in the previous proof, we have shown
that $\phi^{\pm}$ are classical solutions of $H\phi=0$ in
${\mathbb{R}}^{n}\setminus\\{0\\}$ if $\\{H\leq 0\\}$ is strictly convex.

######  Lemma 4.10.

Let $H\in C({\mathbb{R}}^{n})$ be a first-order quasi-convex operator with
bounded sub-level sets. Let $\Omega\subseteq{\mathbb{R}}^{n}$ be open,
$\\{u_{k}\\}\subseteq C^{1}(\Omega)$ be an equicontinuous sequence of
functions satisfying $Hu\leq 0$ in $\Omega$, and assume that $u(x):=\inf
u_{k}(x)$ finite valued. Then $u$ is a viscosity sub-solution of $Hu\leq 0$ in
$\Omega$.

This result is also valid for Lipschitz solutions and appears for instance in
[45, Corollary 2.36 and Exercise 30] as a consequence of the characterization
of viscosity solutions for convex Hamiltonians by Barron and Jensen in [8].

###### Proof.

The function $u$ as a uniform limit of functions the form
$v_{k}:=\min(u_{1},\ldots,u_{k})$. Hence the result will follow by the
stability of viscosity sub-solutions once we show that each $v_{k}$ is a sub-
solution of $Hu=0$ in $\Omega$. By an inductive argument we can further reduce
the analysis to the case $k=2$.

Let $\varphi\in C^{1}(B_{r}(x_{0}))$ be a test function touching $v_{2}$
strictly from above at $x_{0}$ with $B_{r}(x_{0})\subseteq\Omega$. If
$u_{1}(x_{0})<u_{2}(x_{0})$ we get that $\varphi$ touches $u_{1}$ from above
at $x_{0}$ over a neighborhood of $x_{0}$, so we conclude using that $u_{1}$
is a viscosity sub-solution. The case $u_{2}(x_{0})<u_{1}(x_{0})$ can be
treated in the same way and then we are left with the alternative
$u_{1}(x_{0})=u_{2}(x_{0})$.

If $p:=Du_{1}(x_{0})\neq q:=Du_{2}(x_{0})$, then ${u_{1}=u_{2}}$ forms a
$C^{1}$ surface around $x_{0}$ with normal given by $Du_{1}-Du_{2}\neq 0$. We
can then choose a system of coordinates centered at $x_{0}$ such that
$p^{\prime}=q^{\prime}$ and $p_{n}<q_{n}$. Consequently, the exterior normal
of $\\{u_{1}>u_{2}\\}$ at the origin is the vector $e_{n}$.

As $\varphi$ touches $\min(u_{1},u_{2})=u_{1}=u_{2}$ over $\\{u_{1}=u_{2}\\}$
from above at $x_{0}$, we have that the tangential derivatives must coincide

$(D\varphi(x_{0}))^{\prime}=p^{\prime}=q^{\prime}.$

On the other hand, for the normal derivatives we must also have that
$p_{n}\leq\partial_{n}\varphi(x_{0})\leq q_{n}$. For instance,
$p_{n}\leq\partial_{n}\varphi(x_{0})$ follows because
$\varphi-\varphi(x_{0})\geq u_{1}-u_{1}(x_{0})$ over $\\{u_{1}<u_{2}\\}$,
which has $e_{n}$ as a interior normal vector at $x_{0}$.

Now we notice that for
$\lambda:=(\partial_{n}\varphi(x_{0})-p_{n})/(q_{n}-p_{n})\in[0,1]$ we get
that by the convexity of $\\{H\leq 0\\}$ and using that $u_{1}$ and $u_{2}$
are classical sub-solutions

$D\varphi(x_{0})=\lambda Du_{1}(x_{0})+(1-\lambda)Du_{2}(x_{0})\in\\{H\leq
0\\}.$

If instead, $Du_{1}(x_{0})=Du_{2}(x_{0})$ we must also have
$D\varphi(x_{0})=Du_{1}(x_{0})=Du_{2}(x_{0})\in\\{H\leq 0\\}$. Otherwise, we
have that $\\{\varphi<u_{1}\\}$ and $\\{\varphi<u_{2}\\}$ have the same
exterior normal vector at $x_{0}$, so that $\\{\varphi\geq
u_{1}\\}\cup\\{\varphi\geq u_{2}\\}$ can not cover a neighborhood of $x_{0}$.
This contradicts $\varphi\geq\min(u_{1},u_{2})$ around $x_{0}$. ∎

######  Corollary 4.11.

Let $H\in C({\mathbb{R}}^{n})$ be a first-order quasi-convex operator with
bounded sub-level sets. Then $\phi^{-}$ is a viscosity solution of the
equation $H\phi=0$ in ${\mathbb{R}}^{n}$.

The following lemmas are the boundary estimates we will use for the eikonal
problem. The first result says that under certain conditions a paraboloid
$\varphi$ over $B_{1}\cap\\{x_{n}=0\\}$ can be extended to a paraboloid with
$H(D\varphi(0))\leq 0$.

######  Lemma 4.12.

Let $H\in C({\mathbb{R}}^{n})$ and let $u\in C^{0,1}(B_{1}^{n-1}\times(-1,0])$
be a viscosity sub-solution of $Hu\leq 0$ in $B_{1}^{n-1}\times(-1,0)$ such
that for some $p^{\prime}\in{\mathbb{R}}^{n-1}$ and $\varepsilon>0$

$u\leq u(0)+\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}\text{ on }B_{1}\cap\\{x_{n}=0\\}.$

Then there exists $p=(p^{\prime},p_{n})\in{\mathbb{R}}^{n}$ such that
$H(p)\leq 0$.

###### Proof.

For $L:=[u]_{C^{0,1}(B_{1}\cap\\{x_{n}\leq 0\\})}$ and $r\in(0,1/2)$ small, we
get that

$u\leq u(0)+\tfrac{r^{2}}{2\varepsilon}|p^{\prime}|r+\sqrt{2}Lr\text{ in
}B_{r}^{n-1}\times[-r,0].$

Then, by letting $\rho:=r/\varepsilon+|p^{\prime}|+\sqrt{2}L$, we obtain that

$\varphi(x):=u(0)+\tfrac{1}{\varepsilon}|x^{\prime}|^{2}+p^{\prime}\cdot
x^{\prime}-\rho x_{n}-\tfrac{r^{2}}{4\varepsilon}>u\text{ on }(\partial
B_{r}^{n-1}\times[-r,0])\cup(B_{r}^{n-1}\times\\{-r\\}).$

Given that $\varphi(0)<u(0)$ we must have that for some $\eta>0$ the function
$\psi=\varphi-\eta x_{n}^{-1}$ touches $u$ from above at some $x_{r}\in
B_{r}^{n-1}\times(-r,0)$. At such contact point we have

$p_{r}:=D\psi(x_{r})=(\tfrac{2}{\varepsilon}x_{r}^{\prime}+p^{\prime},-\rho+\eta(x_{r})_{n}^{-2})\in
B_{L}$

and
$H(\tfrac{2}{\varepsilon}x_{r}^{\prime}+p^{\prime},-\rho+\eta(x_{r})_{n}^{-2})\leq
0$.

Then $|p_{r}^{\prime}-p^{\prime}|\leq 2r/\varepsilon\to 0$ as $r\to 0$ and by
compactness there is an accumulation value such that $(p_{r})_{n}\to p_{n}$
for some sequence. We can now conclude the proof thanks to the continuity of
$H$. ∎

To finish this section let us consider $\eta>0$ and

$\phi_{1}^{+}(x):=\max_{p\in\\{H+\eta\leq 0\\}}p\cdot x.$

For any sub-solution $u$ of $Hu+\eta\leq 0$ in $B_{1}\cap\\{x_{n}<0\\}$ with
boundary data $u\leq\varphi$ over $B_{1}\cap\\{x_{n}=0\\}$, the lower envelope
given by the Hopf-Lax formula

$\psi(x):=\inf_{y^{\prime}\in{\mathbb{R}}^{n-1}}\varphi(y^{\prime})+\phi_{1}^{+}(x-y^{\prime})$

is an upper bound for $u$. This is a consequence of Corollary 4.9.

The following lemma gives us a further bound from above for $\psi$ whenever
$\varphi$ can be extended to a paraboloid with $H(D\varphi(0))=0$.

######  Lemma 4.13.

Let $H\in C({\mathbb{R}}^{n})$ be a first-order quasi-convex operator with
bounded sub-level sets. Let $\eta>0$, and assume that there exists
$p_{1}=(p_{*}^{\prime},\rho_{1})\in{\mathbb{R}}^{n}$ such that
$H(p_{1})+\eta\leq 0$. Let
$p_{0}=(p_{*}^{\prime},\rho_{0})\in{\mathbb{R}}^{n}$ such that

$\rho_{0}=\min\\{\rho\in{\mathbb{R}}\ |\ H((p_{*}^{\prime},\rho))\leq
0\\}<\rho_{1}.$

Let $\varepsilon>0$ and

$\varphi(x):=\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{0}\cdot
x,\qquad\psi(x):=\inf_{y^{\prime}\in{\mathbb{R}}^{n-1}}\varphi(y^{\prime})+\phi_{1}^{+}(x-y^{\prime}).$

Then for some $r>0$ sufficiently small $\psi\leq\varphi$ in
$B_{r}\cap\\{x_{n}\leq 0\\}$.

###### Proof.

The goal is to see that we can take $r$ sufficiently small such that for any
$x\in B_{r}\cap\\{x_{n}\leq 0\\}$ there exists some $y^{\prime}\in
B_{1}\cap\\{x_{n}=0\\}$ for which

$\frac{1}{2\varepsilon}|y^{\prime}|^{2}+p_{*}^{\prime}\cdot
y^{\prime}+\max_{p\in\\{H+\eta\leq
0\\}}p\cdot(x-y^{\prime})\leq\frac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{0}\cdot
x.$

Let $r_{0}\in(0,1)$ such that $B_{2r_{0}}(p_{0})\cap\\{H+\eta\leq
0\\}=\emptyset$. By Hahn-Banach there exists some $z\in\partial B_{1}$ such
that

$\min_{p\in\\{H+\eta\leq 0\\}}z\cdot p\geq\max_{q\in B_{r_{0}}(p_{0})}z\cdot
q+r_{0}.$

Notice that $z_{n}>0$. Moreover, we can even give a strictly positive lower
bound for $z_{n}$. Using that $p_{0}\in B_{r_{0}}(p_{0})$ and
$p_{1}\in\\{H+\eta\leq 0\\}$ are separated by the previous application of
Hanh-Banach we obtain that

$\displaystyle z\cdot p_{1}=z^{\prime}\cdot p_{*}^{\prime}+z_{n}\rho_{1}\geq
z\cdot p_{0}+r_{0}=z^{\prime}\cdot p_{*}^{\prime}+z_{n}\rho_{0}+r_{0},$
$\displaystyle\Rightarrow\qquad$ $\displaystyle z_{n}\geq
r_{0}/(\rho_{1}-\rho_{0})>0.$

For any $\lambda\in(0,\varepsilon r_{0})$ the vector $\bar{z}=\lambda z$
satisfies

$\min_{p\in\\{H+\eta\leq 0\\}}\bar{z}\cdot p\geq\max_{q\in
B_{r_{0}}(p_{0})}\bar{z}\cdot q+\tfrac{1}{\varepsilon}\bar{z}_{n}^{2}.$

If we take $r$ smaller than $\varepsilon
r_{0}\min(r_{0}/(\rho_{1}-\rho_{0}),1/2)$ and $x\in B_{r}\cap\\{x_{n}\leq
0\\}$, we get that $\lambda:=-x_{n}/z_{n}\in(0,\varepsilon r_{0})$ so the
previous estimate holds. Let $y^{\prime}=x+\bar{z}=x-(x_{n}/z_{n})z\in
B_{1}\cap\\{x_{n}=0\\}$ so that
$|y^{\prime}+x|=|\bar{z}+2x|\leq|\bar{z}|+2|x|<2\varepsilon r_{0}$ and then

$\displaystyle\min_{p\in\\{H+\eta\leq 0\\}}(y^{\prime}-x)\cdot p$
$\displaystyle\geq\max_{q\in B_{r_{0}}(p_{0})}(y^{\prime}-x)\cdot
q+\tfrac{1}{\varepsilon}x_{n}^{2},$
$\displaystyle\geq(y^{\prime}-x)\cdot(\tfrac{1}{2\varepsilon}(y^{\prime}+x)+p_{0})+\tfrac{1}{2\varepsilon}x_{n}^{2},$
$\displaystyle=\tfrac{1}{2\varepsilon}|y^{\prime}|^{2}-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{0}\cdot(y^{\prime}-x).$

This is the desired inequality which concludes the proof. ∎

###  4.3. Proof of the comparison principle

As a first step towards the proof of Theorem 4.1 we show that the comparison
principle holds under the following assumptions for the solutions $u,v\in
C(B_{1}^{n-1}\times(-1,1))\cap C^{0,1}(B_{1}^{n-1}\times(-1,0])$ with respect
to $\varepsilon\in(0,1)$, $[\lambda,\Lambda]\subseteq(0,\infty)$, $H\in
C({\mathbb{R}}^{n})$, and $\eta>0$:

  * (C’)

The following functions are convex

$x^{\prime}\in B_{1}^{n-1}\mapsto
u(x^{\prime},0)+\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}\quad\text{ and }\quad
x^{\prime}\in
B_{1}^{n-1}\mapsto-v(x^{\prime},0)+\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}.$

  * (UE’)

Both of the functions $u$ and $v$ satisfy in the viscosity sense

$\begin{cases}{\mathcal{M}}^{+}_{\lambda,\Lambda}(D^{2}u)+\Lambda|Du|+\Lambda
u_{+}+\Lambda\geq 0\text{ in }B_{1}^{n-1}\times(0,1),\\\
{\mathcal{M}}^{-}_{\lambda,\Lambda}(D^{2}u)-\Lambda|Du|-\Lambda
u_{-}-\Lambda\leq 0\text{ in }B_{1}^{n-1}\times(0,1).\end{cases}$

  * (Sub’)

The function $u$ satisfies in the viscosity sense

$Hu+\eta\leq 0\text{ in }B_{1}^{n-1}\times(-1,0].$

  * (Sup’)

The function $v$ satisfies in the viscosity sense

$Hv\geq 0\text{ in }B_{1}^{n-1}\times(-1,0].$

######  Lemma 4.14.

Let $\eta>0$ and let $H\in C({\mathbb{R}}^{n})$ a first-order quasi-convex
operator with bounded sub-level sets. Let $\varepsilon\in(0,1)$,
$[\lambda,\Lambda]\subseteq(0,\infty)$, and let $u,v\in
C(B_{1}^{n-1}\times(-1,1))\cap C^{0,1}(B_{1}^{n-1}\times(-1,0])$ satisfy (C’),
(UE’), (Sub’), (Sup’) above. Then $u$ can not touch $v$ from below at the
origin.

In the following proof we will use once again the notation $u_{\pm}$ for the
restriction of a given function to $\Omega_{\pm}\cup\Gamma$.

###### Proof.

Assume by contradiction that $u\leq v$ with $u(0)=v(0)=0$ (without loss of
generality). By the convexity hypothesis on $u$ and $v$ we get that for some
$p_{*}^{\prime}\in{\mathbb{R}}^{n-1}$

(4.8)
$\displaystyle-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{*}^{\prime}\cdot
x^{\prime}\leq u(x^{\prime},0)\leq
v(x^{\prime},0)\leq\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{*}^{\prime}\cdot
x^{\prime}\text{ in }B_{1}^{n-1}.$

Given $t\in[0,1]$ consider

$\displaystyle\rho_{t}$ $\displaystyle:=\min\\{\rho\in{\mathbb{R}}\ |\
H(p_{*}^{\prime},\rho)+\eta t\leq 0\\},$ $\displaystyle\phi^{\pm}_{t}(x)$
$\displaystyle:=\pm\max_{p\in\\{H+\eta t\leq 0\\}}(\pm p\cdot x),$
$\displaystyle\varphi^{\pm}_{t}(x)$
$\displaystyle:=\pm\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{*}^{\prime}\cdot
x^{\prime}+\rho_{t}x_{n}.$

Thanks to the Lemma 4.12 we known that $\rho_{t}$ are well defined as finite
numbers, and by the continuity of $H$

$0\leq t_{1}<t_{2}\leq
1\qquad\Rightarrow\qquad\rho_{t_{1}}<\rho_{t_{2}}\qquad\Rightarrow\qquad\begin{cases}\varphi_{t_{1}}^{\pm}>\varphi_{t_{2}}^{\pm}\text{
in }\\{x_{n}<0\\},\\\ \varphi_{t_{1}}^{\pm}<\varphi_{t_{2}}^{\pm}\text{ in
}\\{x_{n}>0\\}.\end{cases}$

The functions $\phi^{\pm}_{t}$ also have a monotonicity property

$0\leq t_{1}<t_{2}\leq
1\qquad\Rightarrow\qquad\begin{cases}\phi^{-}_{t_{1}}<\phi^{-}_{t_{2}}\text{
in }{\mathbb{R}}^{n}\setminus\\{0\\},\\\
\phi^{+}_{t_{1}}>\phi^{+}_{t_{2}}\text{ in
}{\mathbb{R}}^{n}\setminus\\{0\\}.\end{cases}.$

By Corollary 4.11 we have that $\phi^{-}_{t}$ satisfies in the viscosity
sense,

(4.9) $\displaystyle H(D\phi_{t}^{-})+\eta t=0\text{ in }{\mathbb{R}}^{n}.$

Meanwhile, $\phi^{+}_{t}$ satisfies instead

$H(D\phi_{t}^{+})+\eta t\leq 0\text{ in }{\mathbb{R}}^{n}.$

By Corollary 4.9 we obtain that for every $x,y\in B_{1}\cap\\{x_{n}\leq 0\\}$

(4.10) $\displaystyle\phi^{-}_{1}(x-y)\leq u(x)-u(y)\leq\phi^{+}_{1}(x-y).$

Finally we also point out that
$\varphi_{t}^{\pm}(x)=-\tfrac{1}{2\varepsilon}|x^{\prime}|^{2}+p_{*}^{\prime}\cdot
x^{\prime}$ on $\\{x_{n}=0\\}$ such that

(4.11) $\displaystyle\varphi_{t}^{-}\leq u\leq v\leq\varphi_{t}^{+}\text{ in
}B_{1}^{n-1}$

and at the origin we get that $H(D\varphi_{t}^{\pm}(0))=-\eta t$.

1. Our first goal is to show that for some radius $r\in(0,1)$ sufficiently small,

(4.12) $\displaystyle\varphi^{-}_{1/2}\overset{?}{\leq}v\text{ in
}B_{r}\cap\\{x_{n}\leq 0\\}.$

Let $r\in(0,1)$ be a sufficiently small radius such that
$H(D\varphi^{-}_{1/2})\in(-\eta,0)$ in $B_{r}$, which is possible because of
continuity and $H(D\varphi^{-}_{1/2})=-\eta/2$. Consider now the function

$\psi:=\min(\varphi^{-}_{1/2},\phi^{-}_{0}+c),\qquad c:=\inf_{\partial
B_{r}}(\phi^{-}_{1}-\phi^{-}_{0})>0.$

By construction and the use of (4.11) and (4.10) we have that $\psi\leq v$ on
$\partial(B_{r}\cap\\{x_{n}<0\\})$. Over $B_{r}\cap\\{x_{n}=0\\}$ we use that
$\psi\leq\varphi^{-}_{1/2}\leq v$. Over $\partial B_{r}\cap\\{x_{n}<0\\}$ we
use that $\psi\leq\phi^{-}_{1}\leq u\leq v$. Moreover, $\psi$ is also sub-
solution of the equation $H\psi\leq 0$ in $B_{r}^{n-1}\times(-r,0)$. This is a
consequence of Lemma 4.10. By comparison, we conclude that $\psi\leq v$ in
$B_{r}\cap\\{x_{n}\leq 0\\}$.

Finally notice that by continuity $\psi=\varphi_{1/2}^{-}$ in some
neighborhood of the origin. With this we have shown that (4.12) must hold for
some sufficiently small radius.

2. Now we look at the other side, $\\{x_{n}\geq 0\\}$. The new goal is to see that

$\partial_{n}v_{+}(0)\overset{?}{\leq}\rho_{1/2}.$

This derivative is well defined thanks to the Lemma 4.7. If we assume by
contradiction that the opposite inequality holds, then Lemma 4.7 and the
previous lower bound for $v_{-}$ would imply that the paraboloid
$\varphi^{-}_{1/2}$ touches $v$ from below at the origin. This contradicts $v$
being a super-solution of $Hv\geq 0$ given that
$H(D\varphi^{-}_{1/2}(0))=-\eta/2<0$.

3. Let us show now that for some radius $r\in(0,1)$

(4.13) $\displaystyle\varphi^{+}_{3/4}\overset{?}{\geq}u\text{ in
}B_{r}\cap\\{x_{n}\geq 0\\}.$

Lemma 4.7 applies to $u_{+}$, so we get that $\partial_{n}u_{+}(0)$ is well
defined. From the contact of $v_{+}$ and $u_{+}$ at the origin we get that
$\partial_{n}u_{+}(0)\leq\partial_{n}v_{+}(0)\leq\rho_{1/2}$. Moreover, by the
same lemma we conclude that for some $r\in(0,1)$ sufficiently small

$u_{+}-\varphi^{+}_{1/2}\leq(\rho_{3/4}-\rho_{1/2})x_{n}\text{ in
}B_{r}\cap\\{x_{n}\geq 0\\}.$

This inequality is equivalent to (4.13).

4. As a final step we will extend the bound in (4.13) to the other side

$\varphi^{+}_{3/4}\overset{?}{\geq}u\text{ in }B_{r}.$

By (4.10) we know that

$u(x)\leq\psi(x):=\inf_{y^{\prime}\in
B_{1}\cap\\{x_{n}=0\\}}\varphi^{+}_{1}(y^{\prime})+\phi^{+}_{1}(x-y^{\prime})\text{
for }x\in B_{1}\cap\\{x_{n}\leq 0\\}.$

By Lemma 4.13 we know that $\psi\leq\varphi^{+}_{3/4}$ in $B_{r}$.

This last step concludes the proof as it contradicts $u$ being a sub-solution
of $Hu+\eta\leq 0$. The test function $\varphi^{+}_{3/4}$ satisfies at the
contact point that $H(D\varphi_{3/4}^{+}(0))=-3\eta/4>-\eta$. ∎

####  4.3.1. Proof of Theorem 4.1

Assume by contradiction that $u$ and $v$ satisfy the hypotheses of the
theorem, however $u\leq v$ on $\partial\Omega$ and $m:=\sup_{\Omega}(u-v)>0$.

Recall the constructions from Section 4.1 and consider the sup and inf-
convolutions (in the directions parallel to $\\{x_{n}=0\\}$) of $u$ and $v$
respectively and with respect to some parameter $\varepsilon>0$. Let
$M=\max(\|u\|_{L^{\infty}(\overline{\Omega})},\|v\|_{L^{\infty}(\overline{\Omega})})$
and $r=2\sqrt{M\varepsilon}$. By the uniform convergence convergence of
$u^{\varepsilon}\searrow u$ and $v_{\varepsilon}\nearrow v$, we have that for
$\varepsilon$ sufficiently small we can enforce
$\max_{\partial\Omega^{r}}(u^{\varepsilon}-v_{\varepsilon})\leq m/4$ meanwhile
$\sup_{\Omega^{r}}(u^{\varepsilon}-v_{\varepsilon})\geq m/2$.

The hypotheses ($H_{-}$) and ($H_{+}$), together with the assumptions $u,v\in
C(\overline{\Omega})\cap C^{0,1}(\Omega_{-})$ allows to get that for
$\varepsilon$ even smaller, $u^{\varepsilon}$ is a sub-solution of

$\displaystyle\begin{cases}H_{-}u+3\eta/4\leq 0\text{ in
}\Omega^{r}_{-}\cup\Gamma^{r},\\\ H_{+}u+3\eta/4\leq 0\text{ in
}\Omega^{r}_{+},\end{cases}$

meanwhile $v_{\varepsilon}$ is a super-solution of

$\displaystyle\begin{cases}H_{-}v+\eta/4\geq 0\text{ in
}\Omega^{r}_{-}\cup\Gamma^{r},\\\ H_{+}v+\eta/4\geq 0\text{ in
}\Omega^{r}_{+}.\end{cases}$

If
$\operatorname*{argmax}_{\Omega_{r}}(u^{\varepsilon}-v_{\varepsilon})\cap(\Omega_{-}^{r}\cup\Omega_{+}^{r})\neq\emptyset$,
then we get a contradiction by the standard comparison principle. Let us
assume then that there is some
$x_{0}\in\operatorname*{argmax}_{\Omega_{r}}(u^{\varepsilon}-v_{\varepsilon})\cap\Gamma^{r}$
and $c=(u^{\varepsilon}-v_{\varepsilon})(x_{0})>0$. Therefore, the function
$u^{\varepsilon}$ touches $v_{\varepsilon}+c$ from below at $x_{0}\in\Gamma$
over some small neighborhood $\overline{B_{\rho}(x_{0})}\subseteq\Omega^{r}$.

Let $H_{0}(p):=H_{-}(p,u(x_{0}),x_{0})$. By the continuity of the solutions
and the operators, we get that for $\rho>0$ sufficiently small, the function
$u^{\varepsilon}$ is a viscosity sub-solution of

$H_{0}u+5\eta/8\leq 0\text{ in }B_{\rho}(x_{0})\cap\\{x_{n}\leq 0\\},$

and $v_{\varepsilon}+c$ is a viscosity super-solution of

$H_{0}v+3\eta/8\geq 0\text{ in }B_{\rho}(x_{0})\cap\\{x_{n}\leq 0\\}.$

By taking $H:=H_{0}+3\eta/8$ we get that $u^{\varepsilon}$ and
$v_{\varepsilon}+c$ satisfy the translation invariant equations (Sub’) and
(Sup’) respectively for the gap $\eta/4>0$.

As in Lemma 2.6, we consider now $\bar{u}\in C(\overline{B_{\rho}(x_{0})})$ to
be the lifting of $u^{\varepsilon}$ constructed from the boundary value
problem

$\begin{cases}H_{+}\bar{u}+3\eta/4=0\text{ in
}B_{\rho}(x_{0})\cap\\{x_{n}>0\\},\\\ \bar{u}=u^{\varepsilon}\text{ on
}\overline{B_{\rho}(x_{0})}\setminus(B_{\rho}(x_{0})\cap\\{x_{n}>0\\}).\end{cases}$

In a similar way we let $\bar{v}\in C(\overline{B_{\rho}(x_{0})})$ to be given
by

$\begin{cases}H_{+}\bar{v}+\eta/4=0\text{ in
}B_{\rho}(x_{0})\cap\\{x_{n}>0\\},\\\ \bar{v}=v_{\varepsilon}+c\text{ on
}\overline{B_{\rho}(x_{0})}\setminus(B_{\rho}(x_{0})\cap\\{x_{n}>0\\}).\end{cases}$

The comparison principle for uniformly elliptic equations guarantee that we
still have that $\bar{u}$ touches $\bar{v}$ from below at $x_{0}$. The
advantage of this construction is that now both $\bar{u}$ and $\bar{v}$ now
satisfy (UE’) in $B_{r}(x_{0})$, meanwhile they keep satisfying equations
(Sub’) and (Sup’), by the same argument towards the end in the proof of Lemma
2.6.

Now that we have all the hypotheses from Lemma 4.14, we we have the desired
contradiction as $\bar{u}$ is not allowed to touch $\bar{v}$ from below at
$x_{0}\in\Gamma$.∎

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