Discontinuous viscosity solutions of first order Hamilton-Jacobi equations

TL;DR

This paper establishes the uniqueness and existence of discontinuous viscosity solutions for a class of first order Hamilton-Jacobi equations with initial data having finite jump discontinuities, and analyzes their evolution.

Contribution

It proves the uniqueness and existence of discontinuous viscosity solutions for Hamilton-Jacobi equations with jump discontinuities in initial data, using comparison theorems and barrier effects.

Findings

Uniqueness of solutions with finite jump discontinuities.

Existence of viscosity solutions under given conditions.

Characterization of the evolution of discontinuities over time.

Abstract

We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the initial function has a finite number of jump discontinuities, the corresponding discontinuous viscosity solution of the corresponding Cauchy problem on the real line is unique. Uniqueness follows from a comparison theorem for semicontinuous viscosity sub- and supersolutions, using the barrier effect of spatial discontinuities of a solution. We also prove an existence theorem, as well as a comparison theorem for viscosity solutions with different initial data. In addition, we describe several properties of the evolution of the jump discontinuities.

Equations

$$u_t+H(u_x)=0$$

$$\{\begin{array}{ll}{u}_t+H(u_x)=0& \text{in }(-\infty ,c)\times (0,\tau )\\ u_x(c,t)=\infty & \text{for }0<t<\tau \\ u(x,0)=u_0(x)& \text{for }x<c,\end{array}\{\begin{array}{ll}{u}_t+H(u_x)=0& \text{in }(c,\infty )\times (0,\tau )\\ u_x(c,t)=\infty & \text{for }0<t<\tau \\ u(x,0)=u_0(x)& \text{for }x>c.\end{array}$$

$$[u{]}_{\pm }=\max \{\pm u,0\},{\mathrm{sgn}}_{\pm }(u)=\pm {\chi }_{{\mathbb{R}}_{\pm }}(u),\mathrm{sgn}(u)={\mathrm{sgn}}_{-}(u)+{\mathrm{sgn}}_{+}(u).$$

$$\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}z(x,t):=\underset{\delta >0}{\inf }\left(\mathrm{ess}\underset{(x,t)\in D\cap B_{\delta }(x_0,t_0)}{\sup }z(x,t)\right)=\underset{\delta \to 0^{+}}{\lim }\left(\mathrm{ess}\underset{(x,t)\in D\cap B_{\delta }(x_0,t_0)}{\sup }z(x,t)\right),$$

$$\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{liminf}}z(x,t):=\underset{\delta >0}{\sup }\left(\mathrm{ess}\underset{(x,t)\in D\cap B_{\delta }(x_0,t_0)}{\inf }z(x,t)\right)=\underset{\delta \to 0^{+}}{\lim }\left(\mathrm{ess}\underset{(x,t)\in D\cap B_{\delta }(x_0,t_0)}{\inf }z(x,t)\right),$$

$$B_r(x_0,t_0)=\{(x,t)\in {\mathbb{R}}^2|(x-x_0{)}^2+(t-t_0{)}^2<r^2\}(r>0).$$

$$\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\lim }z(x,t):=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}z(x,t)=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{liminf}}z(x,t).$$

$$\text{ess}\underset{D\ni (x,t)\to (x_0,t_0^{\pm })}{\mathrm{limsup}}z(x,t),\text{ess}\underset{D\ni (x,t)\to (x_0,t_0^{\pm })}{\mathrm{liminf}}z(x,t)$$

$$\text{ess}\underset{D\ni (x,t)\to (x_0^{\pm },t_0)}{\mathrm{limsup}}z(x,t),\text{ess}\underset{D\ni (x,t)\to (x_0^{\pm },t_0)}{\mathrm{liminf}}z(x,t)$$

$$z^{\ast }(x_0,t_0):=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}z(x,t)\text{for any }(x_0,t_0)\in \overline{D}.$$

$$z_{\ast }(x_0,t_0):=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{liminf}}z(x,t)\text{for any }(x_0,t_0)\in \overline{D}.$$

$$z^{\ast }(x_0,t_0^{\pm }):=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0^{\pm })}{\mathrm{limsup}}z(x,t),z_{\ast }(x_0,t_0^{\pm }):=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0^{\pm })}{\mathrm{liminf}}z(x,t),$$

$$z^{\ast }(x_0^{\pm },t_0):=\text{ess}\underset{D\ni (x,t)\to (x_0^{\pm },t_0)}{\mathrm{limsup}}z(x,t),z_{\ast }(x_0^{\pm },t_0):=\text{ess}\underset{D\ni (x,t)\to (x_0^{\pm },t_0)}{\mathrm{liminf}}z(x,t).$$

$$z^{\ast }(x_0,t_0^{\pm })\le z^{\ast }(x_0,t_0),z_{\ast }(x_0,t_0^{\pm })\ge z_{\ast }(x_0,t_0),$$

$$z^{\ast }(x_0^{\pm },t_0)\le z^{\ast }(x_0,t_0),z_{\ast }(x_0^{\pm },t_0)\ge z_{\ast }(x_0,t_0).$$

$$\underset{\overline{D}\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}{z}^{\ast }(x,t)\le z^{\ast }(x_0,t_0),\underset{\overline{D}\ni (x,t)\to (x_0,t_0)}{\mathrm{liminf}}{z}_{\ast }(x,t)\ge z_{\ast }(x_0,t_0).$$

$$(z^{\ast }{)}^{\ast }=(z_{\ast }{)}^{\ast }=z^{\ast },(z^{\ast }{)}_{\ast }=(z_{\ast }{)}_{\ast }=z_{\ast }\text{in }\overline{D}.$$

$$(z_{\ast }{)}^{\ast }(x_0,t_0)$$

$$=\text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}{z}^{\ast }(x,t)\le \underset{\overline{D}\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}{z}^{\ast }(x,t)\le z^{\ast }(x_0,t_0)$$

$$z^{\ast }(x_0,t_0)\le \text{ess}\underset{\overline{D}\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}{z}_{\ast }(x,t).$$

$$\mathrm{ess}\underset{(x,t)\in B_r(x_0,t_0)\cap D}{\sup }z(x,t)>z^{\ast }(x_0,t_0)-\frac{1}{2}\epsilon \text{mbox}forallr\le r_{\epsilon }.$$

$$z(x,t)\geq{\rm ess}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\sup_{(x,t)\in B_{r}(x_{0},t_{0})\cap D}\!\!z(x,t)-\tfrac{1}{2}\varepsilon>z^{*}(x_{0},t_{0})-\varepsilon\quad\ \mbox{for a.e.~{}$(x,t)\in\mathcal{B}_{r,\varepsilon}$}\,.$$

$$z>z^{\ast }(x_0,t_0)-\epsilon \text{a.e. in }{\mathcal{B}}_{\epsilon }.$$

$$z>z^{\ast }(x_0,t_0)-\epsilon \text{a.e. in }{\mathcal{B}}_{\epsilon }\cap B_{\delta }(x,t),(|{\mathcal{B}}_{\epsilon }\cap B_{\delta }(x,t)|>0\text{mbox}forall\delta >0),$$

$$\text{ess}\underset{\overline{D}\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}{z}_{\ast }(x,t)\ge z^{\ast }(x_0,t_0)-\epsilon .$$

$$(z_1{)}^{\ast }(x_0,t_0)=\text{ess}\underset{D_1\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}z(x,t)\le \text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{limsup}}z(x,t)=z^{\ast }(x_0,t_0),$$

$$(z_1{)}_{\ast }(x_0,t_0)=\text{ess}\underset{D_1\ni (x,t)\to (x_0,t_0)}{\mathrm{liminf}}z(x,t)\ge \text{ess}\underset{D\ni (x,t)\to (x_0,t_0)}{\mathrm{liminf}}z(x,t)=z_{\ast }(x_0,t_0)$$

Full text

1. Introduction

After the introduction of continuous viscosity solutions of first order Hamilton-Jacobi (HJ) equations ([15, 16]), it was readily understood that the basic concepts and comparison results of the theory could be extended to the case of semicontinuous viscosity sub- and supersolutions. A systematic study of discontinuities of both the Hamiltonian itself and solutions of HJ equations, started in [23, 24], is an important issue since discontinuous solutions arise in many applications (e.g., optimal control problems, differential game theory; see [3, 14] and references therein).

Existence of possibly discontinuous viscosity solutions was proven in [24] by Perron’s method, but uniqueness of such solutions remained unclear since in general the classical comparison result for semicontinuous viscosity sub- and supersolutions does not imply uniqueness of viscosity solutions - apart from the trivial case of continuous data, when the unique viscosity solution is continuous ([12]). In addition, the literature contains interesting examples of nonuniqueness of viscosity solutions for non-convex Hamiltonians with explicit space and/or time dependence and discontinuous initial data ([3, 6, 22]).

Motivated by these difficulties, several different notions of discontinuous solutions of HJ equations have been proposed ([1, 5, 7, 13, 22, 30]) to prove existence, comparison and uniqueness results under various assumptions (for instance, if the Hamiltonian is convex). Although interesting in their own right and for specific applications (e.g. to control problems), the relationship between different notions of solution have been elucidated only in some specific cases (see [14, 22]).

In the present paper we consider discontinuous viscosity solutions, defined in the spirit of [24], in the simplest example of a time-dependent HJ equation, i.e. the one-dimensional equation

[TABLE]

with a bounded and uniformly Lipschitz continuous Hamiltonian:

[TABLE]

We do not require convexity conditions on $H$, nor the existence of $\lim\limits_{s\to\pm\infty}H(s)$. The boundedness assumption for $H$ is suggested by a model for ion etching ([21, 28, 29]).

Given $u_{0}\in L^{\infty}_{{\rm loc}}(\mathbb{R})$, we consider in particular the Cauchy problem

[TABLE]

As in Ishii’s paper ([24]), the definition of (possibly discontinuous) viscosity sub- and supersolutions is based on the concept of semicontinuous envelopes of functions, but we use essential limits to define semicontinuous envelopes (see Section 2). The use of essential limits excludes some unnatural and artificial examples of admissible viscosity solutions (see for example to the discussion on page 27 of [17]). As in [24], existence of viscosity solutions of $(CP)$ is based on Perron’s method, which can be adapted to our definitions (see Section 6). Also the basic comparison result continues to hold for discontinuous viscosity sub- and supersolutions (Section 4); since, by $(H_{1})$, the speed of propagation is bounded by the Lipschitz constant $\|H^{\prime}\|_{\infty}$ ([16, 25]), the comparison result is formulated locally.

Since $H$ is bounded, it is rather intuitive that solutions are Lipschitz continuous with respect to time, and that spatial jump discontinuities of the solution do not disappear instantaneously. In Section 5 we show that this is indeed the case, and in addition we prove that spatial jump discontinuities are non-increasing in time and satisfy an explicit decay rate if $\limsup_{s\to\pm\infty}H(s)\!>\!\liminf_{s\to\pm\infty}H(s)$.

The main purpose of the paper is to show that the viscosity solution of $(CP)$ is unique if $u_{0}$ has a finite number of jump discontinuities:

[TABLE]

If $u_{0}$ is continuous, the comparison result implies uniqueness of the viscosity solution. In Section  7 we present the proof of uniqueness if $u_{0}$ has jump discontinuities.

We briefly describe our approach. By the boundedness of $H$, a jump discontinuity of $u_{0}$ at a certain point $c$ does not disappear instantaneously. On the other hand it is known ([19]) that spatial discontinuities of a solution produce a barrier effect : if a viscosity solution $u$ has a spatial jump discontinuity at $c\in\mathbb{R}$ for $t\in[0,\tau)$, the evolution of $u$ in $(-\infty,c)\times(0,\tau)$ is independent of that in $(c,\infty)\times(0,\tau)$. More precisely (see Lemma 5.2), if the jump $u(c^{+},t)-u(c^{-},t)$ is positive for $t\in(0,\tau)$, then $u$ satisfies on either side of $y$ the singular Neumann problems

[TABLE]

Similarly if $u(c^{+},t)-u(c^{-},t)<0$, with $u_{x}(c,t)=\infty$ replaced by $u_{x}(c,t)=-\infty$.

This leads to the following procedure to prove uniqueness of a discontinuous viscosity solution of $(CP)$. Let $x_{1},\ldots,x_{p}$ be points where $u_{0}$ has a jump discontinuity, let $u$ and $v$ be two viscosity solutions of $(CP)$ and let $\tau_{1}\in(0,T]$ be the maximal time for which the jump continuities of $u$ and $v$ at $x_{k}$ persist in $[0,\tau_{1})$ for all $k=1,\ldots,p$. We define the intervals $I_{0}=(-\infty,x_{1})$, $I_{k}=(x_{k},x_{k-1})$ for $k=1,\ldots,p-1$ and $I_{p}=(x_{p},\infty)$. Then the restrictions of $u$ and $v$ to $I_{k}\times(0,\tau_{1})$ solve the same singular Neumann problem in $I_{k}\times(0,\tau_{1})$ with continuous initial data. Also this singular problem satisfies a comparison principle for viscosity sub- and supersolutions, and since the initial function restricted to $I_{k}$ is continuous, this implies that $u=v$ a.e. in $I_{k}\times(0,\tau_{1})$ for all $k$, and thus a.e. in $\mathbb{R}\times(0,\tau_{1})$. If $\tau_{1}<T$, a finite iteration of this procedure proves uniqueness of the viscosity solution of $(CP)$.

Reassuming, we introduce a procedure, based on the barrier effect of spatial discontinuities, which indicates how the comparison result for semicontinuous sub- and supersolutions can be used to prove uniqueness of suitably defined viscosity solutions with discontinuous initial data. Although we have only done this for a particularly simple problem, preliminary calculations suggest that the procedure can be adapted to more general problems (to be addressed in future papers), namely the cases of initial data with infinitely many jump discontinuities and Hamiltonians with linear growth and explicit $x$ and $t$ dependence. The latter case is particularly interesting since it includes equations for which uniqueness of discontinuous viscosity solutions fails, as mentioned in the beginning of the Introduction. In particular our approach seems to suggest a mathematical uniqueness criterium. Many other problems concerning discontinuous solutions of first order HJ equations remain to be solved, in particular the multidimensional case.

Finally we observe that, setting $v=u_{x}$ and $v_{0}=u_{0}^{\prime}$, problem $(CP)$ is formally related to the Cauchy problem for a scalar conservation law,

[TABLE]

although it is not trivial to make the correspondence rigorous ([11]; see also [4] for a statement in this direction). If $v_{0}=u_{0}^{\prime}$ is a signed Radon measure, it is possible to prove existence of suitably defined measure-valued entropy solutions of problem $(CL)$ ([10]; see also [8, 9] for the case of positive initial measures). Remarkably, if the singular part $v_{0s}$ of $v_{0}$ (with respect to the Lebesgue measure) is a finite superposition of Dirac masses, the uniqueness of such solutions requires some additional compatibility conditions to be satisfied near the support of $v_{0s}$. The singularities of $v_{0}$ in $(CL)$ have a barrier effect which corresponds to that produced by discontinuities of $u_{0}$ in $(CP)$, and well-posedness of $(CL)$ can be proven using singular Dirichlet problems which are the natural counterpart of singular Neumann problems for HJ equations (see [9, 10] for details).

2. Semicontinuous envelopes

Let $\chi_{E}$ denote the characteristic function of $E\subseteq\mathbb{R}$. For all $u\in\mathbb{R}$ we set

[TABLE]

Let $D\subset\mathbb{R}^{2}$ be open, $z:D\mapsto\mathbb{R}$ a measurable function, and $(x_{0},t_{0})\in\overline{D}$. We set

[TABLE]

[TABLE]

where

[TABLE]

If $\text{\rm ess}\limsup_{D\ni(x,t)\to(x_{0},t_{0})}z(x,t)=\text{\rm ess}\liminf_{D\ni(x,t)\to(x_{0},t_{0})}z(x,t)$, we also set

[TABLE]

The quantities

[TABLE]

are defined replacing $B_{r}(x_{0},t_{0})$ by $B_{r}(x_{0},t_{0})\cap\{(x,t)\in\mathbb{R}^{2}\,|\,t\gtrless t_{0}\}$. Similarly,

[TABLE]

are defined replacing $B_{r}(x_{0},t_{0})$ by $B_{r}(x_{0},t_{0})\cap\{(x,t)\in\mathbb{R}^{2}\,|\,x\gtrless x_{0}\}$.

Let $z\in L^{\infty}_{{\rm loc}}(\overline{D})$. By the essential upper semicontinuous envelope of $z$ we mean the function $z^{*}:\overline{D}\to\mathbb{R}$,

[TABLE]

The essential lower semicontinuous envelope of $z$ is the function $z_{*}:\overline{D}\to\mathbb{R}$,

[TABLE]

We also set

[TABLE]

[TABLE]

Observe that

[TABLE]

[TABLE]

Similar definitions hold for any measurable function $z:F\subseteq\mathbb{R}\to\mathbb{R}$. For shortness, we shall say “upper (respectively lower) envelope” instead of “essential upper (respectively lower) semicontinuous envelope”.

It is easily checked that the upper (lower) envelope $z^{*}$ ($z_{*}$) is indeed upper (lower) semicontinuous in $\overline{D}$, namely for any $(x_{0},t_{0})\in\overline{D}$

[TABLE]

The inequalities in (2.7) can be replaced by equalities. More generally we have:

Lemma 2.1**.**

Let $D\subseteq\mathbb{R}^{2}$ be open and $z\in L^{\infty}_{{\rm loc}}(\overline{D})$. Then

[TABLE]

Proof.

Let $(x_{0},t_{0})\in\overline{D}$. We only prove that $(z^{*})^{*}(x_{0},t_{0})=(z_{*})^{*}(x_{0},t_{0})=z^{*}(x_{0},t_{0})$. Since

[TABLE]

(see (2.1), (2.2) and (2.7)), it is enough to show that

[TABLE]

For every $\varepsilon>0$ there exists $r_{\varepsilon}>0$ such that

[TABLE]

Therefore, for every such $r$ there exists $\mathcal{B}_{r,\varepsilon}\subseteq B_{r}(x_{0},t_{0})\cap D$, $|\mathcal{B}_{r,\varepsilon}|>0$, such that

[TABLE]

Setting $\mathcal{B}_{\varepsilon}=\bigcup_{r\leq r_{\varepsilon}}\mathcal{B}_{r,\varepsilon}$, it follows that $|\,\mathcal{B}_{\varepsilon}\cap B_{r}(x_{0},t_{0})|>0$ for every $r>0$ (hence $|\mathcal{B}_{\varepsilon}|>0$), and

[TABLE]

Let $(x,t)\in\mathcal{B}_{\varepsilon}$ satisfy $|B_{\delta}(x,t)\cap\mathcal{B}_{\varepsilon}|>0$ for every $\delta>0$; this choice is possible up to a null set $\mathcal{N}_{\varepsilon}\subset\mathcal{B}_{\varepsilon}$, since $|\mathcal{B}_{\varepsilon}|>0$ and almost every $(x,t)\in\mathcal{B}_{\varepsilon}$ is a Lebesgue point of $f(x,t)=\chi_{\mathcal{B}_{\varepsilon}}(x,t)$ ($e.g.$, see [20, Subsection 1.7.1]). Then we have

[TABLE]

whence $z_{*}(x,t)\geq z^{*}(x_{0},t_{0})-\varepsilon$ for a.e. $(x,t)\in\mathcal{B}_{\varepsilon}$. Since $|\,\mathcal{B}_{\varepsilon}\cap B_{r}(x_{0},t_{0})\,|>0$ for all $r>0$, this implies that

[TABLE]

and (2.9) follows from the arbitrariness of $\varepsilon$. ∎

Similar results hold if $D\subseteq\mathbb{R}$ is open and $z\in L^{\infty}_{{\rm loc}}(\overline{D})$.

Remark 2.1**.**

Since the definition of $z^{*},z_{*}$ depends on the domain of definition of $z$, different restrictions of $z$ can have different upper and lower envelopes. In fact, let $D_{1}\subseteq D$ be an open set, and let $z_{1}=z\lefthalfcup D_{1}$ be the restriction of $z$ to $D_{1}$. If $(x_{0},t_{0})\in\partial D_{1}$, then:

[TABLE]

[TABLE]

(of course, if $(x_{0},t_{0})\in\mathring{D}_{1}$ these inequalities become equalities). Observe that (2.5)-(2.6) are a particular case of these inequalities. If $D=\Omega\times(0,T)$ and $t_{0}=0$, inequalities (2.5) from above become equalities, namely

[TABLE]

More precisely, let $D_{1}\subseteq D$ be an open set, and let $z_{1}=z\lefthalfcup D_{1}$. Set

[TABLE]

Then for any $(x_{0},t_{0})\in\tilde{D}_{1}$ there holds

[TABLE]

In fact, if $(x_{0},t_{0})\in\tilde{D}_{1}$ then $D\cap B_{\delta_{1}}(x_{0},t_{0})=D_{1}\cap B_{\delta_{1}}(x_{0},t_{0})$ and, by (2.1),

[TABLE]

For further reference we consider some specific cases of $D_{1}\subseteq D=(a,b)\times(0,T)$, with $-\infty<a<b<\infty$. The first example concerns a trapezoidal domain.

• $(a)$

Let $D_{1}=\{(x,t)\,|\,x\in(a,b-\|H^{\prime}\|_{\infty}t),\,t\in(0,\tau)\}$ for $\tau\in(0,\tau_{1}]$, where

[TABLE]

Then $\tilde{D}_{1}=\{(x,t)\,|\,x\in[a,b-\|H^{\prime}\|_{\infty}t),\,t\in[0,\tau)\}$ if $\tau\in(0,\tau_{1})$, and $\tilde{D}_{1}=\{(x,t)\,|\,x\in[a,b-\|H^{\prime}\|_{\infty}t),\,t\in[0,T]\}$ if $\tau=\tau_{1}=T$;

• $(b)$

let $D_{1}=Q_{\tau,T}\,:=\,\{(x,t)\in D\,|\,t\in(\tau,T)\}$ for any $\tau\in(0,T)$. Then $\tilde{D}_{1}=\{(x,t)\,|\,x\in[a,b],\,t\in(\tau,T]\}$;

• $(c)$

let $D_{1}=\{(x,t)\in D\,|\,x\in(a,c),\,t\in(0,T)\}$ for any $c\in(a,b)$. Then $\tilde{D}_{1}=\{(x,t)\,|\,x\in[a,c),\,t\in[0,T]\}$.

3. Definitions and results

As we have explained in the Introduction, the proof of the main uniqueness result requires the introduction of singular Neumann problems. Given $a,b\in\mathbb{R}$, $a<b$, we consider the problems

[TABLE]

In addition we consider equation (1.1) in the following trapezoidal domains:

[TABLE]

Due to their slope, no boundary conditions are required on the oblique sides:

[TABLE]

All these problems require an initial condition $u_{0}$, which is assumed to be locally bounded in the main existence and regularity results (see Proposition 3.2 and Theorem 3.3 below) and, in addition, piecewise continuous in the uniqueness Theorem 3.4. More precisely, a function $f\in L^{\infty}(\Omega)$ defined in a bounded interval $\Omega=(a,b)$ is said to be piecewise continuous in $\Omega$ if

• $\Omega=\bigcup_{j=1}^{p+1}I_{j}$ $(p\in\mathbb{N})$ with $I_{1}=(a,x_{1})$, $I_{j}=(x_{j-1},x_{j})$ if $2\leq j\leq p$, $I_{p+1}=(x_{p},b)$;

• the restriction $f\lefthalfcup I_{j}$ has a representative $f_{j}\in C(\overline{I}_{j})$ for $1\leq j\leq p+1$, $f_{j}(x_{j})\neq f_{j+1}(x_{j})$ for $1\leq j\leq p$.

Observe that a function $f\in C(\overline{\Omega})$ is piecewise continuous (the case $p=0$). If instead $\Omega$ is an unbounded interval, a function $f\in L^{\infty}_{{\rm loc}}(\overline{\Omega})$ is piecewise continuous in $\Omega$ if it is piecewise continuous in every bounded interval $(a_{0},b_{0})\subset\Omega$.

3.1. Definitions

The following definitions are used throughout the paper.

Definition 3.1**.**

Let $D\subseteq\mathbb{R}^{2}$ open and $u\in L^{\infty}_{{\rm loc}}(\overline{D})$.

$(i)$ $u$ is a viscosity subsolution of equation (1.1) in $D$ if for all $\varphi\in C^{1}(D)$ the following condition holds: if $(x,t)\!\in\!D$ is a local maximum point of $u^{*}\!-\varphi$ in $D$, then

[TABLE]

$(ii)$ $u$ is a viscosity supersolution of equation (1.1) in $D$ if for all $\varphi\in C^{1}(D)$ the following condition holds: if $(x,t)\in D$ is a local minimum point of $u_{*}-\varphi$ in $D$, then

[TABLE]

If $D\subseteq\mathbb{R}^{2}$ is bounded, viscosity sub and supersolutions of (1.1) in $D$ belong to $L^{\infty}(D)$.

We first define the solution concept for problem $(N_{a,\pm}^{b,\pm})$, and then explain how to extend it to other problems.

Definition 3.2**.**

Let $-\infty<a<b<\infty$, $Q=(a,b)\times(0,T)$ and $\hat{Q}=[a,b]\times(0,T]$.

$(i)$ A viscosity subsolution $u$ of (1.1) in $Q$ is a viscosity subsolution of:

• problem $(N_{a,-}^{b,+})$;

• problem $(N_{a,+}^{b,+})$, if for all $\varphi\in C^{1}(\hat{Q})$

[TABLE]

• problem $(N_{a,-}^{b,-})$, if for all $\varphi\in C^{1}(\hat{Q})$

[TABLE]

• problem $(N_{a,+}^{b,-})$, if for all $\varphi\in C^{1}(\hat{Q})$ both (3.4) and (3.5) are satisfied.

$(ii)$ A viscosity supersolution $u$ of (1.1) in $Q$ is a viscosity supersolution of:

• problem $(N_{a,+}^{b,-})$;

• problem $(N_{a,+}^{b,+})$, if for all $\varphi\in C^{1}(\hat{Q})$

[TABLE]

• problem $(N_{a,-}^{b,-})$, if for all $\varphi\in C^{1}(\hat{Q})$

[TABLE]

• problem $(N_{a,-}^{b,+})$, if for all $\varphi\in C^{1}(\hat{Q})$ both (3.6) and (3.7) are satisfied.

$(iii)$ A function $u$ is a viscosity solution of problem $(N_{a,\pm}^{b,\pm})$ if it is both a viscosity subsolution and a viscosity supersolution of $(N_{a,\pm}^{b,\pm})$.

$(iv)$ Let $u_{0}\in L^{\infty}(a,b)$. A viscosity solution of $(N_{a,\pm}^{b,\pm})$ with initial condition $u_{0}$ is a viscosity solution of $(N_{a,\pm}^{b,\pm})$ such that

[TABLE]

Observe that Definition 3.2 makes sense for any $H\in C(\mathbb{R})$.

It is straightforward to modify Definition 3.2 in case of the other problems we mentioned before. For example, a viscosity subsolution of problem (${\rm T}_{a,-}$) is a viscosity subsolution of (1.1) in the set $A$, a viscosity subsolution of problem (${\rm T}_{a,+}$) is a viscosity subsolution of (1.1) in $A$ which satisfies (3.4) for all $\varphi\in C^{1}(\hat{A})$ (where $\hat{A}=\overline{A}\cap(\mathbb{R}\times(0,T])$), and a viscosity subsolution of problem (${\rm T}_{C}$) is a viscosity subsolution of (1.1) in the set $C$.

If $u_{0}$ is defined in an unbounded interval, in point $(iv)$ we only require boundedness of $u_{0}$ in all bounded subintervals. In particular:

Definition 3.3**.**

For every $u_{0}\in L_{\rm loc}^{\infty}(\mathbb{R})$, $u$ is a viscosity solution of the Cauchy problem $(CP)$ if it is a viscosity sub- and supersolution of (1.1) in $\mathbb{R}\times(0,T)$ and satisfies (3.8) in $\mathbb{R}$.

Remark 3.1**.**

Let $Q=(a,b)\times(0,T)$ with $-\infty<a<b<\infty$, and $Q_{1}=(a,b)\times(\tau_{1},\tau_{2})$, for $0\leq\tau_{1}<\tau_{2}\leq T$. It is easily seen that if $u$ is a viscosity subsolution of $(N_{a,\pm}^{b,\pm})$ in $Q$, its restriction $u_{1}=u\lefthalfcup Q_{1}$ is a viscosity subsolution of $(N_{a,\pm}^{b,\pm})$ in $Q_{1}$. In fact, if $(x_{0},t_{0})$ is a local maximum point of $(u_{1})^{*}-\varphi$ in $\hat{Q}_{1}$ and $t_{0}\in(\tau_{1},\tau_{2})$, by (2.12) it is also a local maximum point of $u^{*}-\varphi$ in $\hat{Q}$, whereas in the case $t_{0}=\tau_{2}$ it suffices to argue as in [18, Section 5.2].

Similar remarks hold for viscosity subsolutions of equation (1.1) in $Q$ or of any of the other problems mentioned before, as well as for viscosity supersolutions.

3.2. Main results

In Section 4 we prove the basic comparison result for viscosity sub- and supersolutions:

Theorem 3.1**.**

Let $(H_{1})$ hold. Let $u$ be a viscosity subsolution and $v$ a viscosity supersolution of problem $(N_{a,\pm}^{b,\pm})$. Then

[TABLE]

Similar results are valid for problems $({\rm T}_{a,\pm})$, $({\rm T}^{b,\pm})$ and $({\rm T}_{C})$:

[TABLE]

Theorem 3.1 will be proven by a method of doubling variables adapted from [27], where only the case of homogeneous Neumann boundary conditions was considered (see also [2] for more general Neumann boundary conditions). It implies uniqueness and continuity of viscosity solutions of, for example, problem $(CP)$ if $u_{0}$ is continuous in $\mathbb{R}$ (since $u^{*}\equiv u_{*}$ in $\mathbb{R}\times[0,T]$). If instead $u_{0}$ is not continuous at a point $x=c$, the following regularity results give information about the temporal evolution of $u^{*}-u_{*}$ at $c$. These result, proved in Section 5, are crucially based on the boundedness of $H$.

Proposition 3.2**.**

Let $(H_{1})$ hold and let $\Omega=(a,b)$, with $-\infty\leq a<c<b\leq\infty$, let

[TABLE]

and let $u_{0}\in L^{\infty}_{\rm loc}(\overline{\Omega})$ be such that the one-sided essential limits $u_{0}(c^{\pm})$ exist.

$(i)$* Let $u$ and $v$ be a viscosity sub- and supersolution of equation (1.1) in $(a,b)\times(0,T)$. Then for any $t_{1},t_{2}\in[0,T]$, $t_{1}\neq t_{2}$, and $x\in\overline{\Omega}$*

[TABLE]

$(ii)$* Let $u$ be a viscosity solution of equation (1.1) in $(a,b)\times(0,T)$ with initial function $u_{0}$. Let $u_{0}(c^{+})\neq u_{0}(c^{-})$ and set*

[TABLE]

Then, for all $t\in(0,\min\{\underline{t},T\})$,

[TABLE]

[TABLE]

If (3.13) or (3.14) holds for $0<t<t_{c}$ for some $t_{c}\in[\min\{\underline{t},T\},T]$, then for every $0\leq t_{0}<t_{1}<t_{c}$ we have

[TABLE]

where

[TABLE]

Part $(i)$, a sort of Lipschitz continuity with respect to $t$, might be easily guessed from the boundedness of $H$. Part $(ii)$ quantifies the fact that initial jump discontinuities do not disappear instantaneously; in addition, if $A_{\pm}>0$ it will be seen below that for initial data as in $(H_{2})$ spatial jumps may disappear in the interval $(0,T)$ (see Theorem 3.4 and Remark 3.2; see also [11]).

In Section 6 we use Perron’s method to prove the existence of viscosity solutions:

Theorem 3.3**.**

If $H$ satisfies $(H_{1})$ and $u_{0}\in L^{\infty}_{{\rm loc}}(\mathbb{R})$, problem $(CP)$ has a viscosity solution. Similarly, if $-\infty<a<b<\infty$, there exists a viscosity solution of:

- problem $(N_{a,\pm})$ with initial function $u_{0}\in L^{\infty}_{\rm loc}([a,\infty))\,;$

- problem $(N^{b,\pm})$ with initial function $u_{0}\in L^{\infty}_{\rm loc}(-\infty,b])\,;$

- problem $(N_{a,\pm}^{b,\pm})$ with initial function $u_{0}\in L^{\infty}((a,b))\,$.

Theorem 3.1 will be used to prove uniqueness of discontinuous viscosity solutions of problems $(CP)$, $(N_{a,\pm})$, $(N^{b,\pm})$ and $(N_{a,\pm}^{b,\pm})$ with initial data which have a finite number of jump discontinuities. Below we indicate these problems generically by

[TABLE]

So $-\infty\leq a<b\leq\infty$, $a=-\infty$ in case of problems $(CP)$ and $(N^{b,\pm})$, $b=\infty$ in case of problems $(CP)$ and $(N_{a,\pm})$, and in condition $(H_{2})$ the set $\mathbb{R}$ is replaced by $\Omega$.

We denote by $x_{j}$ the points where $u_{0}$ is discontinuous and set $Q=(a,b)\times(0,T)$ and $Q_{j}=I_{j}\times(0,T)$ for $j=1,\dots,p+1$, where

[TABLE]

The main statement of the following result is part $(i)$, the uniqueness of viscosity solutions.

Theorem 3.4**.**

Let $H$ satisfy $(H_{1})$, and let $K$, $k$ be defined by (3.11). Let $u_{0}$ satisfy $(H_{2})$ in $(a,b)$ $(-\infty\leq a<b\leq\infty)$.

$(i)$* If $u$ and $v$ are viscosity solutions of problem $(N)$ with initial condition $u_{0}$, then $u=v$ a.e. in $Q$.*

$(ii)$* If $u$ is the viscosity solution of problem $(N)$ with initial condition $u_{0}$, then:*

• $(a)$

for all $j=1,\dots,p+1$ the restriction $u_{j}=u\lefthalfcup Q_{j}$ has a continuous representative $\tilde{u}_{j}$ in $\overline{Q}_{j}$;

• $(b)$

for all $j=1,\dots,p$ there exists a unique $\tau_{j}\in(0,T]$ such that

[TABLE]

• $(c)$

for all $j=1,\dots,p+1$ the representative $\tilde{u}_{j}$ is Lipschitz continuous with respect to $t$ in $\overline{Q}_{j}$:

[TABLE]

• $(d)$

for all $j=1,\dots,p$ and $0\leq t_{0}<t_{1}<\tau_{j}$,

[TABLE]

[TABLE]

where $A_{\pm}$ is defined by (3.16) with $c$ replaced by $x_{j}$.

Remark 3.2**.**

Let $(H_{1})$-$(H_{2})$ be satisfied and let $u$ be the viscosity solution of problem $(N)$ with initial function $u_{0}$. By claim $(a)$ in Theorem 3.4$(ii)$,

[TABLE]

Conversely, if $u_{0}$ has a jump discontinuity at $x_{j}$, then from claim $(b)$ in Theorem 3.4$(ii)$ it follows that for every $t\in(0,T]$

[TABLE]

and there exists $\tau_{j}\in(0,T]$ such that

[TABLE]

More precisely, if $u_{0}$ has a jump discontinuity at $x_{j}$, then initial jumps cannot change sign: for all $t\in[0,\tau_{j})$ there holds

[TABLE]

[TABLE]

(see also Proposition 3.2 and Lemma 5.1 below).

An almost immediate consequence of Theorems 3.3 and 3.1 is a comparison principle for piecewise continuous viscosity solutions:

Corollary 3.5**.**

Let $H$ satisfy $(H_{1})$, and let $u_{0}$ and $v_{0}$ satisfy $(H_{2})$ in $\Omega=(a,b)$. If $u$ and $v$ are viscosity solutions of problem $(N)$ with initial data $u_{0}\leq v_{0}$ a.e. in $\Omega$, then $u\leq v$ a.e. in $Q$.

4. Comparison between viscosity sub- and supersolutions

In this Section we prove Theorem 3.1. To do so, we need two preliminary results of independent interest.

Proposition 4.1**.**

Let $H\in C(\mathbb{R})$. Let $Q_{1}=(c,d)\times(t_{1},t_{2})$, $\hat{Q}_{1}=[c,d]\times(t_{1},t_{2}]$ with $-\infty<c<d<\infty$, $0\leq t_{1}<t_{2}\leq T$.

$(i)$* Let $u$ be a viscosity subsolution of equation (1.1) in $Q_{1}$, and let $\varphi\in C^{1}(\hat{Q}_{1})$.*

• $(a)$

Let $(c,t_{0})\in\hat{Q}_{1}$ be a local maximum point of $u^{*}-\varphi$ in $\hat{Q}_{1}$. Then

[TABLE]

• $(b)$

Let $(d,t_{0})\in\hat{Q}_{1}$ be a local maximum point of $u^{*}-\varphi$ in $\hat{Q}_{1}$. Then

[TABLE]

$(ii)$* Let $u$ be a viscosity supersolution of equation (1.1) in $Q_{1}$, and let $\varphi\in C^{1}(\hat{Q}_{1})$.*

• $(a^{\prime})$

Let $(c,t_{0})\in\hat{Q}_{1}$ be a local minimum point of $u_{*}-\varphi$ in $\hat{Q}_{1}$. Then

[TABLE]

• $(b^{\prime})$

Let $(d,t_{0})\in\hat{Q}_{1}$ be a local minimum point of $u_{*}-\varphi$ in $\hat{Q}_{1}$. Then

[TABLE]

Proof.

We only prove (4.1). The proofs of (4.2)-(4.4) are similar.

Let $(c,t_{0})$ be a local maximum point of $u^{*}-\varphi$ in $\hat{Q}_{1}$. Then

[TABLE]

whence

[TABLE]

for all $\xi\geq\varphi_{x}(c^{+},t_{0})$ and

[TABLE]

First we consider the case that $\bar{\xi}>-\infty$. Then (4.5) holds with $\xi=\bar{\xi}$ and there exists $\psi\in C^{1}(\hat{Q}_{1})$ such that $\psi_{x}(c,t_{0})=\bar{\xi}$, $\psi_{t}(c,t_{0})=\varphi_{t}(c,t_{0})$, and $u^{*}-\psi$ has a strict maximum at $(c,t_{0})$ ($e.g.$, see [26, Proposition 2.6]). For all sufficiently small $\delta>0$ the function $u^{*}-\psi+\delta(x-c)$ has a maximum at some point $(x_{\delta},t_{\delta})\in(c,d)\times(t_{1},t_{2}]$; observe that $x_{\delta}>c$ by the minimality of $\bar{\xi}$, and $(x_{\delta},t_{\delta})\to(c,t_{0})$ as $\delta\to 0^{+}$. By (3.2), for such values of $\delta$ there holds $\psi_{t}(x_{\delta},t_{\delta})+H(\psi_{x}(x_{\delta},t_{\delta})-\delta)\leq 0$. Letting $\delta\to 0^{+}$ we obtain that $\varphi_{t}(c,t_{0})+H(\bar{\xi})\leq 0$. Since $\bar{\xi}\leq\varphi_{x}(c^{+},t_{0})$, we obtain (4.1).

Now let $\bar{\xi}=-\infty$. Then there exists a sequence $\xi_{n}\to-\infty$ such that (4.5) holds for $\xi=\xi_{n}$, thus for all $\xi\geq\xi_{n}$. By the arbitariness of $n$, (4.5) holds for all $\xi\leq\varphi_{x}(c,t_{0})$. Hence for any $\xi\leq\varphi_{x}(c,t_{0})$ there exists $\psi\in C^{1}(\hat{Q}_{1})$ such that $\psi_{x}(c,t_{0})=\xi$, $\psi_{t}(c,t_{0})=\varphi_{t}(c,t_{0})$, and $u^{*}-\psi$ has a strict maximum at $(c,t_{0})$.

Since, by Lemma 2.1, $\displaystyle{u^{*}(c,t_{0})=\!\!\!\limsup_{\overline{Q}_{1}\ni(y,\tau)\to(c,t_{0})}u^{*}(y,\tau)={\rm ess}\!\!\!\!\!\limsup_{Q_{1}\ni(y,\tau)\to(c,t_{0})}u^{*}(y,\tau),}$ there exists $\{(y_{n},\tau_{n})\}\subseteq\overline{Q}_{1}$ such that

[TABLE]

In particular, for every $\varepsilon>0$ there exists $n_{\varepsilon}\in\mathbb{N}$ such that

[TABLE]

Let $\varepsilon>0$ be fixed, and for $n>n_{\varepsilon}$ set

[TABLE]

Then $\psi_{\varepsilon,n}\in C^{1}(\hat{Q}_{1})$, and

[TABLE]

[TABLE]

[TABLE]

Without loss of generality we may assume that $t_{0}<t_{2}$. We fix a positive constant $\sigma<\min\{d-c,t_{0}-t_{1},t_{2}-t_{0}\}$ and set $Q_{2}=(c,c+\sigma)\times(t_{0}-\sigma,t_{0}+\sigma)$.

Claim: If $\varepsilon>0$ is sufficiently small and $n>n_{\varepsilon}$, then $u^{*}-\psi_{\varepsilon,n}$ has a maximum in $\hat{Q}_{1}$ which is attained at a point $(x_{n,\varepsilon},t_{n,\varepsilon})=(x_{n},t_{n})\in Q_{2}$.

In fact, since $u^{*}-\psi$ has a strict maximum at $(c,t_{0})$ and (4.11) holds, for all $\varepsilon>0$ small enough and $n>n_{\varepsilon}$ the function $u^{*}-\psi_{\varepsilon,n}$ has a maximum in $\hat{Q}_{1}$, attained at a point $(x_{n},t_{n})\in[c,c+\sigma)\times(t_{0}-\sigma,t_{0}+\sigma)$. It remains to prove that $x_{n}>c$. Suppose that $x_{n}=c$. Then $t_{n}=t_{0}$, since $u^{*}-\psi$ has a strict maximum at $(c,t_{0})$ and $\psi_{\varepsilon,n}(c,t)=\psi(c,t)$ for every $t\in(t_{0}-\sigma,t_{0}+\sigma)$ (see (4.10)). On the other hand, by (4.7) and the equality $\psi_{\varepsilon,n}(y_{n},\tau_{n})=\psi(y_{n},\tau_{n})-2\varepsilon$ (see (4.8)), we have that

[TABLE]

Since $\psi(y_{n},\tau_{n})\to\psi(c,t_{0})$ (see (4.6)), for all sufficiently large $n$ we also get

[TABLE]

Since $y_{n}>c$ by (4.6), we have found a contradiction and proved the Claim.

Since $u$ is a viscosity subsolution of $u_{t}+H(u_{x})=0$ in $Q_{1}$, it follows from the Claim, (3.2) and the first equality in (4.9), that

[TABLE]

Since $(\psi_{\varepsilon,n})_{x}(x_{n},t_{n})\leq\psi_{x}(x_{n},t_{n})$ (see (4.9)), it follows from (4.12) that

[TABLE]

On the other hand, letting first $n\to\infty$ and then $\varepsilon\to 0^{+}$ in (4.11) we find that $(x_{n},t_{n})=(x_{n,\varepsilon},t_{n,\varepsilon})\to(c,t_{0})$ (recall that $c\leq x_{n,\varepsilon}\leq c+\sigma$ and $\sigma>0$ is arbitrarily fixed). Since $\psi_{x}(c,t_{0})=\xi$, we obtain from (4.13) that

[TABLE]

Hence inequality (4.1) also holds if $\bar{\xi}=-\infty$. ∎

Proposition 4.2**.**

Let $H\in{\rm Lip}(\mathbb{R})$. Let the trapezoid $A$ be defined by (3.1) and let $\varphi\in C^{1}(\hat{A})$, where $\hat{A}\,:=\,\overline{A}\cap(\mathbb{R}\times(0,T])$.

$(i)$* Let $u$ be a viscosity subsolution of equation (1.1) in $A$. Let \big{(}d-\|H^{\prime}\|_{\infty}t_{0},t_{0}\big{)}, with $t_{0}\in(0,T]$, be a local maximum point of $u^{*}-\varphi$ in $\hat{A}$. Then*

[TABLE]

$(ii)$* Let $v$ be a viscosity supersolution of equation (1.1) in $A$. Let \big{(}d-\|H^{\prime}\|_{\infty}t_{0},t_{0}\big{)}, $t_{0}\in(0,T]$, be a local minimum point of $v_{*}-\varphi$ in $\hat{A}$. Then*

[TABLE]

Similar results hold for the trapezoids $B$ and $C$.

Proof.

We only prove claim $(i)$ for $A$. The remaining proofs are similar. Set

[TABLE]

(observe that under the inverse transformation $(y,t)\mapsto(x,t)=(y-\|H^{\prime}\|_{\infty}t,t)$ the set $\mathcal{A}$ is mapped onto $\{(x,t)\,|\,x\in(a+\|H^{\prime}\|_{\infty}(T-t),d-\|H^{\prime}\|_{\infty}t),\,t\in(0,T)\}$, which is a proper subset of $A$).

We set $\tilde{H}(p)=H(p)+\|H^{\prime}\|_{\infty}p$ for $p\in\mathbb{R}$. Then $\tilde{H}$ is Lipschitz continuous on $\mathbb{R}$ and nondecreasing. We claim that $w$ is a viscosity subsolution in $\mathcal{A}$ of the equation

[TABLE]

In fact, fix any $\psi=\psi(y,t)\in C^{1}\left(\hat{\mathcal{A}}\right)$, where $\hat{\mathcal{A}}=\overline{\mathcal{A}}\cap(\mathbb{R}\times(0,T])$, and let $w^{*}-\psi$ have a local maximum at $(\bar{y},\bar{t})\in\mathcal{A}$; here by definition (see (2.1))

[TABLE]

It is easily seen that $w^{*}(y,t)=u^{*}(y-\|H^{\prime}\|_{\infty}t,t)$ for every $(y,t)\in\overline{\mathcal{A}}$ such that $a+\|H^{\prime}\|_{\infty}T<y\leq d$. Therefore $u^{*}-\varphi$, where $\varphi=\varphi(x,t)=\psi(x+\|H^{\prime}\|_{\infty}t,t)$, has a local maximum at $(\bar{y}-\|H^{\prime}\|_{\infty}\bar{t},\bar{t})\in A$. Since $u$ is a viscosity subsolution of equation (1.1) in $A$, by (3.2), we obtain the claim:

[TABLE]

Now let $\varphi\in C^{1}(\hat{A})$, and let $(d-\|H^{\prime}\|_{\infty}t_{0},t_{0})$ $(t_{0}\in(0,T])$ be a local maximum point of $u^{*}-\varphi$ in $\hat{A}$. Then $(d,t_{0})$ is a local maximum point of $w^{*}-\psi$ in $\hat{\mathcal{A}}$, where $\psi\in C^{1}\left(\hat{\mathcal{A}}\right)$ is defined by $\psi=\psi(y,t)=\varphi(y-\|H^{\prime}\|_{\infty}t,t)$. Since $w$ is a viscosity subsolution in $\mathcal{A}$ of (4.16) and $\tilde{H}$ is nondecreasing, by (4.2) we have that

[TABLE]

∎

Now we can prove Theorem 3.1 for trapezoidal domains. We give the proof for problem $(T_{a,\pm})$ in the domain $A$. The proof is similar for problem $(T^{a,\pm})$ in $B$ and easier for problem $(T_{C})$.

Proof of Theorem 3.1 for problem $(T_{a,\pm})$. Arguing by contradiction we suppose that

[TABLE]

for some $\sigma>0$. Consider the function $F:\overline{A}\times\overline{A}\mapsto\mathbb{R}$ defined by

[TABLE]

where $\lambda\in\left(0,\frac{\sigma}{4T}\right)$ is fixed, $p\in(1,2]$ and $\varepsilon\in(0,\min\{d-a,1\})$ with

[TABLE]

This implies that $|\,\alpha|\varepsilon\leq\frac{d-a}{2}$, and so $\frac{a+d}{2}+\alpha\varepsilon\in[a,d]$.

Since $F$ is upper semicontinuous, it attains the maximum in $\overline{A}\times\overline{A}$ at some point $(\bar{x},\bar{t},\bar{y},\bar{s})$. Observe that $F$ and $(\bar{x},\bar{t},\bar{y},\bar{s})$ depend on $\varepsilon$ and $p$, but for notational simplicity we suppress this dependence.

Since $\frac{a+d}{2}+\alpha\varepsilon\in[a,d]$, there holds

[TABLE]

which implies that

[TABLE]

Hence there holds $|\,\bar{t}-\bar{s}\,|\,\leq\,(4M)^{\frac{1}{p}}\,\varepsilon$ and

[TABLE]

Observe that both estimates can be made independent of $p\in(1,2]$.

Now consider the function $g:\overline{A}\mapsto\mathbb{R}$,

[TABLE]

Observe that $g$ is upper semicontinuous, thus its maximum in $\overline{A}$ exists.

Set $A_{\tau}=\{(x,t)\in\overline{A}\,|\,t\in[0,\tau]\}$ for any $\tau\in(0,T]$. Below we prove the following claims.

Claim 1: There exists $\tau\in(0,T)$ such that

[TABLE]

Claim 2: There exists $\varepsilon_{1}\in(0,d-a)$ which does not depend on $p\in(1,2]$ such that for all $\varepsilon\in(0,\varepsilon_{1})$ and $p\in(1,2]$

[TABLE]

To prove Claim 1 set $G(t)\,:=\,\max\limits_{A_{t}}g$  $(t\in(0,T])$. By (4.17) and since $\lambda<\frac{\sigma}{4T}$,

[TABLE]

Since $G$ is nondecreasing, there exists $L_{0}\,:=\,\lim\limits_{t\to 0^{+}}G(t)$, and Claim 1 follows if we prove that

[TABLE]

In fact, by (4.24)-(4.25) there exists $\tau\in(0,T)$ such that

[TABLE]

To prove (4.25), let $\{t_{n}\}$ be a decreasing sequence such that $t_{n}\to 0^{+}$, and let $(x_{n},\tau_{n})\in A_{t_{n}}$ be a maximum point - namely, $G(t_{n})=g(x_{n},\tau_{n})$. Clearly, there exists a converging subsequence (not relabelled) of $\{(x_{n},\tau_{n})\}\subseteq A_{t_{1}}$ and a point $(\overline{x},0)$, $\overline{x}\in[a,d]$, such that $(x_{n},\tau_{n})\to(\overline{x},0)$ as $n\to\infty$ (observe that $\lim_{n\to\infty}\tau_{n}=\lim_{n\to\infty}t_{n}=0$). Then, by the upper semicontinuity of $g$,

[TABLE]

This proves (4.25) and Claim 1 follows.

To prove Claim 2, we preliminarily observe that, by (4.18) and (4.21), for every maximum point $(\bar{x},\bar{t},\bar{y},\bar{s})$ of $F$ there holds

[TABLE]

In view of (4.19), this implies that

[TABLE]

Now we argue by contradiction. Were Claim 2 false, there would exist a sequence $\{\varepsilon_{n}\}\subset(0,\varepsilon_{0})$ such that $\varepsilon_{n}\to 0^{+}$, a sequence $\{p_{n}\}\subset(1,2]$ and a sequence of maximum points of $F$

[TABLE]

By the boundedness of $\{(\bar{x}_{n},\bar{t}_{n},\bar{y}_{n},\bar{s}_{n})\}$ and (4.20), there would exist a converging subsequence (not relabelled) of $\{(\bar{x}_{n},\bar{t}_{n},\bar{y}_{n},\bar{s}_{n})\}$ and a point $(\tilde{x},\tilde{t},\tilde{x},\tilde{t})\in\left(A_{\tau}\right)^{2}$, such that $(\bar{x}_{n},\bar{t}_{n},\bar{y}_{n},\bar{s}_{n})\to(\tilde{x},\tilde{t},\tilde{x},\tilde{t})$ as $n\to\infty$. Rewriting (4.27) with $\varepsilon=\varepsilon_{n}$, $(\bar{x},\bar{t},\bar{y},\bar{s})=(\bar{x}_{n},\bar{t}_{n},\bar{y}_{n},\bar{s}_{n})$ and letting $n\to\infty$, it follows from the upper semicontinuity of the function at the right-hand side of (4.27) that $\max\limits_{\overline{A}}g\leq g(\tilde{x},\tilde{t})$, which contradicts Claim 1 since $(\tilde{x},\tilde{t})\in A_{\tau}$. Hence we have also proved Claim 2.

Now we complete the proof. Henceforth we assume that $\varepsilon\in(0,\varepsilon_{1})$, so that we can use Claim 2. Then the function

[TABLE]

has a maximum at some point $(\bar{x},\bar{t})\in\overline{A}\setminus A_{\tau}$ with $\tau\in(0,T)$ given by Claim 1. Similarly, the function

[TABLE]

has a minimum at some point $(\bar{y},\bar{s})\in\overline{A}\setminus A_{\tau}$ with $\tau\in(0,T)$ as above. Since $u$ is a viscosity subsolution and $v$ a viscosity supersolution of $({\rm T}_{a,\pm})$, by definition $\phi$ and $\chi$ must satisfy suitable differential inequalities at $(\bar{x},\bar{t})$ and $(\bar{y},\bar{s})$ (see Subsection 3.1 and Proposition 4.1). We show below that these inequalities always lead to a contradiction, whence the result follows. Before proceeding observe that

[TABLE]

[TABLE]

Problem $({\rm T}_{a,+}).$ Then $\alpha=\tfrac{1}{2}\sqrt{\varepsilon}$ (see (4.19)) and we distinguish two cases:

$(1)\;\bar{x}=\bar{x}_{\varepsilon,p}\geq a,\,\,\bar{y}=\bar{y}_{\varepsilon,p}>a$ for some $\varepsilon\in(0,\varepsilon_{1})$ and $p\in(1,2]$: Since $u$ is a viscosity subsolution and $(\bar{x},\bar{t})$ is a maximum point of $u^{*}-\phi$, it follows from (3.4) and (3.2) that

[TABLE]

Similarly, since $v$ is a viscosity supersolution and $(\bar{y},\bar{s})$ is a minimum point of $v_{*}-\chi$, it follows from (3.3) (if $\overline{y}<d-\|H^{\prime}\|_{\infty}\overline{s}$) and (4.15) (if $\overline{y}=d-\|H^{\prime}\|_{\infty}\overline{s}$) that

[TABLE]

Subtracting (4.29) from (4.28) we find that $2\lambda\leq 0$, which is a contradiction.

$(2)\;\bar{x}=\bar{x}_{p}\geq a,\,\bar{y}=\bar{y}_{p}=a$ for all $p\in(1,2]$ and $\varepsilon\in(0,\varepsilon_{1})$: we fix $\varepsilon=\tilde{\varepsilon}\in(0,\varepsilon_{1})$ so small that

[TABLE]

Since we have chosen $\varepsilon=\tilde{\varepsilon}$, $\bar{x}=\bar{x}_{p}$ and $\bar{y}=\bar{y}_{p}$ only depend on $p$. Now (4.28) reads

[TABLE]

(since $\bar{x}-a+\alpha\tilde{\varepsilon}>0$, $\,{\rm sgn}(\bar{x}-a+\alpha\tilde{\varepsilon})=1$). On the other hand, by (4.3) there holds

[TABLE]

Subtracting (4.32) from (4.31) we find that

[TABLE]

whence, as $p\to 1^{+}$,

[TABLE]

By (4.30) this implies that $\lambda\leq 0$, and again we have found a contradiction.

Problem $({\rm T}_{a,-})$. Then $\alpha=-\tfrac{1}{2}\sqrt{\varepsilon}$ (see (4.19)) and again we distinguish two cases:

$(1)\;\bar{x}=\bar{x}_{\varepsilon,p}>a,\,\,\bar{y}=\bar{y}_{\varepsilon,p}\geq a$ for some $\varepsilon\in(0,\varepsilon_{1})$ and $p\in(1,2]$: in this case (4.28)-(4.29) follow from (3.2), (3.3) and (3.7), whence again $2\lambda\leq 0$.

$(2)\;\bar{x}=\bar{x}_{p}=a,\,\bar{y}=\bar{y}_{p}\geq a$ for all $p\in(1,2]$ and $\varepsilon\in(0,\varepsilon_{1})$: we fix $\varepsilon=\tilde{\varepsilon}\in(0,\varepsilon_{1})$ so small that

[TABLE]

(so $\bar{x}=\bar{x}_{p}$ and $\bar{y}=\bar{y}_{p}$ only depend on $p$). By (4.1) there holds

[TABLE]

(since $a-\bar{y}+\alpha\tilde{\varepsilon}<0$, $\,{\rm sgn}(a-\bar{y}+\alpha\tilde{\varepsilon})=-1$). On the other hand, since $\bar{y}>a$,

[TABLE]

Subtracting (4.36) from (4.35) gives

[TABLE]

Letting $p\to 1^{+}$ this implies that

[TABLE]

whence by (4.34) we get $\lambda\leq 0$, again a contradiction. $\square$

It remains to prove Theorem 3.1 for problem $(N_{a,\pm}^{b,\pm})$.

Proof of Theorem 3.1 for problem $(N_{a,\pm}^{b,\pm})$. Let $Q=(a,b)\times(0,T)$ with $-\infty<a<b<\infty$, and $Q_{\tau}=\{(x,t)\in Q\,|\,t\in[0,\tau]\}$ for any $\tau\in(0,T]$. Let $u^{*},v_{*}$ be defined in $\overline{Q}$ by (2.1)-(2.2). As before we set

[TABLE]

for all $\tau\in(0,\tau_{1}]$, with $\tau_{1}$ defined by (2.13). By Remark 3.1, the restrictions $u_{1,A}=u\lefthalfcup A_{\tau_{1}}$, $v_{1,A}=v\lefthalfcup A_{\tau_{1}}$ are viscosity sub- and supersolutions of problem $({\rm T}_{a,\pm})$ in $A_{\tau_{1}}$ (similarly for $u_{1,B}=u\lefthalfcup B_{\tau_{1}}$, $v_{1,B}=v\lefthalfcup B_{\tau_{1}}$). Since we have already proved Theorem 3.1 for the trapezoidal domains $A$ and $B$, we have that

[TABLE]

(notice that $(u_{1,A})^{*}(x,t)\leq u^{*}(x,t)$ and $(v_{1,A})_{*}(x,t)\geq v_{*}(x,t)$, since $A_{\tau_{1}}\subseteq Q$),

[TABLE]

whence, by Remark 2.1, $\displaystyle{\max_{[a,b]\times[0,\tau_{1}-\delta]}\,[u^{*}-v_{*}]_{+}\leq\,\max_{[a,b]}\,\left[u^{*}(\cdot,0)-v_{*}(\cdot,0)\right]_{+}}\,\,$ for all $\delta\in(0,\tau_{1})$. By the arbitrariness of $\delta$ this means that

[TABLE]

Let $\delta\in(0,\tau_{1})$ be arbitrary and fixed. Arguing as before in the rectangle $\overline{Q}_{\tau_{2}-\delta}\setminus\overline{Q}_{\tau_{1}-\delta}$, where $\tau_{2}=\min\left\{(b-a)/\|H^{\prime}\|_{\infty},\,T\right\}$, we obtain

[TABLE]

for all $(x,t)\in[a,b]\times(\tau_{1}-\delta,\tau_{2}-\delta)$. Since

[TABLE]

(see (2.5)), from the above inequality and (4.38) we obtain that

[TABLE]

whence, by the arbitrariness of $\delta$,

[TABLE]

It is now clear that in a finite number of steps the claim follows. $\square$

5. Regularity

In this section we prove Proposition 3.2. We begin with the proof of the first part, which concerns one-sided Lipschitz bounds in $t$ for sub- and supersolution of equation (1.1).

Proof of Proposition 3.2 $(i)$. To prove (3.12)1, i.e. the first inequality in (3.12), it is enough to show that

[TABLE]

Indeed, taking the $\limsup$ as $\bar{x}\to a^{+}$ (if $a\in\mathbb{R}$) or $\bar{x}\to b^{-}$ (if $b\in\mathbb{R}$) it follows from (2.12) that (5.1) is also satisfied if $\bar{x}\in\partial\Omega$, and then (3.12)1 follows from (2.5).

We prove (5.1). By Lemma 2.1 applied to the restriction of $u$ to $(a,b)\times(t_{1},T)$, for any $\varepsilon>0$ there exists $\delta>0$ such that

[TABLE]

and $(\bar{x}-\delta,\bar{x}+\delta)\subset\Omega$. Setting $D_{\delta}\,:=\,(\bar{x}-\delta,\bar{x}+\delta)\times(t_{1},T)$ and $u_{1}\,:=\,u\lefthalfcup D_{\delta}$, it follows from Remark 2.1 that

[TABLE]

and from Remark 3.1 that $u_{1}$ is a viscosity subsolution of equation (1.1) in $D_{\delta}$. Observe that, by Definition 3.2$-(i)$, this implies that $u_{1}$ is also a viscosity subsolution of problem $(N_{\bar{x}-\delta,-}^{\bar{x}+\delta,+})$ in $D_{\delta}$.

Let

[TABLE]

It is easy to prove that $w$ is a viscosity supersolution of problem $(N_{\bar{x}-\delta,-}^{\bar{x}+\delta,+})$ in $D_{\delta}$: if $\varphi\in C^{1}(\overline{D}_{\delta})$ and $(x_{0},t_{0})\in\overline{D}_{\delta}$ (with $t_{0}>t_{1}$) is a local minimum point of $w_{*}-\varphi=w-\varphi$ in $\overline{D}_{\delta}$, then $\varphi_{t}(x_{0},t_{0})+H(\varphi_{x}(x_{0},t_{0}))\geq w_{t}(x_{0},t_{0})-K=0$.

Applying Theorem 3.1 in $D_{\delta}$, and observing that, by (5.2)-(5.3), $(u_{1})^{*}(\cdot,t_{1})=u^{*}(\cdot,t_{1}^{+})\leq w(\cdot,t_{1})$ in $[\bar{x}-\delta,\bar{x}+\delta]$, we obtain that $u^{*}\leq w$ in $D_{\delta}$. In particular,

[TABLE]

and (5.1) follows from the arbitrariness of $\varepsilon$.

The proof of (3.12)2 is similar: arguing as before one shows that

[TABLE]

$\square$

Formulas (3.13) and (3.14) in Proposition 3.2$-(ii)$ quantify the fact that initial jump discontinuities cannot disappear instantaneously. They are a special case of the following result, with $x_{0}=c$, $t_{0}=0$ and $G=u_{0}$:

Lemma 5.1**.**

Let $(H_{1})$ hold, let $K,k$ be defined by (3.11), let $u$ be a viscosity solution of equation (1.1) and let $(x_{0},t_{0})\in\Omega\times[0,T)$. Let $G\in L^{\infty}_{{\rm loc}}(\overline{\Omega})$ be such that

[TABLE]

[TABLE]

and set

[TABLE]

Then for all $t\in(t_{0},\min\{\underline{t},T\})$

[TABLE]

[TABLE]

Proof.

We only prove (5.7), the proof of (5.8) is similar. By assumption, for any $\varepsilon>0$ there exists $\delta\in(0,\bar{\delta})$ such that

[TABLE]

[TABLE]

whence, since $G(x_{0}^{+})>G(x_{0}^{-})$,

[TABLE]

[TABLE]

Then we get that for all $t\in(t_{0},T]$ and $x\in(x_{0}-\delta,x_{0})$

[TABLE]

Similarly, using (5.10) instead of (5.9), for all $t\in(t_{0},T]$ and $y\in(x_{0},x_{0}+\delta)$

[TABLE]

In particular we obtain from the above inequalities that for all $t\in(t_{0},T]$

[TABLE]

Now set

[TABLE]

Observe that $\underline{t}_{\varepsilon}\to\underline{t}$ as $\varepsilon\to 0$. Then for all $\varepsilon\in(0,\varepsilon_{0})$ and $t\in[t_{0},\underline{t}_{\varepsilon})$ there holds

[TABLE]

whence, by (5.11), for all $\varepsilon\in(0,\varepsilon_{0})$, $t\in(t_{0},\underline{t}_{\varepsilon})$, $x\in(x_{0}-\delta,x_{0})$ and $y\in(x_{0},x_{0}+\delta)$

[TABLE]

[TABLE]

Plainly this implies that for all $t\in(t_{0},\underline{t}_{\varepsilon})$

[TABLE]

(the equalities in these estimates follow from Lemma 2.1 applied to $(a,x_{0})\times(0,T)$, respectively $(x_{0},b)\times(0,T)$). Therefore it follows from Lemma 2.1 that

[TABLE]

[TABLE]

for all $t\in(t_{0},\underline{t}_{\varepsilon})$. Since $\varepsilon$ is arbitrary and $\underline{t}_{\varepsilon}\to\underline{t}$ as $\varepsilon\to 0$, the conclusion follows. ∎

The concept of barrier effect of a discontinuity, discussed in the Introduction, is made precise by the following lemma.

Lemma 5.2**.**

Let $-\infty<a<c<b<\infty$, and let $Q=(a,b)\times(0,T)$, $Q^{-}=(a,c)\times(0,T)$ and $Q^{+}=(c,b)\times(0,T)$. Let $u$ be a viscosity solution of problem $(N_{a,\pm}^{b,\pm})$.

$(i)$* If $u^{*}(c^{+},\cdot)>u^{*}(c^{-},\cdot)$ in $(0,T)$, then $u\lefthalfcup Q^{+}$ is a viscosity solution of $(N_{c,+}^{b,\pm})$.*

$(ii)$* If $u_{*}(c^{+},\cdot)>u_{*}(c^{-},\cdot)$ in $(0,T)$, then $u\lefthalfcup Q^{-}$ is a viscosity solution of $(N_{a,\pm}^{c,+})$.*

$(iii)$* If $u^{*}(c^{+},\cdot)<u^{*}(c^{-},\cdot)$ in $(0,T)$, then $u\lefthalfcup Q^{-}$ is a viscosity solution of $(N_{a,\pm}^{c,-})$.*

$(iv)$* If $u_{*}(c^{+},\cdot)<u_{*}(c^{-},\cdot)$ in $(0,T)$, then $u\lefthalfcup Q^{+}$ is a viscosity solution of $(N_{c,-}^{b,\pm})$.*

Similar statements hold for viscosity solutions of problems $(N_{a,\pm})$, $(N^{b,\pm})$ and equation (1.1) in $\mathbb{R}$.

Remark 5.1**.**

Let $0\leq t_{1}<t_{2}\leq T$ and $\Omega=(a,b)$, with $-\infty\leq a<b\leq\infty$. Then, for any $c\in(a,b)$, it can be easily checked that the conclusions of Lemma 5.2 hold true for viscosity solutions of problems $(N_{a,\pm}^{b,\pm})$, $(N_{a,\pm})$ or $(N^{b,\pm})$ in $\Omega\times(t_{1},t_{2})$.

Proof.

We only prove $(i)$, since the other proofs are similar. We set $\tilde{u}=u\lefthalfcup Q^{+}$. Since $\tilde{u}^{*}=u^{*}$ in $Q^{+}$ (see Remark 2.1) and $u$ is a viscosity solution of $(N_{a,\pm}^{b,\pm})$ in $Q$, $\tilde{u}$ is a viscosity solution of (1.1) in $Q^{+}$. By Definition 3.2$-(i)$, it remains to prove that if $\varphi\in C^{1}([c,b]\times(0,T])$ and $(c,t_{0})$ is a local maximum point of $\tilde{u}^{*}-\varphi$ in $[c,b]\times(0,T]$, then

[TABLE]

To prove (5.14), let $t_{0}\in(0,T)$ (if $t_{0}=T$ we argue as in [18, Section 10.2]) and observe first that, by assumption, there holds

[TABLE]

If $(c,t_{0})$ is a strict local maximum point of $\tilde{u}^{*}-\varphi$ in $[c,b]\times(0,T]$, then

[TABLE]

for some $r>0$, where $B^{+}_{r}(c,t_{0})=\{(y,\tau)\in B_{r}(c,t_{0})\,|\,\,y>c\}$. Here $r$ is chosen such that $0<t_{0}-r<t_{0}+r<T$. In view of (5.15) this implies that

[TABLE]

From (5.16) and (5.17) we obtain that

[TABLE]

On the other hand, by (5.15) and the upper semicontinuity of $u^{*}$ we also have that

[TABLE]

thus for some $r>0$ there holds

[TABLE]

Hence we can extend the definition of $\varphi$ to $[a,b]\times(0,T]$ so that $\varphi_{x}(c_{-},t_{0})=\varphi_{x}(c_{+},t_{0})$, and

[TABLE]

By (5.18)-(5.19) $(c,t_{0})$ is a local maximum point of $u^{*}-\varphi$ in $Q$, thus by (3.2) we obtain (5.14). ∎

To complete the proof of Proposition 3.2$-(ii)$, we show that $u^{*}-u_{*}$ is nonincreasing with respect to $t$ for viscosity solutions $u$ of equation (1.1) and prove (3.15).

Proposition 3.2 $(ii)$: proof of (3.15). We prove (3.15a) assuming that $u_{0}(c^{+})>u_{0}(c^{-})$ and

[TABLE]

Observe that (3.15) follows at once (by subtraction) if we show that

[TABLE]

for $0\leq t_{0}<t_{1}<t_{c}$.

We only prove the first inequality of (5.21). Let $b_{0}=b$ if $b<\infty$ and let $b_{0}\in(c,\infty)$ if $b=\infty$. We set $P=(c,b_{0})\times(t_{0},t_{c})$ and $u_{1}=u\lefthalfcup P$. By Remark 2.1 and (5.20)

[TABLE]

By Lemma 2.1 applied to $(u_{1})^{*}$, for all $\varepsilon>0$ there exists $\delta>0$ such that

[TABLE]

We set, for all $p>0$ and $(x,t)\in\overline{P}$,

[TABLE]

Observe that there exist $p_{\varepsilon}\geq 0$ and $\xi\in(c,b_{0})$ such that for all $p>p_{\varepsilon}$

[TABLE]

and

[TABLE]

One easily checks that if $p>p_{\varepsilon}$, then, by (5.25), $v(x,t;p)$ is a viscosity supersolution of problem $(N_{c,+}^{b_{0},+})$ in $P$.

On the other hand, if $b=b_{0}\in\mathbb{R}$ it follows from Lemma 5.2$-(i)$ (see also Remark 5.1) that $u_{1}$ is a solution of $(N_{c,+}^{b_{0},\pm})$ in $P$, and hence, by Definition 3.2$-(i)$, $u_{1}$ is a viscosity subsolution of problem $(N_{c,+}^{b_{0},+})$ in $P$ (if $b=\infty$ we argue similarly: the restriction $\tilde{u}=u\lefthalfcup\tilde{P}$, with $\tilde{P}=(c,\infty)\times(t_{0},t_{c})$, is a viscosity solution of $(N_{c,+})$ in $\tilde{P}$ and $u_{1}$, which coincides with the restriction of $\tilde{u}$ to $P$, is a viscosity subsolution of $(N_{c,+}^{b_{0},+})$ in $P$). By (5.24) and Theorem 3.1, this implies that $(u_{1})^{*}\leq v$ in $\overline{P}$ for all $p>p_{\varepsilon}$. In particular

[TABLE]

(here we have used the second equality in (5.22)). Choosing $p=p_{n}$ such that $p_{n}\to+\infty$ and $H(p_{n})\to\limsup\limits_{s\to\infty}H(s)$ as $n\to\infty$, we obtain that

[TABLE]

and the first inequality in (5.21) follows from the arbitrariness of $\varepsilon$. $\square$

6. Proof of existence: Perron’s method revisited

Proof of Theorem 3.3. $(i)$ We first prove the existence of a viscosity solution of the Cauchy problem $(CP)$. We adapt Perron’s method used by Ishii ([24]) to our definition of semi-continuous envelopes based on essential limits.

Let $u_{0}\in L^{\infty}_{\text{loc}}(\mathbb{R})$, and let $K,\,k$ be defined in (3.11). Set

[TABLE]

Then $\underline{u}$ is a viscosity subsolution of $(CP)$, $\overline{u}$ a viscosity supersolution, and

[TABLE]

Let $\mathcal{S}$ be the set of viscosity subsolutions $v$ of $(CP)$ such that $v^{*}\leq\overline{u}^{*}$ and $v_{*}\leq\overline{u}_{*}$ in $\mathbb{R}\times[0,T]$ (observe that these two inequalities are equivalent: if for example $v^{*}\leq\overline{u}^{*}$, then $(v^{*})_{*}\leq(\overline{u}^{*})_{*}$ and, by Lemma 2.1, $v_{*}\leq\overline{u}_{*}$). We set

[TABLE]

Since $\underline{u}\in\mathcal{S}$, we have $u\geq\underline{u}^{*}$ in $\mathbb{R}\times[0,T]$. In particular, it follows that $u^{*}\geq\underline{u}^{*}$ in $\mathbb{R}\times[0,T]$, hence $u^{*}(x,0)\geq(u_{0})^{*}(x)$ for $x\in\mathbb{R}$ (see (6.1)). On the other hand $u\leq\overline{u}^{*}$, whence $u^{*}\leq\overline{u}^{*}$ in $\mathbb{R}\times[0,T]$ and, by (6.1), $u^{*}(x,0)\leq(u_{0})^{*}(x)$ for $x\in\mathbb{R}$. So we have $u^{*}(x,0)=(u_{0})^{*}(x)$. Analogously, since $u^{*}\geq\underline{u}^{*}$ and $u^{*}\leq\overline{u}^{*}$ in $\mathbb{R}\times[0,T]$, from Lemma 2.1 we get $u_{*}=(u^{*})_{*}\geq(\underline{u}^{*})_{*}=\underline{u}_{*}$ and $u_{*}=(u^{*})_{*}\leq(\overline{u}^{*})_{*}=\overline{u}_{*}$ in $\mathbb{R}\times[0,T]$, thus (see (6.1))

[TABLE]

Then we have proved that $u_{*}(x,0)=(u_{0})_{*}(x)$ for any $x\in\mathbb{R}$.

We claim that the function $u$ defined in (6.2) is a viscosity solution of $(CP)$. By the above remarks, it is enough to show that $u$ is a viscosity solution of $u_{t}+H(u_{x})=0$ in $\mathbb{R}\times(0,T)$. We shall prove this in two steps.

Step 1: $u$ is a viscosity subsolution of $u_{t}+H(u_{x})=0$ in $\mathbb{R}\times(0,T)$.

Let $v\in\mathcal{S}$, and fix $(x,t)\in\mathbb{R}\times(0,T)$. Then

[TABLE]

Since there exists a sequence $\{v_{n}^{*}\}\subseteq\mathcal{S}$ such that $v_{n}^{*}(x,t)\to u(x,t)$, we have that

[TABLE]

Since $(x,t)$ is arbitrary this statement holds for all $(x,t)\in\mathbb{R}\times(0,T)$.

Let $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$ and $\varphi\in C^{1}(\mathbb{R}\times(0,T))$ be such that $u^{*}-\varphi$ has a strict local maximum at $(x_{0},t_{0})$. Let $(y_{n},\tau_{n})\to(x_{0},t_{0})$ be such that

[TABLE]

(this is possible by the definition of $u^{*}$). By the definition of $u$, for all $n\in\mathbb{N}$ there exists $v_{n}\in\mathcal{S}$ such that

[TABLE]

Let $r>0$ be so small that $u^{*}-\varphi<u^{*}(x_{0},t_{0})-\varphi(x_{0},t_{0})$ in $\overline{B}_{r}(x_{0},t_{0})\setminus\{(x_{0},t_{0})\}$, and let $(x_{n},t_{n})\in\overline{B}_{r}(x_{0},t_{0})$ be a point at which $v_{n}^{*}-\varphi$ has its maximum in $\overline{B}_{r}(x_{0},t_{0})$. Hence

[TABLE]

and, by (6.3),

[TABLE]

This means that

[TABLE]

Up to subsequences we have that $(x_{n},t_{n})\to(\overline{x},\overline{t})\in\overline{B}_{r}(x_{0},t_{0})$. Recalling that, by Lemma 2.1,

[TABLE]

for any $\varepsilon>0$ there exists $\delta>0$ such that

[TABLE]

Since $(x_{n},t_{n})\to(\overline{x},\overline{t})$, for fixed $\delta$ there exists $N\in\mathbb{N}$ such that

[TABLE]

So we have found that for all $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that

[TABLE]

Combining this with (6.4) we find that

[TABLE]

Passing to the limit $n\to\infty$ and using the arbitrariness of $\varepsilon$ we conclude that

[TABLE]

Since the local maximum of $u^{*}-\varphi$ at $(x_{0},t_{0})$ is strict, this means that $(\overline{x},\overline{t})=(x_{0},t_{0})$. Finally, since $v_{n}^{*}-\varphi$ has a maximum at $(x_{n},t_{n})$ we have that

[TABLE]

Passing to the limit $n\to\infty$ this implies that $\varphi_{t}(x_{0},t_{0})+H(\varphi_{x}(x_{0},t_{0}))\leq 0$, and we have completed Step 1.

Step 2: $u$ is a viscosity solution of $u_{t}+H(u_{x})=0$ in $\mathbb{R}\times(0,T)$.

Arguing by contradiction we suppose the $u$ is not a viscosity solution of $u_{t}+H(u_{x})=0$ in $\mathbb{R}\times(0,T)$. In view of Step 1 this means that $u$ is not a viscosity supersolution of $u_{t}+H(u_{x})=0$ in $\mathbb{R}\times(0,T)$. Hence there exist $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$ and $\varphi\in C^{1}(\mathbb{R}\times(0,T))$ such that $u_{*}-\varphi$ has a strict local minimum at $(x_{0},t_{0})$ and

[TABLE]

Observe that by definition we have $u\leq\overline{u}^{*}$ in $\mathbb{R}\times[0,T]$, thus $u_{*}\leq(\overline{u}^{*})_{*}=\overline{u}_{*}$ by Lemma 3.2. So let us assume that

[TABLE]

(otherwise also $\overline{u}_{*}-\varphi$ would have a minimum at $(x_{0},t_{0})$, and so $\varphi_{t}+H(\varphi_{x})\geq 0$ at $(x_{0},t_{0})$, a contradiction). Since $\varphi\in C^{1}$ and $\overline{u}_{*}$ is lower semicontinuous, there exist $r>0$ and $0<\delta_{0}\leq\tfrac{1}{2}[\overline{u}_{*}(x_{0},t_{0})-u_{*}(x_{0},t_{0})]$ such that

[TABLE]

Since the minimum of $u_{*}-\varphi$ at $(x_{0},t_{0})$ is strict and $u_{*}$ is lower semicontinuous, we may choose $\delta\in(0,\delta_{0}]$ so small that

[TABLE]

So

[TABLE]

Now we set

[TABLE]

Since $\varphi_{t}+H(\varphi_{x})\leq 0$ in B_{2r}(x_{0},t_{0})\cap\big{(}\mathbb{R}\times[0,T]\big{)}, $w^{*}\leq\overline{u}^{*}$ and $u\in\mathcal{S}$, the function $w$ belongs to $\mathcal{S}$. This implies that for all $(x,t)\in B_{r}(x_{0},t_{0})$

[TABLE]

Since the right-hand side of the above inequality is a smooth function, we also have that

[TABLE]

Choosing $(x,t)=(x_{0},t_{0})$ we obtain $\delta\leq 0$, a contradiction. This completes the proof of Theorem 3.3 in the case of problem $(CP)$.

$(ii)$ Let us now consider the initial-boundary value problem $(N)$. We shall only prove the result for problem $(N_{a,+}^{b,-})$ in $Q=(a,b)\times(0,T)$ with initial data $u_{0}\in L^{\infty}((a,b))$ $(-\infty<a<b<\infty);$ the proof is analogous in the other cases.

Let $\tilde{u}_{0}\in L^{\infty}(\mathbb{R})$ be defined by setting

[TABLE]

where $l_{1},l_{2}\,<\,0$ are chosen so that

[TABLE]

and $K,k$ are the constants in (3.11). Let $\tilde{u}$ be the viscosity solution of the Cauchy problem $(CP)$ with initial condition $\tilde{u}_{0}$, given by step $(i)$ above. By (3.8) and (3.12) for all $t\in(0,T)$ there holds

[TABLE]

(here $\tilde{u}_{*}\equiv(\tilde{u})_{*}$ , $\tilde{u}^{*}\equiv(\tilde{u})^{*}$ for notational simplicity). Moreover,

[TABLE]

Then for all $t\in(0,T)$ we have

[TABLE]

[TABLE]

whence by (6.6)

[TABLE]

[TABLE]

Arguing as in the last part of the proof of Lemma 5.1, it follows from the above inequalities that for all $t\in(0,T)$

[TABLE]

Hence by Lemma 5.2$-(i)$, $(iii)$ $u:=\tilde{u}\lefthalfcup Q$ is a viscosity solution of problem $(N_{a,+}^{b,-})$ with initial data $u_{0}$. This proves the result. $\square$

7. Proof of uniqueness

In step $(ii)$ of the following proof we use Lemma 5.2, which describes the barrier effect of spatial discontinuities, to handle possible discontinuities of viscosity solutions if $u_{0}$ is piecewise continuous.

Proof of Theorem 3.4. The result is easily proven if $u_{0}\in C(\overline{\Omega})$: if $\Omega$ is bounded, then $(u_{0})^{*}=(u_{0})_{*}=u_{0}$ in $\overline{\Omega}$ and, by (3.8) and (3.9), $u^{*}=u_{*}=v^{*}=v_{*}$ in $\overline{Q}$; hence $u$ has a continuous representative $\tilde{u}$ in $\overline{Q}$ and, by Proposition 3.2, $\tilde{u}$ is Lipschitz continuous with respect to $t$ in $\overline{Q}$ and satisfies (3.17). If $\Omega$ is unbounded we argue similarly, using (3.10) instead of (3.9).

Let us consider the general case of initial data as in assumption $(H_{2})$. For simplicity we suppose that $u_{0}$ has a single jump discontinuity at $x_{1}\in\Omega=(a,b)$, and that

[TABLE]

If $u_{0}(x_{1}^{+})<u_{0}(x_{1}^{-})$ or the number of jumps is finite, the proof is similar.

Let $u$ and $v$ be two viscosity solutions of $(N)$ with initial datum $u_{0}$. By (7.1) and (3.13) there exists $\underline{t}_{1}\in(0,T]$ such that

[TABLE]

Therefore $\tau_{1}\,:=\,\sup\left\{\underline{t}_{1}\in(0,T)\,|\,\text{\eqref{dimox} holds}\right\}>0$.

Consider the set $Q_{1,\tau_{1}}\,:=\,(a,x_{1})\times(0,\tau_{1})$ and the restrictions $u_{1}\,:=\,u\lefthalfcup Q_{1,\tau_{1}}$ and $v_{1}\,:=\,v\lefthalfcup Q_{j,\tau_{1}}$. By Lemma 5.2 (see also Remark 5.1), $u_{1}$ and $v_{1}$ are viscosity solutions in $Q_{1,\tau_{1}}$ of a problem with lateral boundary condition $u_{1x}=v_{1x}=\infty$ at $x=x_{1}$, with continuous initial function $u_{0}\lefthalfcup(a,x_{1})$. As already observed, Theorem 3.4 holds for continuous initial data, hence there holds $u_{1}=v_{1}=\tilde{u}_{1}$ a.e. in $Q_{1,\tau_{1}}$, where $\tilde{u}_{1}\in C(\overline{Q}_{1,\tau_{1}})$.

Similarly, the restrictions of $u$ and $v$ to $Q_{2,\tau_{1}}\,:=\,(x_{1},b)\times(0,\tau_{1})$ coincide a.e. in $Q_{2,\tau_{1}}$ with a continuous solution $\tilde{u}_{2}\in C(\overline{Q}_{2,\tau_{1}})$. In particular we have found that $u=v$ a.e. in $Q_{\tau_{1}}=(a,b)\times(0,\tau_{1})$, and (3.17) is satisfied in $Q_{\tau_{1}}$.

It follows from (7.2) that for all $t\in(0,\tau_{1})$

[TABLE]

If $\tau_{1}=T$ the proof is complete, otherwise we claim that $\tilde{u}_{2}(x_{1},\tau_{1})=\tilde{u}_{1}(x_{1},\tau_{1})$. Arguing by contradiction, it follows from the continuity of $\tilde{u}_{j}$ in $\overline{Q}_{j,\tau_{1}}$ ($j=1,2$) that there exists $\eta>0$ such that $\tilde{u}_{2}(x_{1},\tau)-\tilde{u}_{1}(x_{1},\tau)\geq\eta$ for all $\tau\in(0,\tau_{1})$ sufficiently close to $\tau_{1}$. Then by Lemma 5.1 there exists $\delta>0$, independent of $\tau$ (see (5.7)), such that (7.2) holds for every $t\in(\tau,\tau+\delta)$, a contradiction for $\tau>\tau_{1}-\delta$ with the definition of $\tau_{1}$.

Since $u^{*}=u_{*}=v^{*}=v_{*}=\tilde{u}_{j}$ in $Q_{j,\tau_{1}}$ and $\tilde{u}_{j}\in C(\overline{Q}_{j,\tau_{1}})$, we have that

[TABLE]

Let $Q_{t,T}=(a,b)\times(t,T)$ $(t\in(0,\tau_{1}))$. For the sake of simplicity, we assume that $(a,b)$ is bounded (otherwise we argue similarly and consider suitable trapezoidal domains as in (3.1) instead of $Q_{t,T}$). By Theorem 3.1 (applied in $Q_{t,T}$) and (7.4)

[TABLE]

[TABLE]

Since $J(t)\to 0$ as $t\to\tau_{1}^{-}$, we conclude that $u^{*}=u_{*}=v^{*}=v_{*}$ in $\overline{Q}_{\tau_{1},T}$, whence $u=v$ $a.e.$ in $Q$. Moreover, by the above considerations, both the restrictions $u_{1}=u\lefthalfcup Q_{1}$ and $u_{2}=u\lefthalfcup Q_{2}$ ($Q_{1}=(a,x_{1})\times(0,T)$, $Q_{2}=(x_{1},b)\times(0,T)$) of the viscosity solution $u$ of problem $(N)$ admit continuous representatives $\tilde{u}_{i}\in C(\overline{Q}_{i})$ ($i=1,2$), and $\tilde{u}_{1}(x_{1},t)\neq\tilde{u}_{2}(x_{1},t)$ if and only if $t\in[0,\tau_{1})$. This proves claims $(a)$ and $(b)$ of Theorem 3.4$(ii)$. Finally, (3.17) and claim $(d)$ follow from (3.12) and (3.15), since by (7.3) we have $u^{*}(x_{1},t)=\tilde{u}_{2}(x_{1},t)$ and $u_{*}(x_{1},t)=\tilde{u}_{1}(x_{1},t)$ for all $t\in[0,\tau_{1})$ (observe also that, if $\tau_{1}<T$, $u^{*}(x_{1},t)=\tilde{u}_{2}(x_{1},t)=u_{*}(x_{1},t)=\tilde{u}_{1}(x_{1},t)$ for $t\in[\tau_{1},T]$). $\square$

Finally we show that the existence and uniqueness of piecewise continuous viscosity solutions implies a comparison principle for these solutions.

Proof of Corollary 3.5. Let $u_{0}\leq v_{0}$ a.e. in $(a,b)$. By the uniqueness of viscosity solutions (Theorem 3.4), it is enough to show that the corresponding viscosity solutions $u$ and $v$ given by Perron’s method satisfy $u\leq v$ a.e. in $(a,b)\times(0,T)$. But this follows trivially from the pointwise definition in the proof of Theorem 3.3 of $u$ and $v$ in terms of the corresponding sets $\mathcal{S}$, say $\mathcal{S}_{u}$ and $\mathcal{S}_{v}$, and the observation that $\mathcal{S}_{u}\subseteq\mathcal{S}_{v}$ since $u_{0}\leq v_{0}$ a.e. in $(a,b)$. $\square$

Acknowledgement. MB acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, and the project Beyond Borders of the University of Rome Tor Vergata, CUP E84I19002220005.