# Viscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal
equation

Siyao Zhu, Xiaojun Cui and Tianqi Shi  Department of Mathematics, Nanjing
University, Nanjing 210093, China<EMAIL_ADDRESS>Department of
Mathematics, Nanjing University, Nanjing 210093, China<EMAIL_ADDRESS>Department of Mathematics, Nanjing University, Nanjing 210093, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we propose a Cauchy type problem to the timelike Lorentzian
eikonal equation on a globally hyperbolic space-time. For this equation, as
the value of the solution on a Cauchy surface is known, we prove the existence
of viscosity solutions on the past set (future set) of the Cauchy surface.
Furthermore, when the time orientation of viscosity solution is consistent,
the uniqueness and stability of viscosity solutions are also obtained.

###### Key words and phrases:

Lorentzian eikonal equation, globally hyperbolic space-time, viscosity
solution, weak KAM theory

###### 2010 Mathematics Subject Classification:

53B30, 53C22, 49L25, 35F21

##  1\. Introduction

In the study of global Lorentzian geometry, timelike Lorentzian eikonal
equation

$\displaystyle g(\nabla u,\nabla u)=-1$

plays a vital role, where $g$ is the Lorentzian metric. The integral curves of
a gradient field of a smooth solution, if exists, are timelike geodesics.
Geometrically, many distance-like functions, such as Busemann functions, are
essentially the solutions to timelike eikonal equation in some sense. Owing to
this fact, distance-like functions are one of the key points to construct the
compactification and completion of a manifold and to prove splitting theorem,
even for finding the solutions to Einstein field equations. For more details,
readers can refer [3, 21].

It is well known an eikonal equation, as a particular kind of Hamilton-Jacobi
equation, has no global smooth solutions in general. Actually, the expectable
maximal regularity of these solutions is the local semiconcavity (see [5, 7,
32] for more information). For the well-posedness of Hamilton-Jacobi
equations, a more natural choice in some sense is the notion of viscosity
solutions, systematically introduced by M-G. Crandall, L-C. Evans and P-L.
Lions in [9, 10] during the early 1980s, although the discussion on well-
posedness of evolutionary Hamilton-Jacobi equations

$\displaystyle\frac{\partial u}{\partial t}+H\left(x,\frac{\partial
u}{\partial x}\right)=0,\,\,(t,x)\in\mathbb{R}^{+}\times M$

had been developed in [13, 23, 24] before the theory of viscosity solutions.
Until now, many monographs have done rich work on this topic. Since late
1990s, for Tonelli Hamiltonian $H$ which includes the eikonal case, Fathi
established weak KAM theory, which made the connection between Aubry-Mather
theory ([26, 25, 27]) and the theory of viscosity solution of stationary
Hamilton-Jacobi equation

$\displaystyle H(x,Du(x))=c,\,\,x\in M,$

on Riemannian manifold ([20, 14, 16, 15, 18, 17]). Cauchy and Dirichlet
problems of Hamilton-Jacobi equations are partly mentioned in [19, 6].

To our knowledge, the convexity (positive definiteness), coercivity (even
superlinearity) of Hamiltonians are needed essentially in the classical theory
of viscosity solutions, Mather theory and weak KAM theory, but these
properties cannot apply to the general space-time. Even so, there are also a
few research papers on viscosity solutions to eikonal equation of various
space-time in recent years. Following from [2], the negative regular cosmology
time function is the viscosity solution of eikonal equation, which means a
kind of viscosity solutions is found in a specific globally hyperbolic space-
time ([12]). Moreover, in the case of class A torus, [22] and [30] developed
the well-posedness of eikonal equations, weak KAM theory and Aubry-Mather
theory for corresponding geodesic flows. [31] is also a good reference for
Aubry-Mather theory for compact space-time. More recently, the work in [29]
dealt the related problems in globally hyperbolic space-time. Authors gave the
existence of global-defined viscosity solutions and the local variational
representation of a solution which has a consistent time-orientation.

Based on the ideas above, we consider a Cauchy type problem of eikonal
equation on a globally hyperbolic space-time. Main results of this paper are
as follows.

Suppose $(M,g)$ is a globally hyperbolic space-time, then there exists a
unique viscosity solution of the following problem:

$\displaystyle\left\\{\begin{array}[]{ll}g(\nabla u(x),\nabla u(x))=-1,&x\in
I^{-}[\Gamma],\\\
u|_{\Gamma}=\phi\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(\Gamma),&\end{array}\right.$
(1.3)

with always past-directed directed time orientation, where $\Gamma$ is a
Cauchy surface. Moreover, this solution can also be represented by a
variational formula as

$\displaystyle u_{\phi}(x)=\inf_{y\in J_{\Gamma}^{+}(x)}\\{\phi(y)-d(x,y)\\}$

and it yields that the solution is semiconcave. In the end, we show that (1.3)
continuously depends on the initial value function on fixed $\Gamma$.

The paper is organized as follows. In Section 2, we introduced some basic
materials involving the Lorentzian geometry and causality theory, semiconcave
functions and viscosity solutions. In Section 3, we answered affirmatively on
the well-posedness of (1.3) (Theorem 3.9, 3.14 and 3.16). The general
regularity and the variational formula of the solution were also given in
Proposition 3.6 and Theorem 3.9.

##  2\. Preliminaries

###  2.1. Lorentzian causality theory

First, we recall some basic concepts and facts on Lorentzian geometry
mentioned in [3, 11, 12, 21, 22], as the setting of this paper.

Let $(M,g)$ be a space-time, i.e., a connected, time-oriented, smooth
Lorentzian manifold with the Lorentzian metric $g$, where the signature for
$g$ is $(-,+,\cdots,+)$. A point $p\in M$ is usually called an event from the
viewpoint of general relativity. For each $p\in M$, we use $T_{p}M$ and $TM$
to denote the tangent space at $p$ and tangent bundle of $M$, respectively.

A tangent vector $V\in T_{p}M$ is called timelike, spacelike or lightlike if
$g(V,V)<0$, $g(V,V)>0$ or $g(V,V)=0$ ($V\neq 0$), respectively. Also, a
tangent vector $V\in T_{p}M$ is called causal if $V$ is either timelke or
lightlike and is called non-spacelike if $g(V,V)\leq 0$. In other words, the
set of non-spacelike vectors consists of zero vector and causal vectors.

Thus, the property of time-orientation of $(M,g)$ ensures that $M$ admits a
continuous and nowhere vanishing timelike vector field $X$, which is used to
separate the causal vectors at each base point into past-directed and future-
directed ones. More precisely, a causal tangent vector $V\in T_{p}M$ is said
to be past-directed (resp. future-directed), if $g(X(p),V)>0$ (resp.
$g(X(p),V)<0$).

Furthermore, a piecewise $C^{1}$ curve $\gamma:I\rightarrow M$
($I\subset\mathbb{R}$ is an interval) is called causal, timelike, lightlike
and past-directed or future-directed if the tangent vector $\dot{\gamma}(s)$
is causal, timelike, lightlike and past-directed or future-directed at every
differentiable point $\gamma(s)$.

For two events $p,q\in M$, if there exists a future-directed timelike (resp.
non-spacelike) curve from $p$ to $q$, we say $p$ and $q$ are chronologically
related (resp. causally related) and denote by $p\ll q$ (resp. $p\leq q$). For
any fixed $p\in M$ and $A\subset M$, it is common practice to define
chronological past $I^{-}(p),I^{-}[A]$, chronological future
$I^{+}(p),I^{+}[A]$, causal past $J^{-}(p),J^{-}[A]$, and causal future
$J^{+}(p),J^{+}[A]$ as follows:

$\displaystyle I^{-}(p)$ $\displaystyle:=\\{q\in M:q\ll
p\\},\,I^{-}[A]:=\\{q\in M:q\ll s~{}~{}\text{for some}~{}~{}s\in A\\},$
$\displaystyle I^{+}(p)$ $\displaystyle:=\\{q\in M:p\ll
q\\},\,I^{+}[A]:=\\{q\in M:s\ll q~{}~{}\text{for some}~{}~{}s\in A\\},$
$\displaystyle J^{-}(p)$ $\displaystyle:=\\{q\in M:q\leq
p\\},\,\,J^{-}[A]:=\\{q\in M:q\leq s~{}~{}\text{for some}~{}~{}s\in A\\},$
$\displaystyle J^{+}(p)$ $\displaystyle:=\\{q\in M:p\leq
q\\},\,\,J^{+}[A]:=\\{q\in M:s\leq q~{}~{}\text{for some}~{}~{}s\in A\\}.$

We call a subset $A$ of $M$ achronal (resp. acausal) if no two points of $A$
are chronologically related (resp. causally related). A space-time $(M,g)$ is
said to be causal if there are no causal loops. A space-time $(M,g)$ is
globally hyperbolic if it is causal and the sets $J^{+}(p)\cap J^{-}(q)$ are
compact for all $p,q\in M$.

Throughout this paper, we always assume that $(M,g)$ is a globally hyperbolic
space-time. Simultaneously, we are going to use two metrics on $M$: one is the
Lorentzian metric $g$, the other is an auxiliary complete Riemannian metric
$h$. The length functional, distance function, the norm and gradient
associated to $g$ are denoted by $L(\cdot)$, $d(\cdot,\cdot)$, $|\cdot|$ and
$\nabla$ respectively. Analogously, the corresponding length functional,
distance function, norm and gradient associated to $h$ are denoted by
$L_{h}(\cdot)$, $d_{h}(\cdot,\cdot)$, $|\cdot|_{h}$ and $\nabla_{h}$,
respectively. More specifically, for a piecewise $C^{1}$ causal curve
$\gamma:[0,1]\rightarrow M$, $L(\gamma)$ is given by

$\displaystyle
L(\gamma):=\int_{0}^{1}\sqrt{-g(\dot{\gamma}(s),\dot{\gamma}(s))}\,ds.$

For any $x,y\in M$ with $x\leq y$, $d(x,y)$ is defined as

$\displaystyle d(x,y):=\sup\left\\{L(\gamma):\gamma\in\Omega_{x,y}\right\\},$

where $\Omega_{x,y}$ denotes the path space of all future-directed causal and
constant curves $\gamma:[0,1]\rightarrow M$ with $\gamma(0)=x$ and
$\gamma(1)=y$. For a non-spacelike vector $V\in TM$, the associated Lorentzian
norm is defined by $|V|:=\sqrt{-g(V,V)}$.

So far, the results on curvature in Lorentz geometry have been abundant. Here
we only extract the following conclusions as the theoretical support of this
paper, which will be revisited in Sections 3.1 and 3.3. For more details, we
refer to [1].

######  Lemma 2.1 ([1, Lemma 3.1]).

Let $M^{n+1}$ be an $(n+1)$-dimensional spacetime such that the sectional
curvatures of timelike planes of $M$ are bounded from above by some constant
$c\in\mathbb{R}$. Assume that there exists $p\in M$ such that
$\mathcal{I}^{+}(p)\neq\varnothing$, and let $q\in\mathcal{I}^{+}(p)$, (with
$d(p,q)<\pi/\sqrt{-c}$ when $c<0$). Then for every spacelike vector $V\in
T_{q}M$ orthogonal to the gradient of $d(p,q)$ it holds that

$\displaystyle(\mathbf{Hess}\,d(p,q))\,(V,V)\geqslant-f_{c}(d(p,q))\langle
V,V\rangle,$ (2.1)

where $\mathbf{Hess}$ stands for the Hessian operator on $M$. When $c<0$ and
$d(p,q)\geqslant\pi/\sqrt{-c}$, then it still holds that

$\displaystyle(\mathbf{Hess}\,d(p,q))\,(V,V)\geqslant-\frac{1}{d(p,q)}\langle
V,V\rangle\geqslant-\frac{\sqrt{-c}}{\pi}\langle V,V\rangle.$ (2.2)

Here, the function $f_{c}$ above is defined as

$\displaystyle
f_{c}(s):=\left\\{\begin{array}[]{ll}\sqrt{c}\coth(\sqrt{c}s),&c>0,s>0,\\\
1/s,&c=0,s>0,\\\
\sqrt{-c}\cot(\sqrt{-c}s),&c<0,0<s<\pi/\sqrt{-c}.\end{array}\right.$

###  2.2. Semiconvex functions and generalized gradient

######  Definition 2.2 ([12, Definition 2.6], Semiconvex/Semiconcave
function).

Let $O$ be an open subset of $M$. A function $\psi:O\to\mathbb{R}$ is said to
be semiconvex (with constant $C$), if there exists $C>0$ such that for any
constant-speed geodesic path $\gamma(t)$, $t\in[0,1]$, $\gamma(t)\in O$,

$\displaystyle\psi(\gamma(t))\leq(1-t)\psi(\gamma(0))+t\psi(\gamma(1))+C\frac{t(1-t)}{2}d^{2}_{h}(\psi(\gamma(0)),\psi(\gamma(1))).$
(2.3)

A function $\psi:O\to\mathbb{R}$ is said to be semiconcave, if $-\psi$ is
semiconvex.

A function $\psi:M\to\mathbb{R}$ is called locally semiconvex if for each
$p\in M$ there is a neighborhood $O$ of $p$ in $M$ such that (2.3) holds true
as soon as $\gamma(t)\in O$ ($t\in[0,1]$); or equivalently if (2.3) holds for
some fixed positive number $C$ which only depends on compact subset $K$ of
$O$.

Readers can refer [7] for more information on the classic definitions and
properties of semiconcave functions.

The following definitions and results in this subsection are well known when
the gradient is induced by the Riemannian metric $h$. In the Lorentzian case,
these can be guaranteed by bilinearity and nondegeneracy of $g$.

######  Definition 2.3 ([7, Definition 3.1.6], Dini gradient).

Assume that $f\in C(M)$. A vector $V\in T_{p}M$ is called to be a subgradient
(resp. supergradient) of $f$ at $p\in M$, if there exist a neighborhood $O$ of
$p$ and a $C^{1}$ function $\phi:O\rightarrow\mathbb{R}$, differentiable at
$p$, with $f(p)=\phi(p)$, $\phi(p)\leqslant f(p)$ (resp. $\phi(p)\geqslant
f(p)$) for every $x\in O$ and $\nabla\phi(p)=V$.

Suoopse $f:M\to\mathbb{R}$ and we denote by $\nabla^{-}f(p)$ the set of
subgradients of $f$ at $p$ and $\nabla^{+}f(p)$ the set of supergradients of
$f$ at $p$. It is easy to check that $\nabla^{\pm}f(p)$ are closed convex sets
(possibly empty). $\nabla^{-}f(p)$ and $\nabla^{+}f(p)$ are both nonempty if
and only if $f$ is differentiable at $p$; in this case,

$\displaystyle\nabla^{-}f(p)=\nabla^{+}f(p)=\\{\nabla f(p)\\}.$

Let ($\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(M)$) $\mbox{\rm Lip}\,(M)$ be the
set of (locally) Lipschitz functions on $M$ with respect to the Riemannian
metric $h$. For a locally Lipschitz function, as it is differentiable almost
everywhere, the following generlized gradient makes sense.

######  Definition 2.4 ([7, Definition 3.1.10], Reachable gradient).

Let $f\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(M)$, then a vector $V\in T_{p}M$
is called a reachable gradient if there exists a sequence $\\{p_{k}\\}\subset
M\setminus\\{p\\}$, such that $f$ is differentiable at $p_{k}$ for each
$k\in\mathbb{N}$, and

$\displaystyle\lim_{k\to\infty}p_{k}=p,\,\,\,\,\lim_{k\to\infty}\nabla
f(p_{k})=V,$ (2.4)

where, the first limit in (2.4) is taken in the sense of manifold topology on
$M$; the second one is taken in the sense of any fixed chart that contains
$p$.

The set of all reachable gradients of $\phi$ at $p$ is denoted by
$\nabla^{*}f(p)$.

######  Remark 2.5.

Since the first limit is taken, we know that when $k$ is sufficiently large,
$p_{k}$ goes into that chart. The second limit does not depend on the choice
of chart.

######  Proposition 2.6 ([7, Proposition 3.3.4]).

If $\psi$ is locally semiconvex (resp. semiconcave) on $M$, then
$\psi\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(M)$ and we have the following
properties:

  1. (1)

$\nabla^{-}\psi(p)\neq\varnothing$ (resp. $\nabla^{+}\psi(p)\neq\varnothing$)
for any $p\in M$;

  2. (2)

$\nabla^{-}\psi(p)=\mathrm{co}\nabla^{*}\psi(p)$ (resp.
$\nabla^{+}\psi(p)=\mathrm{co}\nabla^{*}\psi(p)$) for any $p\in M$, where
$\mathrm{co}$ means the convex hull of a set;

  3. (3)

$\nabla^{-}\psi(p)$ (resp. $\nabla^{+}\psi(p)$) is a singleton if and only if
$\psi$ is differentiable at $p$;

  4. (4)

if $\\{p_{k}\\}$ converges to $p$ and $V_{k}\in\nabla^{-}\psi(p_{k})$ (resp.
$V_{k}\in\nabla^{+}\psi(p_{k})$) converges to $V$, then
$V\in\nabla^{-}\psi(p)$ (resp. $V\in\nabla^{+}\psi(p)$);

  5. (5)

if $\nabla^{-}\psi(p)$ (resp. $\nabla^{+}\psi(p)$) is a singleton for all
$p\in M$, then $\psi\in C^{1}(M)$.

The following lemma is a standard description about the gradients of a family
of uniformly semiconcave functions and their limit functions.

######  Lemma 2.7 ([8, Lemma 2.1]).

Let $\\{\psi_{k}\\}$ be a sequence of semiconcave functions on a bounded
neighborhood $U$ with a uniform semiconcave constant $C$, $\\{x_{k}\\}\subset
U$, $V_{k}\in\nabla^{+}\psi_{k}(x_{k})$, $k\in\mathbb{N}$. If $x_{k}\to x$ and
$\psi_{k}\rightrightarrows\psi$ as $k\to\infty$, then $\psi$ is semiconcave
with constant $C$ and

$\displaystyle\lim_{k\to\infty}d_{E}(V_{k},D^{+}\psi(x))=0,$

where $d_{E}$ is the Euclidean distance. In particular, if $\psi_{k}$ is
differentiable at $x_{k}$ and $\psi$ is differentiable at $x$, then
$\lim_{k\to\infty}\nabla\psi_{k}(x_{k})=\nabla\psi(x)$.

###  2.3. Lorentzian eikonal equation and viscosity solution

We recall the timelike Lorentzian eikonal equation with respect to the
Lorentzian metric $g$:

$\displaystyle g(\nabla u,\nabla u)=-1.$ (Eg)

Lorentzian eikonal equation plays an important role in the study of global
Lorentzian geometry (see [3, 12]). Similar to the Riemannian case, it is
appropriate to introduce the concept of viscosity solution here, which could
help ones better understand the behavior of timelike rays in the setting of
Lorenzian geometry. Following the idea originating from [9, Definition 2], we
can state the definition of viscosity solution as below.

######  Definition 2.8 (Viscosity solution of (Eg)).

$f\in C(M)$ is called a viscosity subsolution of (Eg) if for any $p\in M$, it
satisfies

$\displaystyle g(V,V)\leqslant-1,\,\,\,\,\,\forall\,V\in\nabla^{+}f(p).$

Similarly, $f\in C(M)$ is called a viscosity supersolution of (Eg) if for any
$p\in M$, we have

$\displaystyle g(V,V)\geqslant-1,\,\,\,\,\,\forall\,V\in\nabla^{-}f(p).$

If $f$ satisfies both of the properties above, we call it a viscosity solution
of equation (Eg).

We would naturally consider the following set

$\displaystyle\mathscr{S}(M):=\left\\{f\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(M):f\mbox{
is global viscosity solution to \eqref{E1}}\right\\}.$

In [29], $\mathscr{S}(M)$ has been studied systematically. The first result is
$\mathscr{S}(M)\neq\varnothing$, i.e., any globally hyperbolic space-time
admits at least one globally defined locally Lipschitz viscosity solution.
Moreover, by defining the time orientation of viscosity solution, the set
$\mathscr{S}(M)$ can be divided into some subclasses. The definition of the
time orientation is defined as follows.

######  Definition 2.9.

For $f\in\mbox{\rm Lip}\,(M)$ where $\nabla f$ is timelike as long as it
exists:

  1. (1)

The time orientation of $f$ is called future-directed at $x$, if any limiting
gradient $V\in\nabla^{*}f(x)$ is future-directed timelike; the time
orientation of $f$ is past-directed at $x$ if any limiting gradient
$V\in\nabla^{*}f(x)$ is past-directed timelike;

  2. (2)

We say the time orientation of $f$ changes at $x$ if there exist two timelike
vectors $V,W$ in $\nabla^{*}f(x)$ such that $V$ is past-directed and $W$ is
future-directed;

  3. (3)

We call the time orientation of $f$ is consistent if the time orientation of
$f$ is either future-directed at arbitrary $x\in M$ or past-directed at
arbitrary $x\in M$. Otherwise, we say the time orientation of $f$ is non-
consistent.

When the time orientation of viscosity solution of equation (Eg) is
consistent, we have the following properties in the view of weak KAM theory.
Recall that a future-directed (resp. past-directed) inextendible timelike
curve (see [3, Page 61]) $\gamma:[0,T)\to M$ is called a ray if
$d(\gamma(a),\gamma(b))=b-a$ (resp. $d(\gamma(b),\gamma(a))=b-a$) for all
$0\leqslant a\leqslant b<T$.

######  Theorem 2.10 ([29, Theorem 3]).

Suppose $u\in\mathscr{S}(M)$, the time orientation of which is always past-
directed, then

  1. (1)

$u$ is differentiable at $x\in M$ if and only if there exists a unique future-
directed timelike ray $\gamma_{x}:[0,T)\rightarrow M$ that satisfies

$\displaystyle\gamma_{x}(0)=x,\,\,\,u(\gamma_{x}(t))=u(x)-t,\,\,\,\,\forall\,t\in[0,T)$
(2.5)

and

$\displaystyle\dot{\gamma_{x}}(0)=-\nabla u(x),$ (2.6)

where $T$ is the maximal existence time of $\gamma_{x}$;

  2. (2)

for any $x\in M$, $u$ is differentiable at any $\gamma_{x}(t)$ for $t\in(0,T)$
and $\dot{\gamma_{x}}(t)=-\nabla u(\gamma_{x}(t))$.

Inspired by the work above, this paper continues to investiage the well-
posedness of Cauchy type problem for a timelike Lorentzian eikonal equation.
To pose the Cauchy type problem exactly, we introduce a preliminary concept.

######  Definition 2.11 (Cauchy Surface).

A Cauchy surface $\Gamma$ is a subset of $M$ for which every inextendible
causal curve intersects exactly once.

By [4, Theorem 3], there exists a steep temporal function
$\tau:M\to\mathbb{R}$ such that $\tau$ is strictly increasing along any
future-directed causal curve and

$|d\tau(V)|\geqslant\max\left\\{\sqrt{-g(V,V)},|V|_{h}\right\\}$ (2.7)

for all timelike vector $V\in TM$. Moreover, Corollary 1.8 in [4] implies that
each level set $\tau_{s}:=\\{p\in M|\tau(p)=s\\}$ is a Cauchy surface.

Hereafter, we always use $\Gamma\subset M$ to represent a given Cauchy
surface. Now one can consider a Cauchy type problem for the timelike
Lorentzian eikonal equation on $M$ as

$\displaystyle\left\\{\begin{array}[]{ll}g(\nabla u(x),\nabla u(x))=-1,&x\in
I^{-}[\Gamma],\\\ u(x)=\phi(x),&x\in\Gamma,\end{array}\right.$ (2.10)

where $\phi\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(\Gamma)$.

##  3\. Well-posedness of (2.10)

###  3.1. Existence of the viscosity solutions

For brevity, as $x\in I^{-}[\Gamma]$, we set

$\displaystyle I^{+}_{\Gamma}(x)$ $\displaystyle:=\Gamma\cap
I^{+}(x)=\\{y\in\Gamma:x\ll y\\},$ $\displaystyle J^{+}_{\Gamma}(x)$
$\displaystyle:=\Gamma\cap J^{+}(x)=\\{y\in\Gamma:x\leq y\\}.$

Suppose $\phi\in\mbox{\rm Lip}\,(\Gamma)$. For any $x\in I^{-}[\Gamma]$,
consider the function

$\displaystyle u_{\phi}(x):=\inf_{y\in
J^{+}_{\Gamma}(x)}\\{\phi(y)-d(x,y)\\}.$ (3.1)

The following lemma establishes the existence of minimizer of equation (3.1).

######  Lemma 3.1.

For any $x\in I^{-}[\Gamma]$, there exists $y_{x}\in\Gamma$ such that

$\displaystyle u_{\phi}(x)=\phi(y_{x})-d(x,y_{x}).$ (3.2)

In other words, $y_{x}$ is the minimizer of (3.1) and the infimum is actually
a minimum.

###### Proof.

We reparametrize all future-directed inextendible causal geodesics by
$h$-arclength, then these geodesics satisfy a system of second order ordinary
differential equations. Since $\Gamma$ is a Cauchy surface, any future-
directed inextendible causal curve $\gamma$ with $\gamma(0)=x$ must strike
$\Gamma$ at a unique point. Therefore, we can define a map $T_{\Gamma,x}$
depends on $x$ as

$\displaystyle T_{\Gamma,x}:U_{h}T_{x}M\cap\mathscr{C}_{x}$
$\displaystyle\to\Gamma$ $\displaystyle V$
$\displaystyle\mapsto\gamma_{x,V}\cap\Gamma,$

where $U_{h}T_{x}M$ is the unit tangent sphere at $x$ with respect to
Riemannian metric $h$, $\mathscr{C}_{x}$ is the future-pointing light cone at
$x$ and $\gamma_{x,V}$ is a future-directed inextendible causal curve with
$\gamma(0)=x$ and $\dot{\gamma}(0)=V$. Applying the continuous dependence on
the initial condition of ODE, it is easy to see that $T_{\Gamma,x}$ is
continuous. Since the $U_{h}T_{x}M\cap\text{int}\,(\mathscr{C}_{x})$ is
diffeomorphic to an open ball in Euclidean space,

$\displaystyle
J^{+}_{\Gamma}(x)=T_{\Gamma,x}(U_{h}T_{x}M\cap\mathscr{C}_{x})=\overline{T_{\Gamma,x}(U_{h}T_{x}M\cap\mathrm{int}\,(\mathscr{C}_{x}))}=\overline{I^{+}_{\Gamma}(x)}$

is compact. Then there must be a point $y_{x}$ which minimizes
$\phi(\cdot)-d(x,\cdot)$. ∎

######  Definition 3.2 (Forward calibrated curve).

For any fixed $u\in\mathscr{S}(M)$, a future-directed timelike curve
$\alpha:[0,t]\rightarrow M$ is said to be a forward calibrated curve of $u$ if
$u(\alpha(0))=u(\alpha(s))-s$ for any $s\in[0,t]$.

For any $x\in I^{-}[\Gamma]$, we always use $y_{x}$ to denote the minimizer of
(3.1) . Moreover, from (3.2), we can obtain a future-directed timelike curve
${\gamma}_{x}$ with ${\gamma}_{x}(0)=x$, ${\gamma}_{x}(d(x,y_{x}))=y_{x}$ and

$\displaystyle u_{\phi}({\gamma}_{x}(t))-u_{\phi}({\gamma}_{x}(0))=t$ (3.3)

for any $t\in[0,d(x,y_{x})]$. Indeed, ${\gamma}_{x}$ is a forward calibrated
curve for $u_{\phi}$ by definition. In the rest of this paper, we always use
$y_{x}$ and ${\gamma}_{x}$ to denote a minimizer of equality (3.1) and forward
calibrated curve of $u_{\phi}$ at $x$.

With the help of Lemma 3.1, we can study the regularity of $u_{\phi}$.
Actually, we will show that $u_{\phi}$ is a locally semiconcave function on
$I^{-}(\Gamma)$. As a preparation, we need to introduce the following
concepts.

######  Definition 3.3 (Upper support function).

Let $x\in I^{-}[\Gamma]$ and $\gamma$ be a future-directed timelike maximal
geodesic with $\gamma(0)=x$ and $\gamma(d(x,y_{x}))=y_{x}$. For any fixed
$p\in\gamma$ and a suitable neighborhood $O$ of $x$ satisfying $O\subset
I^{-}(p)$, we call the function

$\displaystyle u_{\phi,x}:O$ $\displaystyle\to[-\infty,+\infty]$
$\displaystyle z$ $\displaystyle\mapsto\phi(y_{x})-d(z,p)-d(p,y_{x}).$ (3.4)

an upper support function of $u_{\phi}$ at $x$.

######  Remark 3.4.

Actually, for any choice of $p\in\gamma$,

$\displaystyle u_{\phi,x}(z)$
$\displaystyle=\phi(y_{x})-d(z,p)-d(p,y_{x})\geqslant\phi(y_{x})-d(z,y_{x})\geqslant\phi(y_{z})-d(z,y_{z})=u_{\phi}(z)$

for any $z\in O$ and the equality holds when $z=x$. Readers may wonder about
the dependence of $u_{\phi,x}$ on $p$ here, but the choice of $p$ does not
prevent the following conclusions.

Relying on the upper support function, we can study the semiconcavity of
$u_{\phi}$.

######  Lemma 3.5 ([2, Lemma 3.2]).

Let $U\subset\mathbb{R}^{n}$ be an open convex domain and $f\in C(U)$. Also,
assume that there exists some $C>0$ such that for any $p\in U$, $f_{p}$, the
upper support function of $f$ at $p$, satisfies $D^{2}f_{p}\leqslant CI$ in
its domain of definition, then $f-\frac{C}{2}\|\cdot\|_{E}^{2}$ is concave in
$U$, which means $f$ is semiconcave with constant $C$, where $\|\cdot\|_{E}$
denotes the Euclidean norm on $\mathbb{R}^{n}$.

######  Proposition 3.6.

$u_{\phi}$ is locally semiconcave on $I^{-}[\Gamma]$.

###### Proof.

Since $\gamma_{x}$ maximizes the distance between any pair of its points,
there is no cut point to $x$ on $\gamma_{x}|_{(0,d(x,p))}$ for appropriately
selected $p$. Then by [3, Proposition 9.29], there exists a neighborhood $O$
of $x$ such that $\bar{O}$ is compact, $\bar{O}\varsubsetneqq I^{-}(p)$ and
$d_{p}(\cdot):=d(\cdot,p)$ is a smooth function on $O$.

Combining with Lemma 3.5, we only need to prove that there exists a constant
$C(O)>0$ depending only on $O$ such that $D^{2}u_{\phi,x}\leqslant C(O)I$ near
$x$ in the setting of Euclidean coordinates. By the definition of
$u_{\phi,x}$, this is equivalent to prove that
$D^{2}d(\cdot,p)\geqslant-C(O)I$ near $x$ on $O$.

Now for any $z\in O$, we denote $\eta_{z}$ the unique speed future-directed
maximal geodesic connecting $z$ to $p$. In the following we will use the
standard comparison principle to give an estimation of Hessian of Lorentzian
distance function $d_{p}$ in terms of upper and lower bounds of timelike
sectional curvatures of 2-planes containing $\dot{\eta}_{z}(0)$ and the length
of maximal segment between $z$ and $p$, where the Hessian of $d_{p}$ is
defined in terms of the Levi-Civita connection with respect to $g$.

Since $\bar{O}$ is a compact subset of $M$,

$\displaystyle\inf_{z\in\bar{O},V\in\partial\mathscr{C}_{z}^{+1}}\left\\{\bigg{|}\frac{\dot{\eta}_{z}(0)}{|\dot{\eta}_{z}(0)|_{h}}-V\bigg{|}_{h}\right\\}>0,$

where $\mathscr{C}_{z}^{+1}:=\\{V\in T_{z}M:V\text{ is a future-directed
lightlike vector and }|V|_{h}=1\\}$, which implies there exists a compact
subset $K_{O}\subseteq TM$ such that $\cup_{z\in
O}\\{(\eta_{z}(0),\dot{\eta}_{z}(0))\\}$ (thus $\cup_{z\in
O}\\{(\eta_{z}(t),\dot{\eta}_{z}(t)):t\in[0,d(z,p)]\\}$) and $\cup_{z\in
O}\\{(z,V)\in TM:g(V,\dot{\eta}_{z}(0))=0,|V|=1\\}$ are contained in $K_{O}$.
Therefore, both the timelike sectional curvatures of 2-planes containing
$\dot{\eta}_{z}(0)|_{z\in O}$ and the length of $\eta_{z}|_{[0,d(z,p)]}$ are
bounded from above by some constant $C>0$ (e.g. [2, Proposition 3.1]). By
Lemma 2.1, there exists a constant $C_{1}(O)$ depending only on $O$ such that

$\displaystyle D^{2}d(\cdot,p)\geqslant C_{1}(O)I.$ (3.5)

For any $z\in O$, let $\\{e_{1},e_{2},...,e_{n-1},\dot{\eta}_{z}(0)\\}$ be a
local orthonormal frame of $T_{z}M$. Then we have

$\displaystyle\frac{\partial^{2}d_{p}(z)}{\partial x_{i}\partial
x_{j}}-\Gamma_{ij}^{k}\frac{\partial d_{p}(z)}{\partial x_{k}}\geqslant
C_{1}(O)I,$

where $\Gamma_{ij}^{k}$ is the Christoffel symbol. Due to the smoothness of
$\Gamma_{ij}^{k}$ and $d_{p}$ on $O$, we can get another constant $C_{2}(O)$,
such that

$\displaystyle\frac{\partial^{2}d(z,p)}{\partial x_{i}\partial x_{j}}\geqslant
C_{2}(O)I,$ (3.6)

for any $z\in O$. From the estimate on Hessian of $d_{p}$ in Euclidean setting
and Lemma 3.5, we have the local semiconcavity of $u_{\phi}$ on
$I^{-}[\Gamma]$. ∎

######  Lemma 3.7 ([12, Lemma 2.5]).

Suppose $V\in T_{p}M$ is a nonzero past-directed non-spacelike vector and

$\displaystyle g(V,W)\geqslant\sqrt{-g(W,W)}$

for any future-directed causal vector $W$, then $V$ is not lightlike.

######  Proposition 3.8.

If $u_{\phi}$ is differentiable at $x$, then $\nabla u_{\phi}(x)$ is a past-
directed timelike vector. Moreover, in this case, $u_{\phi}$ satisfies (Eg) at
$x$.

###### Proof.

Assume that $u_{\phi}$ is differentiable at $x\in I^{-}[\Gamma]$. Choosing any
fixed smooth future-directed causal curve $\gamma:[0,T)\to M$ with some small
$T>0$ and $\gamma(0)=x$, by reverse triangle inequality and the definition of
$u_{\phi}$, we have

$\displaystyle u_{\phi}(\gamma(t))-d(\gamma(0),\gamma(t))$
$\displaystyle=\min_{y\in
J_{\Gamma}^{+}(\gamma(t))}\\{\phi(y)-d(\gamma(t),y)\\}-d(\gamma(0),\gamma(t))$
$\displaystyle\geqslant\min_{y\in
J_{\Gamma}^{+}(\gamma(t))}\\{\phi(y)-d(\gamma(0),y)\\}$
$\displaystyle\geqslant\min_{y\in
J_{\Gamma}^{+}(\gamma(0))}\\{\phi(y)-d(\gamma(0),y)\\}=u_{\phi}(\gamma(0))$

since $J_{\Gamma}^{+}(\gamma(t))\subset J_{\Gamma}^{+}(\gamma(0))$ for every
$t\in[0,T)$. Armed with that information,

$\displaystyle u_{\phi}(\gamma(t))-u_{\phi}(\gamma(0))$
$\displaystyle\geqslant
d(\gamma(0),\gamma(t))\geqslant\int_{0}^{t}\sqrt{-g(\dot{\gamma}(s),\dot{\gamma}(s))}\,ds$

for every $t\in[0,T)$. Dividing by $t$ on both sides, we get

$\frac{u_{\phi}(\gamma(t))-u_{\phi}(x)}{t}\geqslant\frac{1}{t}\int_{0}^{t}\sqrt{-g(\dot{\gamma}(s),\dot{\gamma}(s))}\,ds.$
(3.7)

As $t\searrow 0$, the differentiability of $u_{\phi}$ at $x$ and inequality
(3.7) yield

$du_{\phi}(x)(\dot{\gamma}(0))\geqslant\sqrt{-g(\dot{\gamma}(0),\dot{\gamma}(0))}.$

Note that $du_{\phi}(x)(\dot{\gamma}(0))=g(\nabla
u_{\phi}(x),\dot{\gamma}(0))$, this means

$g(\nabla
u_{\phi}(x),\dot{\gamma}(0))\geq\sqrt{-g(\dot{\gamma}(0),\dot{\gamma}(0))}$
(3.8)

for any future-directed causal vector $\dot{\gamma}(0)$.

We now prove that $\nabla u_{\phi}(x)$ is timelike. First, it is easy to see
that $\nabla u_{\phi}(x)\neq 0$. Furthermore, suppose $\nabla u_{\phi}(x)$ is
spacelike, then there exists a smooth future-directed timelike curve
$\bar{\gamma}:[0,\varepsilon)\to M$ such that $\bar{\gamma}(0)=p$ and
$g(\dot{\bar{\gamma}}(0),\nabla u_{\phi}(x))=0$, which means that $\nabla
u_{\phi}(x)$ and $\dot{\bar{\gamma}}(0)$ are orthogonal with respect to the
Lorentzian metric $g$. On the other hand, from (3.8), we get $g(\nabla
u_{\phi}(x),\dot{\bar{\gamma}}(0))>0$, which contradicts the orthogonal
hypothesis. Thus, $\nabla u_{\phi}(p)$ is a causal vector. In addition,
$g(\nabla u_{\phi}(x),V)\geqslant 0$ for any future-directed vector $V\in
T_{p}M$. This implies that $\nabla u_{\phi}(x)$ is past-directed. By Lemma
3.7, we have $\nabla u_{\phi}(x)$ is indeed a past-directed timelike vector.

For the rest of the proof, we can show that $|\nabla u_{\phi}(x)|=1$
equivalently. First we choose a future-directed smooth causal curve
$\eta:[0,\varepsilon)\rightarrow M$ with $\eta(0)=x$, $\dot{\eta}(0)=-\nabla
u_{\phi}(x)$. Then by inequality (3.8),

$\displaystyle|\nabla u_{\phi}(x)|^{2}=-g(\nabla u_{\phi}(x),\nabla
u_{\phi}(x))=g(\nabla
u_{\phi}(x),\dot{\eta}(0))\geqslant\sqrt{-g(\dot{\eta}(0),\dot{\eta}(0))}=|\nabla
u_{\phi}(x)|.$

Hence, $|\nabla u_{\phi}(x)|\geqslant 1$ since $\nabla u_{\phi}(x)\neq 0$.
Recall that for any $x\in I^{-}[\Gamma]$ there exists a $y_{x}\in\Gamma$ and a
forward calibrated curve ${\gamma}_{x}:[0,d(x,y_{x})]\to M$ that starts at $x$
and ends at $y_{x}$ satisfies

$\displaystyle u_{\phi}({\gamma}_{x}(t))-u_{\phi}({\gamma}_{x}(0))=t$

for any $t\in[0,d(x,y_{x})]$. Then we get

$\displaystyle\frac{u_{\phi}({\gamma}_{x}(t))-u_{\phi}({\gamma}_{x}(0))}{t-0}=1$

for every $t\in[0,d(x,y_{x})]$. Let $t\searrow 0$, and by the the reverse
Cauchy-Schwarz inequality for causal vectors ([28, Section 2.2.1]) and
${\gamma}_{x}$ is a timelike (unit-speed) curve, we have

$1=du_{\phi}(x)(\dot{{\gamma}}_{x}(0))=g(\nabla
u_{\phi}(x),\dot{{\gamma}}_{x}(0))\geqslant|\nabla
u_{\phi}(x)||\dot{{\gamma}}_{x}(0)|=|\nabla u_{\phi}(x)|.$ (3.9)

Combining with $|\nabla u_{\phi}(x)|\geqslant 1$, (3.9) implies $|\nabla
u_{\phi}(x)|=1$. ∎

######  Theorem 3.9.

Let $(M,g)$ be a globally hyperbolic space-time, then $u_{\phi}$ is a
viscosity solution to (Eg) and (2.10) on $I^{-}[\Gamma]$.

###### Proof.

By the definition of viscosity solution, we need to show $u_{\phi}$ is the
subsolution and supersolution of equation (Eg). Firstly, for any
$V\in\nabla^{+}u_{\phi}(x)$, by Lemma 2.6,

$\displaystyle V\in\nabla^{+}u_{\phi}(x)=\mathrm{co}\nabla^{*}u_{\phi}(x).$

Together with Lemma 3.8, we can conclude that $g(V,V)\leqslant-1$. It means
that $u_{\phi}$ is a subsolution of the equation (Eg).

On the other hand, since $u_{\phi}$ is locally semiconcave, $u_{\phi}$ is
differentiable at $x\in M$ if and only if
$\nabla^{-}u_{\phi}(x)\neq\varnothing$ and
$\nabla^{+}u_{\phi}(x)=\nabla^{-}u_{\phi}(x)=\\{\nabla u_{\phi}(x)\\}$. Thus,
Proposition 3.8 implies that $u_{\phi}$ is a supersolution. Therefore
$u_{\phi}$ is indeed a viscosity solution of the equation (Eg). Moreover, the
definition of $u_{\phi}$ promises that $u_{\phi}|_{\Gamma}=\phi$, which means
$u_{\phi}$ is also a viscosity solution to (2.10). ∎

###  3.2. Uniqueness of the solution

For brevity of the following illustration, we first define some subsets of
$\mathscr{S}(M)$ below:

$\displaystyle\mathscr{S}_{\Gamma_{\phi}}(M)$
$\displaystyle:=\\{u\in\mathscr{S}(M):u|_{\Gamma}=\phi\\},$
$\displaystyle\mathscr{S}^{-}_{\Gamma_{\phi}}(M)$
$\displaystyle:=\\{u\in\mathscr{S}(M):u|_{\Gamma}=\phi\text{ and the time
orientation of }u\text{ is always past-directed}\\}.$

And about (2.10), we also define

$\displaystyle\mathscr{C}_{\phi}(I^{-}[\Gamma])$
$\displaystyle:=\left\\{f\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(M):f\mbox{ is
global viscosity solution to \eqref{A1}}\right\\},$
$\displaystyle\mathscr{C}_{\phi}^{-}(I^{-}[\Gamma])$
$\displaystyle:=\left\\{f\in\mathscr{C}_{\phi}(I^{-}[\Gamma]):\text{ the time
orientation of }u\text{ is always past-directed}\right\\}.$

Note that we have no reason here to claim that
$\mathscr{S}_{\Gamma_{\phi}}(M)=\mathscr{C}_{\phi}(I^{-}[\Gamma])$ and
$\mathscr{S}_{\Gamma_{\phi}}^{-}(M)=\mathscr{C}_{\phi}^{-}(I^{-}[\Gamma])$.

######  Lemma 3.10.

Suppose $\Gamma$ is a Cauchy surface of $(M,g)$ and
$\phi\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(\Gamma)$, then

$\displaystyle\left\\{u|_{I^{-}[\Gamma]}:u\in\mathscr{S}_{\Gamma_{\phi}}^{-}(M)\right\\}=\left\\{u_{\phi}\right\\},$
(3.10)

i.e., in some sense, the solutions in $\mathscr{S}_{\Gamma_{\phi}}^{-}(M)$ are
unique on half-space $I^{-}[\Gamma]$.

###### Proof.

Let $u\in\mathscr{S}^{-}_{\Gamma_{\phi}}(M)$. Recall that $u_{\phi}$ has a
forward calibrated curve $\gamma_{x}$ from $x$ with unit speed. By the
definition of $u_{\phi}$ and $u$, for $x\in I^{-}[\Gamma]$, we have
$u(x)\leqslant u({\gamma}_{x}(t))-d(x,{\gamma}_{x}(t))$ for any
$t\in[0,d(x,y_{x})]$, and when $t=d(x,y_{x})$,

$\displaystyle u(x)$ $\displaystyle\leqslant
u(y_{x})-d(x,y_{x})=\phi(y_{x})-d(x,y_{x})=u_{\phi}(x).$

On the other hand, Theorem 2.10 claims $u$ has an inextendiable forward
calibrated curve $\eta_{x}$ from any fixed $x\in M$. There exists some $T>0$
such that $\eta_{x}(T)\in\Gamma$. Then, we have

$\displaystyle
u(x)=u(\eta_{x}(T))-T=\phi(\eta_{x}(T))-d(x,\eta_{x}(T))\geqslant
u_{\phi}(x),$

which means $u(x)=u_{\phi}(x)$. Arbitrariness of $u$ and $x$ implies (3.10)
holds true. ∎

######  Remark 3.11.

It is worth noting that the condition on consistence of time orientation of
solutions is necessary, and the uniqueness of viscosity solution of (2.10)
doesn’t hold true any more in general. To be precise, see the following
counterexample.

Consider the 2-dimensional Minkowski space-time
$(\mathbb{R}^{2},dy^{2}-dx^{2})$, let

$\displaystyle\Gamma=\\{(x,y)\in\mathbb{R}^{2}:x=0\\},\,\,\,\,\phi=0.$

Then for any $c<0$, $u_{c}(x,y):=|x-c|+c$ is the solution to system (2.10).

######  Corollary 3.12.

Following from Theorem 2.10 and Lemma 3.10, we have

  1. (1)

For any $u\in\mathscr{S}^{-}_{\Gamma_{\phi}}(M)$, $u$ is differentiable at
$x\in M$ if and only if there exists a unique forward calibrated curve
$\gamma_{x}:[0,d(x,\Gamma)]\rightarrow M$ that satisfies

$\displaystyle\gamma_{x}(0)=x,~{}~{}u(\gamma_{x}(t))=u(x)-t~{}~{}\mbox{for
every }t\in[0,d(x,\Gamma)]$ (3.11)

and $\dot{\gamma_{x}}(0)=-\nabla u(x)$;

  2. (2)

For any $u\in\mathscr{S}^{-}_{\Gamma_{\phi}}(M)$ and any $x\in M$, $u$ is
differentiable at any $\gamma_{x}(t)$ for $t\in(0,d(x,\Gamma))$ and
$\dot{\gamma_{x}}(t)=-\nabla u(\gamma_{x}(t))$.

######  Theorem 3.13 ([29, Theorem 2]).

Assume that $u\in\mathscr{S}(M)$ and the time orientation of $u$ is always
past-directed, then there exists some $t_{0}>0$, which depends on $x$, such
that:

  1. (1)

for each $s\in\mathrm{Image}\,(u)$, $u^{-1}(s)$ is a partial Cauchy surface;

  2. (2)

for any $t\in(0,t_{0})$,

$\displaystyle u(x)=\inf\limits_{x\leq y,d(x,y)=t}\\{u(y)-t\\};$

  3. (3)

$u$ is locally semiconcave on $(M,g)$.

######  Theorem 3.14.

Suppose $\Gamma$ is a Cauchy surface of $(M,g)$ and
$\phi\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(\Gamma)$, then

$\displaystyle\mathscr{C}_{\phi}^{-}(I^{-}[\Gamma])=\left\\{u_{\phi}\right\\}.$
(3.12)

In other words, $u_{\phi}$ is the unique viscosity solution to $\eqref{A1}$
with always past-directed time orientation.

###### Proof.

By the proof of Theorem 3.13, if the time orientation of $u$ is always past-
directed, $u$ has a local variational representation. Thus, we still have a
forward calibrated curve $\eta_{x}:[0,t_{0}]\rightarrow M$ of $u$ at $x$. By
[29, Subsection 5.2], we can extend this curve $\eta_{x}$ until $\eta_{x}$
intersects $\Gamma$. Then the rest of proof follows from Lemma 3.10. ∎

###  3.3. The continuous dependence of solutions (2.10) on initial values

Suppose $\phi_{n}\in\mbox{$\mathrm{Lip}_{\mathrm{loc}}$}(\Gamma)$ and consider
the system

$\displaystyle\left\\{\begin{array}[]{ll}g(\nabla u_{n}(x),\nabla
u_{n}(x))=-1,&x\in I^{-}[\Gamma],\\\
u_{n}(x)=\phi_{n}(x),&x\in\Gamma.\end{array}\right.$ (3.15)

As a result of two subsections above, $u_{n}:=u_{\phi_{n}}$ is the unique
viscosity solution to (3.15). To study the continuous dependence of system
(2.10) on initial values, i.e., in some sense, we hope

$\displaystyle\lim_{n\to\infty}\eqref{A1n}=\eqref{A1}.$

under the assumption $\phi_{n}\rightrightarrows\phi$ as $n\to\infty$ on
$\Gamma$.

######  Lemma 3.15.

Suppose $\phi_{n}\rightrightarrows\phi$ on $\Gamma$. Then
$\\{u_{n}\\}_{n\geqslant 1}$ has a subsequence of locally equi-Lipschitz and
equi-semiconcave functions.

###### Proof.

Owing to Lemma 3.5, the lemma is equivalent to the claim that for a suitable
neighbourhood $O$ of $x$, the bound of the Hessian of distance function
$d(\cdot,p_{n})$ defined on $O$ is independent on $n$ for some $p_{n}\in
I^{-}[\Gamma]$. Recall that for each $n$ there is a forward calibrated curve
denoted by $\gamma_{x}^{n}:[0,d(x,y_{x}^{n})]\rightarrow M$ connecting $x$ to
$y_{x}^{n}$, where $y_{x}^{n}$ is a minimizer for
$\phi_{n}(\cdot)-d(x,\cdot)$.

For every $n$, we can also extend $\gamma_{x}^{n}$ to a future-directed
inextendible curve and reparameter by $h$-arc length as
$\tilde{\gamma}_{x}^{n}:[0,+\infty)\rightarrow M$. [3, Lemma 14.2] implies
that there exists a subsequence $\\{\tilde{\gamma}_{x}^{n_{k}}\\}$ and an
intextendible future-directed causal curve
$\tilde{\gamma}_{x}:[0,+\infty)\rightarrow M$ with $\tilde{\gamma}_{x}(0)=x$
such that $\\{\tilde{\gamma}_{x}^{n_{k}}\\}$ converges to $\tilde{\gamma}_{x}$
uniformly under the Riemannian metric $h$ on arbitrary compact subset of
$\mathbb{R}^{+}$.

Since $\Gamma$ is a Cauchy surface, there exists $T>0$ such that
$\tilde{\gamma}_{x}(T)\in\Gamma$. Recall that $\gamma_{x}^{n}$ is a maximal
segment on $[0,d(x,y_{x}^{n})]$. There exists $T^{*}<T$ and $\\{T_{n_{k}}\\}$,
which satisfy
$\gamma_{x}^{n_{k}}(T_{n_{k}})=\tilde{\gamma}_{x}^{n_{k}}(T^{*})$ and
$\gamma_{x}^{n_{k}}$ are well-defined on $[0,T_{n_{k}}]$. By the upper
semicontinuity of Lorentzian length functional, we have

$\displaystyle
L(\tilde{\gamma}_{x})|_{[0,T^{*}]}\geqslant\limsup_{k\rightarrow\infty}L(\tilde{\gamma}_{x}^{n_{k}})|_{[0,T^{*}]}=\limsup\limits_{k\rightarrow\infty}L(\gamma_{x}^{n_{k}})|_{[0,T_{n_{k}}]}=\limsup\limits_{k\rightarrow\infty}d(x,\gamma_{x}^{n_{k}}(T_{n_{k}}))=d(x,\tilde{\gamma}_{x}(T^{*})).$

It means $\tilde{\gamma}_{x}|_{[0,T^{*}]}$ is a future-directed timelike
maximal geodesic up to a time reparameteriztion. We use ${\gamma}_{x}$ to
denote the unit speed future-directed maximal geodesic after
reparameteriztion. Then the internal of ${\gamma}_{x}$ is free of cut point.
Let $\\{t_{n_{k}}\\}\subset[0,T^{*}]$ and $t_{n_{k}}$ monotone decrease to
some $t^{*}\in(0,T^{*})$. Also, set
$p_{n_{k}}:=\tilde{\gamma}_{x}^{n_{k}}(t_{n_{k}})$ and
$p:=\tilde{\gamma}_{x}(t^{*})$. Because of Proposition 9.7 and Proposition
9.29 in [3], there exists some neighbourhood $O$ of $x$ contained in all
$I^{-}(p_{n_{k}})$ such that $d(\cdot,p_{n_{k}})$ is smooth on $O$.

It is easy to check that there is a uniform positive lower bound to
$d(\cdot,p_{n_{k}})$ on $O$, called $l$. Then due to the fact in Lemma 2.1,
the lower bound of the Hessian of distance function $d(\cdot,p_{n_{k}})$ are

$\displaystyle\mathbf{Hess}\,d(\cdot,p_{n_{k}})\geqslant\left\\{\begin{array}[]{ll}-f_{c}(l)I,&c>0,\\\
-l^{-1}I,&c=0,\\\ -\sqrt{-c}\pi^{-1}I,&c<0,\end{array}\right.$

which are obviously independent on $n_{k}$. In addition, for each $n_{k}$, we
likewise define the upper support function of $u_{n_{k}}$ at $p_{n_{k}}$ on
$O$ as

$\displaystyle
u_{n_{k},x}(z):=\phi(y_{x}^{n_{k}})-d(z,p_{n_{k}})-d(p_{n},y_{x}^{n_{k}}).$

Then the fact $\\{u_{n_{k}}\\}$ is equi-Lipschitz and equi-semiconcave on
$\bar{O}$ is a direct consequence of [3, Lemma 14.20] and Lemma 3.5. ∎

######  Theorem 3.16.

Under the assumptions of Lemma 3.15, $u_{n}\rightrightarrows u_{\phi}$ on any
compact set of $I^{-}[\Gamma]$, which means (2.10) depends continuously on the
initial function defined on $\Gamma$.

###### Proof.

Invoking Arzela-Ascoli theorem, we can get a subsequence of $\\{u_{n_{k}}\\}$
on $\bar{O}$ metioned in proof of Lemma 3.15 which uniformly converges to some
$u$ on any compact subset $\bar{O}$. Lemma 2.7 claims that $u$ is also
semiconcave with the same constant, whence it is also Lipschitz. Without loss
of generality, this subsequence is still denoted by $\\{u_{n_{k}}\\}$ for the
sake of brevity. Also by Lemma 2.7, for any $y$, the differentiable point of
$u$, there exists $\\{y_{k}\\}_{k\geqslant 1}$ such that $y_{k}\to y$ and
$u_{n_{k}}$ is differentiable at $y_{k}$. In this case, Lemma 2.7 implies
$\nabla u_{n_{k}}(y_{k})\to\nabla u(y)$. Then by the continuity of Lorentz
metric $g$,

$\displaystyle g(\nabla u(x),\nabla u(x))=\lim_{k\to\infty}g(\nabla
u_{n_{k}}(y_{k}),\nabla u_{n_{k}}(y_{k}))=-1.$ (3.16)

Similar to the proof of Theorem 3.9, we can show that $u$ is a viscosity
solution to (2.10) from (3.16).

For any fixed $x\in\Gamma$, we choose a sequence $\\{x_{m}\\}_{m\geqslant
1}\subset I^{-}[\Gamma]$ with $x_{m}\rightarrow x$. Since
$u_{n_{k}}\rightrightarrows u$ and $\phi_{n_{k}}\rightrightarrows\phi$, then
by the continuous extension of $u$ to $\Gamma$,

$\displaystyle
u(x)=\lim_{m\rightarrow\infty}u(x_{m})=\lim_{m\rightarrow\infty}\lim_{k\rightarrow\infty}u_{n_{k}}(x_{m})=\lim_{k\rightarrow\infty}u_{n_{k}}(x)=\lim_{k\rightarrow\infty}\phi_{n_{k}}(x)=\phi(x).$

Combining with the arbitrariness of $x\in\Gamma$, $u|_{\Gamma}=\phi$. Due to
the uniqueness of solution to (2.10), $u=u_{\phi}$. Note that the argument
above does not depend on some fixed $O$ and all convergent subsequences of
$\\{u_{n}\\}$ will converge to $u_{\phi}$, and it yields
$u_{n}\rightrightarrows u_{\phi}$ on any compact subset of $I^{-}[\Gamma]$. ∎

######  Remark 3.17.

All of results on well-posedness in this paper still hold true, when we
consider instead the following Cauchy type problem

$\displaystyle\left\\{\begin{array}[]{ll}g(\nabla u(x),\nabla u(x))=-1,&x\in
I^{+}[\Gamma],\\\ u(x)=\phi(x),&x\in\Gamma.\end{array}\right.$

The only main difference is that the unique solution to the equation above is
given by

$\displaystyle u_{\phi}(x):=-\sup_{y\in
J_{\Gamma}^{-}(x)}\\{\phi(y)+d(y,x)\\},\,\,x\in I^{+}[\Gamma],$

the local semiconcavity of which ensures all the proofs go through with slight
modification.

Acknowledgements. Xiaojun Cui and Siyao Zhu are supported by the National
Natural Science Foundation of China (Grant No. 12171234).

## References

  * [1] Luis J. Alías, Ana Hurtado, and Vicente Palmer.  Geometric analysis of Lorentzian distance function on spacelike hypersurfaces.  Transactions of the American Mathematical Society, 362(10):5083–5106, 2010.
  * [2] Lars Andersson, Gregory J Galloway, and Ralph Howard.  The cosmological time function.  Classical and Quantum Gravity, 15(2):309–322, February 1998.
  * [3] John K. Beem and Paul E. Ehrlich.  Global Lorentzian geometry.  Number 67 in Monographs and textbooks in Pure and applied mathematics. M. Dekker, New York, 1996.
  * [4] Patrick Bernard and Stefan Suhr.  Lyapounov Functions of Closed Cone Fields: From Conley Theory to Time Functions.  Communications in Mathematical Physics, 359(2):467–498, 2018.
  * [5] Piermarco Cannarsa and Wei Cheng.  Generalized characteristics and Lax-Oleinik operators: global theory.  Calc. Var. Partial Differential Equations, 56(5):Art. 125, 31, 2017\.
  * [6] Piermarco Cannarsa, Wei Cheng, Marco Mazzola, and Kaizhi Wang.  Global generalized characteristics for the Dirichlet problem for Hamilton-Jacobi equations at a supercritical energy level.  SIAM J. Math. Anal., 51(5):4213–4244, 2019.
  * [7] Piermarco Cannarsa and Carlo Sinestrari.  Semiconcave functions, Hamilton-Jacobi equations, and optimal control, volume 58 of Progress in Nonlinear Differential Equations and their Applications.  Birkhäuser Boston, Inc., Boston, MA, 2004.
  * [8] Piermarco Cannarsa and Yifeng Yu.  Singular dynamics for semiconcave functions.  J. Eur. Math. Soc. (JEMS), 11(5):999–1024, 2009.
  * [9] Michael G. Crandall, Lawrence C. Evans, and Pierre-Louis Lions.  Some properties of viscosity solutions of Hamilton-Jacobi equations.  Trans. Amer. Math. Soc., 282(2):487–502, 1984.
  * [10] Michael G. Crandall and Pierre-Louis Lions.  Viscosity solutions of Hamilton-Jacobi equations.  Trans. Amer. Math. Soc., 277(1):1–42, 1983.
  * [11] Xiaojun Cui.  Viscosity solutions, ends and ideal boundaries.  Illinois Journal of Mathematics, 60(2), 2016.
  * [12] Xiaojun Cui and Liang Jin.  The negative of regular cosmological time function is a viscosity solution.  Journal of Mathematical Physics, 55(10):102705, 2014.
  * [13] Avron Douglis.  The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data.  Comm. Pure Appl. Math., 14:267–284, 1961.
  * [14] Albert Fathi.  Weak KAM theorem in Lagrangian dynamics.  Cambridge University Press, Cambridge (to appear).
  * [15] Albert Fathi.  Solutions KAM faibles conjuguées et barrières de Peierls.  C. R. Acad. Sci. Paris Sér. I Math., 325(6):649–652, 1997.
  * [16] Albert Fathi.  Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens.  C. R. Acad. Sci. Paris Sér. I Math., 324(9):1043–1046, 1997\.
  * [17] Albert Fathi.  Orbites hétéroclines et ensemble de Peierls.  C. R. Acad. Sci. Paris Sér. I Math., 326(10):1213–1216, 1998\.
  * [18] Albert Fathi.  Sur la convergence du semi-groupe de Lax-Oleinik.  C. R. Acad. Sci. Paris Sér. I Math., 327(3):267–270, 1998.
  * [19] Albert Fathi.  Weak KAM from a PDE point of view: viscosity solutions of the Hamilton-Jacobi equation and Aubry set.  Proc. Roy. Soc. Edinburgh Sect. A, 142(6):1193–1236, 2012.
  * [20] Albert Fathi.  Viscosity solutions of the Hamilton-Jacobi equation on a non-compact manifold.  preprint, 2020.
  * [21] Jose Luis. Flores, J. Herrera, and M. Sánchez.  Gromov, Cauchy and causal boundaries for Riemannian, Finslerian and Lorentzian manifolds.  Memoirs of the American Mathematical Society, 226(1064):1, 2013\.
  * [22] Liang Jin and Xiaojun Cui.  Global viscosity solutions for eikonal equations on class A Lorentzian 2-tori.  Geometriae Dedicata, 193(1):155–192, 2018.
  * [23] S. N. Kružkov.  Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. Statement of the problems; existence, uniqueness and stability theorems; certain properties of the solutions.  Mat. Sb. (N.S.), 98(140)(3(11)):450–493, 496, 1975.
  * [24] Nikolay Vladimirovich Krylov.  Nonlinear elliptic and parabolic equations of the second order, volume 7 of Mathematics and its Applications (Soviet Series).  D. Reidel Publishing Co., Dordrecht, 1987.  Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ].
  * [25] Ricardo Mañé.  On the minimizing measures of Lagrangian dynamical systems.  Nonlinearity, 5(3):623–638, 1992.
  * [26] John N. Mather.  Action minimizing invariant measures for positive definite Lagrangian systems.  Math. Z., 207(2):169–207, 1991.
  * [27] John N. Mather.  Variational construction of connecting orbits.  Ann. Inst. Fourier (Grenoble), 43(5):1349–1386, 1993.
  * [28] Wu H. Sachs, R.K.  General relativity and cosmology.  Bull. Am. Math. Soc., 1977.
  * [29] Xiaojun Cui Siyao Zhu, Hongguang Wu.  Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times.  preprint, arXiv:2309.15006, 2023.
  * [30] Stefan Suhr.  Aubry-Mather theory and Lipschitz continuity of the time separation.  preprint, arXiv:1104.3849, 2011.
  * [31] Stefan Suhr.  Aubry-Mather theory for Lorentzian manifolds.  Journal of Fixed Point Theory and Applications, 21(2):71, 2019.
  * [32] Cédric Villani.  Optimal transport, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 2009.  Old and new.