\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 120, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/120\hfil A comparison principle]
{A comparison principle for singular parabolic  
equations in the Heisenberg group}

\author[P. Ochoa \hfil EJDE-2015/120\hfilneg]
{Pablo Ochoa}

\address{Pablo Ochoa \newline
Facultad de Ciencias Exactas y naturales,
Universidad Nacional de Cuyo,
Mendoza 5500, Argentina}
\email{ochopablo@gmail.com}

\thanks{Submitted March 24, 2015. Published April 30, 2015.}
\subjclass[2010]{35R03, 35R03}
\keywords{Partial differential equations on the Heisenberg group;
\hfill\break\indent viscosity solutions}

\begin{abstract}
 In this work, we prove a  comparison principle for singular parabolic
 equations with boundary conditions in the context of the Heisenberg group.
 In particular, this result applies to interesting equations, such as
 the parabolic infinite Laplacian, the mean curvature flow equation and
 more general homogeneous diffusions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

The notion of viscosity solution was firstly introduced by  Crandall and Lions
 \cite{CL} in the context of scalar nonlinear first order equations.
This concept was  related to some previous work by Evans \cite{E}.
 In general terms, the definition of viscosity solutions for parabolic
problems may be motivated as follows:
consider a $\mathcal{C}^{2}$-regular function
$u: \Omega \times (0,T) \to \mathbb{R}$, where $\Omega \subset \mathbb{R}^{n}$
is an open set and $T> 0$ is given. Suppose that $u$ solves the
differential inequality
\begin{equation}\label{in}
u_t(z)+F(z, u(z),\nabla u(z), \nabla^{2}u(z)) \leq 0
\end{equation}
for all $z = (t,p)\in \Omega \times (0,T)$.
 Here, $F: [0, T]\times \overline{\Omega} \times \mathbb{R}\times \mathbb{R}^{n}
\times S^{n}(\mathbb{R}) \to \mathbb{R}$ is a given function, which is assumed
to be degenerate elliptic:
$$
F(t,p,r, \eta, \mathcal{X}) \leq F(t,p,r,\eta,\mathcal{Y}) \quad
\text{whenever }\mathcal{Y}\leq \mathcal{X},
$$
so that
\begin{equation}\label{eq}
u_t(z)+F(z, u(z),\nabla u(z), \nabla^{2}u(z)) = 0
\end{equation}
is a parabolic equation.
Suppose now that $\varphi$ is a smooth function defined in
$\Omega \times (0,T)$ so that the difference $u-\varphi$ attains a
maximum at a point $\hat{z} \in \Omega \times (0,T)$. Then, it follows that
$$
u_t(\hat{z}) = \varphi_t(\hat{z}), \quad \nabla u(\hat{z})
= \nabla\varphi(\hat{z}), \quad \nabla^{2}u(\hat{z})
 \leq \nabla^{2}\varphi(\hat{z}).
$$
From the degenerate ellipticity of $F$, we derive
\begin{equation}\label{motivation}
\begin{split}
&\varphi_t(\hat{z}) + F(\hat{z}, u(\hat{z}), \nabla \varphi(\hat{z}),
\nabla^{2}\varphi(\hat{z})) \\
&\leq  u_t(\hat{z}) + F(\hat{z}, u(\hat{z}), \nabla u(\hat{z}),
\nabla^{2}u(\hat{z})) \leq 0.
\end{split}
\end{equation}
Observe that the extremes of these inequalities do not depend on the
derivative of $u$. Hence, this suggests to define an arbitrary
function $u$ to be a generalized or weak subsolution of \eqref{in}
in $\Omega \times (0,T)$ if for each $\hat{z} \in \Omega \times (0,T)$,
a test function $\varphi$ that touches  $u$ from above at $\hat{z}$ always
satisfies
$$
\varphi_t(\hat{z}) + F(\hat{z}, \varphi(\hat{z}), \nabla \varphi(\hat{z}),
\nabla^{2}\varphi(\hat{z})) \leq 0.
$$
The notion of viscosity supersolution is defined analogously.
Finally, a viscosity solution of \eqref{eq} is a subsolution and a
supersolution. The primary virtues of this theory are that it allows
 merely continuous functions to be solutions of fully nonlinear equations
of second order, that it provides very general existence and uniqueness
theorems and that it yields precise formulations of general boundary conditions.
Moreover, it has a great flexibility in passing to limits in various settings.
For a more complete treatment of viscosity solutions in the Euclidean
framework see \cite{BC,BS,CL,CEL,CIL} and the references therein.
For extension of the definition to singular equations, see for instance
the book \cite{G}.

In this work, we are concerned with the development of a comparison principle
for a large class of boundary value problems, in  the Heisenberg group
$\mathcal{H}$, of the form
\begin{equation}\label{boundaryVP}
\begin{gathered}
u_t +F(t,p, u,\nabla_\mathcal{H}u, (\nabla_\mathcal{H}^{2}u)^{*}) = 0,
 \quad \text{in } (0, T)  \times \Omega  \quad(E)\\
u(t,p) = g(t,p)  \quad  p \in \partial \Omega, \; t \in [0,T) \quad (BC)\\
 u(0, p)=h(p)  \quad p\in \overline{\Omega}\quad \quad(IC)
\end{gathered}
\end{equation}
Here $\Omega \subset \mathcal{H}$ is open and bounded,  and
$F = F(t,p, r,\eta,\mathcal{X})$ is assumed to be possibly singular at
$\eta = 0$ (extra assumptions on $F$ will be provided in Section
\ref{assumptions}). See Section \ref{preliminaries} for definitions of
the horizontal gradient $\nabla_\mathcal{H}u$ and the symmetrized Hessian
matrix $(\nabla_\mathcal{H}^{2}u)^{*}$ in $\mathcal{H}$.
A general comparison principle for parabolic equations in the Heisenberg
group for  everywhere  continuous $F$ was introduced in \cite{Bi}.
(See \cite{CIL} for the related result in the Euclidean context).
 With respect to singular parabolic equations, we can quote the particular
case of the horizontal mean curvature flow equation treated in \cite{FLM1},
for which a comparison principle for axisymmetric surfaces was proven.
(See also \cite{G,CGG} for the Euclidean treatment of singular parabolic
equations). In this work, we prove that under some extra assumptions on $F$,
a comparison principle for the boundary value problem \eqref{boundaryVP}
holds for  solutions $u$ which are symmetric with respect to some class of
surfaces $p_3=G(p_1,p_2)$  (see \eqref{assumption-on-g}) in the sense that
$$
u(t,p_1,p_2,p_3)=u(t, \hat{p}_1, \hat{p}_2, p_3)
\quad\text{ whenever }G(p_1,p_2) = G(\hat{p}_1, \hat{p}_2).
$$
We would like to point out that some of the arguments used in our proof
 of the comparison principle are similar to those from the works
\cite{FLM1,Bi} and the seminal paper \cite{CIL}, adapted to our framework 
and generality.

The organization of the paper is as follows. In Section \ref{preliminaries},
we provide a brief introduction to the Heisenberg group. In the next
Section \ref{comparison-section}, we discuss the parabolic boundary problem
we intend to study, the notions of viscosity solutions  and we provide
the main assumptions to prove the comparison principle, which is formulated
and proven in Section \ref{comparisonprinciple8}.
We close the paper with Section \ref{examples section}, where we give
examples of applications.


\section{Preliminaries on the Heisenberg group}\label{preliminaries}

In this section, we introduce the definition of the Heisenberg group
$\mathcal{H}$ together with its differential and metric structures.
The notion of parabolic jet on $\mathcal{H}$ and its characterization
in terms of smooth functions are also explained.

\subsection{The symmetric three dimensional Heisenberg group}\label{sec}

We consider the first order Heisenberg group
$\mathcal{H} = (\mathbb{R}^{3}, \cdot)$, where $\cdot$ is the group
operation defined by
$$
p \cdot q = \Big(p_1 + q_1, p_2 + q_2, p_3 + q_3
+ \frac{1}{2}(p_1q_2 - p_2q_1)\Big),
$$
for all $p =(p_1, p_2, p_3)$, $q = (q_1, q_2, q_3) \in \mathbb{R}^{3}$.
 The group $\mathcal{H}$ is a Lie group with Lie algebra $\mathfrak{h}$
generated by the basis
\begin{equation}
\begin{split}
X_1 &= \frac{\partial}{\partial p_1}
 -\frac{p_2}{2}\frac{\partial}{\partial p_3}\\
X_2 &= \frac{\partial}{\partial p_2}
 +\frac{p_1}{2}\frac{\partial}{\partial p_3}\\
X_3 &= \frac{\partial}{\partial p_3},
\end{split}
\end{equation}
where $p = (p_1, p_2, p_3) \in \mathbb{R}^{3}$.
 Observe that the following Heisenberg uncertainty principle holds:
$$
[X_1,X_2] = X_3.
$$
The exponential mapping takes the vector $p_1X_1 + p_2X_2 + p_3X_3$
in the Lie algebra $\mathfrak{h}$ to the point $p$ in the Lie group
$\mathcal{H}$. This allows us to identify vectors in $\mathfrak{h}$ with
points in $\mathcal{H}$.

On the Heisenberg group, an important role is played by the distribution
$\mathcal{H}^{h}$ generated by the linearly independent vector fields
$X_1$ and $X_2$, called the horizontal distribution. Thus, this space at $p$,
denoted by $\mathcal{H}_p^{h}$, is  a two dimensional linear space generated
by the vectors $X_1(p)$ and $X_2(p)$. As
$[X_1,X_2] = X_3 \notin \mathcal{H}^{h}$, the horizontal distribution is not
involutive, and hence, by Frobenius theorem, it is not integrable;
 that is, there is no surface locally tangent to $\mathcal{H}^{h}$.

\subsection{Carnot-Carath\'eodory distance}
A curve $c(s) = (c_1(s),c_2(s),c_3(s)) $ is horizontal if
$c'(s) \in \mathcal{H}^{h}_{c(s)}$. Moreover, by Chow's theorem any two
points $p$ and $q$ in $\mathcal{H}$ can be joined by a smooth horizontal curve.
Hence, the set
$$
S_{p,q}=\{c: c(0)=p,c(1)=q, \,c \text{ is horizontal}\} \neq \emptyset.
$$
The length of a horizontal curve $c$ is given by
$$
l(c)= \int_0^{1}\sqrt{g(c'(s),c'(s))}ds,
$$
where $g$ is the subRiemannian metric. The Carnot-Carath\'eodory distance
is defined as $d_C:\mathbb{R}^{3}\times \mathbb{R}^{3} \to [0,\infty)$,
$$
d_C(p,q) = \inf \{l(c): c \in S_{p,q}\}.
$$
One may verify that $d_C$ satisfies the distance axioms and that it is complete.
This metric induces a homogeneous norm on $\mathcal{H}$, denoted $|\cdot |$, by
$$
|p|= d_C(0,p),
$$
and we have the estimate
\begin{equation}\label{est-carnot-distance}
|p| \sim \|(p_1,p_2)\|_E+ |p_3|^{1/2}.
\end{equation}
Here, $\|\cdot\|_E$ stands for the Euclidean norm in $\mathbb{R}^{n}$.
This estimate leads to define the left-invariant Heisenberg gauge
$|\cdot|_\mathcal{H}$ that is compatible to the Carnot-Carath\'eodory distance,
 and is defined as follows:
$$
|p|_\mathcal{H}= \big[(p_1^{2}+p_2^{2})^{2}+16p_3^{2}\big]^{1/4}.
$$
For the rest of this article, we shall consider all topological notions
with respect to the metric space $(\mathcal{H}, d_{C})$.
Also, for any $p \in \mathcal{H}$ and $\delta > 0$, we write
$$
B_\mathcal{H}(p, \delta) = \big\{q \in \mathcal{H}: |q^{-1}\cdot p|
< \delta\big\},
$$
to denote the ball in the Heisenberg group with center at $p$ and
radius $\delta$.

\subsection{Analysis on $\mathcal{H}$}
The left translation
$L_p:\mathcal{H}\to \mathcal{H}$ is defined by
$$
L_p(q) =p\cdot q.
$$
Observe that $L_p$ is an affine map. Indeed:
$$L_p(q)= \begin{pmatrix}
p_1 \\
p_2 \\
p_3
\end{pmatrix} + \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
-p_2/2 & p_1/2  & 1
\end{pmatrix}\begin{pmatrix}
q_1 \\
q_2 \\
q_3
\end{pmatrix},
$$
and the determinant of the matrix on the right-hand side is $1$.
It follows then that the left-invariant Haar measure of $\mathcal{H}$
is the Lebesgue measure $\mathcal{L}$ of $\mathbb{R}^{3}$ (which is in
fact also right invariant).

For a smooth function $u: \mathcal{H} \to \mathbb{R}$
the horizontal gradient $\nabla_\mathcal{H}$ of $u$ at a point $p$
is the projection of the gradient of $u$ at $p$ onto the horizontal space
$ \mathcal{H}^{h}_p$,
$$
\nabla_\mathcal{H} u = (X_1 u)X_1 + (X_2 u)X_2.
$$
The symmetrized horizontal second derivative matrix, denoted by
$(\nabla_\mathcal{H}^{2}u)^{*}$ is given by
$$
(\nabla_\mathcal{H}^{2}u)^{*} = \begin{pmatrix}
X_1^{2}u & •\frac{1}{2}(X_1X_2u + X_2X_1 u) \\
\frac{1}{2}(X_1X_2u + X_2X_1 u) & •X_2^{2}u
\end{pmatrix}.
$$
With this notation and the estimate \eqref{est-carnot-distance},
the Taylor expansion for a smooth $u$ around $p_0$ reads as
$$
u(p_0)=u(p) + \langle \nabla u(p_0), p_0^{-1}\cdot p \rangle
+ \frac{1}{2}\langle (\nabla_\mathcal{H}^{2}u(p_0))^{*}p_0^{-1}\
cdot p, p_0^{-1}\cdot p\rangle + o(|p_0^{-1}\cdot p|^{2}
$$
For more about the Heisenberg group, the interested reader is referred
to \cite{Bi,CD,M,Ov}, and the references therein.

\subsection{Parabolic subelliptic jets}

 We start by defining the parabolic superjets of a function $u$ at a point
 $(t_0, p_0) \in (0, \infty) \times \mathcal{H} $, denoted by
$P^{2, +}u(t_0, p_0)$, as the set of all triples
$(\tau, \eta, \mathcal{X}) \in \mathbb{R} \times \mathbb{R}^{3}
\times S^{2}(\mathbb{R})$ that satisfies
\begin{equation}
u(t, p) \leq  u(t_0, p_0) + \tau(t - t_0) + \langle \eta, p^{-1}_0\cdot p\rangle
+ \frac{1}{2}\langle \mathcal{X}h, h\rangle + o(|t - t_0|
+ |p_0^{-1} \cdot p|_\mathcal{H}^{2}),
\end{equation}
as $(t, p) \to (t_0, p_0)$. Here, $h$ denotes the horizontal projection of
$p_0^{-1}\cdot p$.  We define the parabolic subject $P^{2, -}u(t_0, p_0)$ by
$$
P^{2, -}u(t_0, p_0) = -P^{2, +}(-u)(t_0, p_0).
$$
As in the subelliptic case (see \cite{C} for the Euclidean case, \cite{Bi}
for the subelliptic case), it was shown in \cite{Bi2} that
$$
P^{2, +}u(t_0, p_0)= \big\{(\varphi_t(t_0, p_0), \nabla \varphi(t_0, p_0),
(\nabla_\mathcal{H}^{2}\varphi(t_0, p_0))^{*}):
\varphi \in \mathcal{A}u(t_0, p_0)\big\},
$$
where
$$
\mathcal{A}u(t_0, p_0) = \{\varphi \in \mathcal{C}^{2}(\mathcal{H} \times (0, T)):
u - \varphi \text{ has a strict local maximum at }(t_0, p_0)\}.
$$
Similarly, one has
$$
P^{2, -}u(t_0, p_0) = \big\{(\varphi_t(t_0, p_0), \nabla \varphi(t_0, p_0),
(\nabla_\mathcal{H}^{2}\varphi(t_0, p_0))^{*}): \varphi \in
\mathcal{B}u(t_0, p_0)\big\},
$$
where
$$
\mathcal{B}u(t_0, p_0) = \{\varphi \in \mathcal{C}^{2}(\mathcal{H}
\times (0, T)): u - \varphi \text{ has a strict local minimum at }(t_0, p_0)\}.
$$

We also define the closure of second order superjets and subjets.

\begin{definition} \label{def2.1} \rm
The closure of the second order superjet of an upper-semicontin\-uous
function $u$ at a point $(t_0, p_0)$, denoted by
$\overline{P}^{2,+}u(t_0, p_0)$, is defined as the set of
$(\tau,\eta, \mathcal{X}) \in \mathbb{R} \times \mathbb{R}^{3}
 \times S^{2}(\mathbb{R})$, such that there exist sequences of points
$(t_n, p_n)$ and  $(\tau_n, \eta_n, \mathcal{X}_n) \in P^{2, +}u(t_n,p_n)$
such that
$$
(t_n, p_n,  u(t_n, p_n),\tau_n,  \eta_n, \mathcal{X}_n) \to (t_0, p_0,u(t_0,p_0),
 \tau, \eta, \mathcal{X}), \quad\text{as }n\to \infty.
$$
Similarly, the closure of the second order subjet of a lower-semicontinuous
function $u$ at a point $(t_0,p_0)$, denoted by
$\overline{P}^{2,-}u(t_0,p_0)$, is defined as the set of
$(\tau, \eta, \mathcal{X}) \in \mathbb{R}\times \mathbb{R}^{3}
 \times S^{2}(\mathbb{R})$,
such that there exist sequences of points $(t_n,p_n)$ and
$(\tau_n, \eta_n, \mathcal{X}_n) \in P^{2,-}u(t_n,p_n)$ such that
$$
(t_n,p_n,  u(t_n,p_n),\tau_n,  \eta_n, \mathcal{X}_n) \to (t_0, p_0,u(t_0,p_0),
\tau, \eta, \mathcal{X}), \,\,as\,\,n\to \infty.
$$
\end{definition}

\section{General setting}\label{comparison-section}

\subsection{The parabolic problem under study and the notions of
viscosity solutions} \label{assumptions}

Let $\Omega$ be an open and bounded domain in $\mathcal{H}$.
We consider the following class of problems:
\begin{equation}\label{boundaryVP2}
\begin{gathered}
u_t +F(t,p, u,\nabla_\mathcal{H}u, (\nabla_\mathcal{H}^{2}u)^{*}) = 0,
 \quad\text{in } (0, T)  \times \Omega\quad (E)\\
 u(t,p) = g(t,p) \quad  p \in \partial \Omega, \; t \in [0,T) \quad(BC)\\
 u(0, p)=h(p) \quad  p\in \overline{\Omega}\quad  (IC)
\end{gathered}
\end{equation}
Here $g \in \mathcal{C}( [0,T)\times \overline{\Omega})$,
$h \in \mathcal{C}(\overline{\Omega})$ and
$F: [0,T] \times \overline{\Omega}\times \mathbb{R}\times
(\mathbb{R}^{2}\setminus \{0\}) \times S^{2}(\mathbb{R}) \to \mathbb{R}$
is assumed to satisfy the following properties:
\begin{enumerate}
\item $F$ is continuous in $[0,T] \times \overline{\Omega}\times
\mathbb{R}\times (\mathbb{R}^{2}\setminus \{0\}) \times S^{2}(\mathbb{R})$,
and there is a modulus of continuity $\omega$ so that
\begin{equation}\label{modulus}
|F(t,p, r, \eta, \mathcal{X}) - F(s, q, r, \eta, \mathcal{X})|
\leq \omega\big( |t-s|+ d_C(p,q)\big),
\end{equation}
for all $r \in \mathbb{R}$, $\eta \in \mathbb{R}^{2}\setminus \{0\}$ and all
$\mathcal{X}\in S^{2}(\mathbb{R})$. (Observe that the modulus $\omega$
is the same for all $r, \eta$ and $\mathcal{X}$.)
\item $F$ is proper, that is,
\begin{equation*}
(r, \mathcal{X}) \to F(t, p,r,\eta,\mathcal{X})
\end{equation*}
is increasing in $r \in \mathbb{R}$ and decreasing in
$\mathcal{X} \in S^{2}(\mathbb{R})$.

\item $F_{*}(t,p,r,0, \mathcal{O}) = F^{*}(t,p,r,0,\mathcal{O})=0$ for all
$t,p, r\in [0,T]\times \overline{\Omega}\times \mathbb{R}$. $F_*$
and $F^{*}$ are locally bounded in the set
$[0,T] \times \overline{\Omega}\times \mathbb{R}\times \mathbb{R}^{2}
\times S^{2}(\mathbb{R})$.

\item $F(t,p,r,\eta_\epsilon, \mathcal{Y}_\epsilon)
-F(t,p, r,\eta_\epsilon,\mathcal{X}_\epsilon) \leq o(1)$,
 uniformly in $t,p$, $r$, and $\eta_\epsilon$ uniformly bounded, and all
$\mathcal{X}_\epsilon, \mathcal{Y}_\epsilon \in S^{2}(\mathbb{R})$ so that
$$
\mathcal{X}_\epsilon - \mathcal{Y}_\epsilon \leq o(1)I,
$$
where $I$ is the identity matrix.
\end{enumerate}

\begin{remark} \label{rmk3.1} \rm
We may replace assumptions (1) and (4) by the following stronger assumption:
there exist constants $K, L > 0$ such that
$$
F(t,p,r,\eta,\mathcal{Y})-F(s,q,r',\beta,\mathcal{X})
\leq K\big( d_C(p,q)+|s-t| + |r-r'|+ |\beta-\eta|\big) + L\sigma,
$$
for all $p,q \in \overline{\Omega}$, $s,t \in [0,T]$, $r,r' \in \mathbb{R}$,
$\eta, \beta \in \mathbb{R}^{2}\setminus \{0\}$ and all
$\mathcal{X}, \mathcal{Y}\in S^{2}(\mathbb{R})$ so that
$$
\mathcal{X}\leq \mathcal{Y}+\sigma I.
$$
\end{remark}

Next, we introduce the definition of viscosity
solution to the singular parabolic equation (E) in the context
of the Heisenberg group.

\begin{definition} \label{visc.parabolic} \rm
An upper (respectively, lower) semicontinuous function
$u: (0, T)  \times \Omega  \to \mathbb{R} \cup \{\pm \infty\}$
is a viscosity subsolution (resp. supersolution) in
$ (0, T)  \times \Omega$ to (E) if for all
$(t_0, p_0) \in  (0, T)  \times \Omega$ and all smooth
$\varphi \in \mathcal{A}u(t_0, p_0)$
(resp. $\varphi \in \mathcal{B}u(t_0,p_0)$) there holds
\begin{gather*}
\varphi_t(t_0, p_0) +  F_{*}(t_0, p_0, u(t_0,p_0),\nabla_\mathcal{H}
\varphi(t_0, p_0), (\nabla_\mathcal{H}^{2}\varphi(t_0, p_0))^{*}) \leq 0.\\
(\text{resp. } \varphi_t(t_0, p_0) +  F^{*}(t_0, p_0, u(t_0,p_0),
\nabla_\mathcal{H} \varphi( t_0, p_0), (\nabla_\mathcal{H}^{2}
\varphi(t_0, p_0))^{*}) \geq 0.)
\end{gather*}
A continuous function $u$ is a viscosity solution if it is a viscosity
subsolution and a viscosity supersolution.
\end{definition}

It is also possible to deal with the singularity of $F$ at
$\eta = 0 \in \mathbb{R}^{2}$ by restricting the set of test functions
to the  set
$$
\mathcal{A}_0 = \{\varphi \in \mathcal{C}^{\infty}( (0, \infty)
 \times\mathcal{H}): \nabla_\mathcal{H}\varphi(t, p) = 0
\text{ implies } (\nabla_\mathcal{H}^{2}\varphi(t, p))^{*} = 0\}.
$$
This is the content of Definition \ref{second-def}.
Another way is to use parabolic jets as in Definition \ref{third-def}
below. We shall see in Lemma \ref{equivalence} that these definitions
are equivalent.

\begin{definition} \label{second-def} \rm
An upper (respectively, lower) semicontinuous function
$u:  (0, T)  \times \Omega\to \mathbb{R} \cup \{\pm \infty\}$
is a subsolution (resp. supersolution) to (E) if:
\begin{enumerate}
\item[(i)] $u < \infty$ (resp. $u > - \infty$) in $ (0, T)  \times \Omega$;

\item[(ii)] For any smooth function $\varphi$ and
$(t_0, p_0) \in  (0, T)  \times \Omega $ such that
$\varphi \in \mathcal{A}u(t_0, p_0)$
(resp. $\varphi \in \mathcal{B}u(t_0,p_0)$), the function $\varphi$ satisfies
\begin{gather*}
\varphi_t + F(t_0,p_0,u(t_0,p_0),\nabla_\mathcal{H}
\varphi, (\nabla_\mathcal{H}^{2}\varphi)^{*}) \leq 0, \text{ at }(t_0, p_0),\\
(\text{resp. } \varphi_t + F(t_0,p_0,u(t_0,p_0),\nabla_\mathcal{H}\varphi,
(\nabla_\mathcal{H}^{2}\varphi)^{*}) \geq 0, \text{ at }(t_0, p_0).)
\end{gather*}
if $\nabla_\mathcal{H}\varphi(t_0, p_0) \neq 0$, and
$ \varphi_t(t_0, p_0) \leq 0$ 
(respectively $\varphi_t(t_0, p_0) \geq 0$.)
when $\nabla_\mathcal{H}\varphi(t_0, p_0) = 0$ and
$(\nabla_\mathcal{H}^{2}\varphi(t_0, p_0))^{*} = 0$.
\end{enumerate}
\end{definition}


\begin{definition} \label{third-def} \rm
An upper (respectively, lower) semicontinuous function
$u:  (0, T)  \times \Omega \to \mathbb{R} \cup \{\pm \infty\}$
is a subsolution (resp. supersolution) to (E) if:
\begin{enumerate}
\item[(i)] $u < \infty$ (resp. $u > - \infty$) in $ (0, T)  \times \Omega$;

\item[(ii)] For any $(t_0, p_0) \in  (0, T)  \times \Omega$ and any
$(\tau,\eta,\mathcal{X}) \in \overline{P}^{2, +}u(t_0,p_0)$
(resp. $(\tau,\eta,\mathcal{X}) \in \overline{P}^{2, -}u(t_0,p_0)$), we have
\begin{gather*}
\tau + F_{*}(t_0,p_0,u(t_0,p_0),\eta,\mathcal{X}) \leq 0 \\
(\text{resp. }\tau + F^{*}(t_0,p_0,u(t_0,p_0),\eta,\mathcal{X})\geq0.
\end{gather*}
\end{enumerate}
\end{definition}



It is straightforward to check that Definition \ref{visc.parabolic}
and Definition \ref{third-def} are equivalent.
Moreover Definition \ref{visc.parabolic} implies Definition
\ref{second-def}. Hence, it is remains to prove the converse.
This is the content of the next result, which is
\cite[Proposition 3.1]{FLM1}.

\begin{lemma}\label{equivalence}
An upper-semicontinuous function $u$ is a subsolution to (E)
in the sense of Definition \ref{visc.parabolic} if and only if
it is a subsolution in the sense of Definition \ref{second-def}.
A similar statement holds for supersolutions.
\end{lemma}

\begin{proof}
The proof is the same as that of \cite[Proposition 3.1]{FLM1}.
We just mention how to treat with the dependence of $F$ on $t$, $p$ and $r$,
 which is not the case considered in \cite{FLM1}. First of all,
it is clear that Definition \ref{visc.parabolic} implies Definition
\ref{second-def}. To prove the converse, assume that $u$ is a
subsolution according to Definition \ref{second-def}.
Let $(\hat{t}, \hat{p} )\in (0, T) \times \Omega$ and let $\varphi$
 a smooth function such that
$$
\max_{(0,T) \times \Omega}(u-\varphi) = (u - \varphi)(\hat{t}, \hat{p}).
$$
As in \cite{FLM1}, we let
$$
\Phi^{\tau}(t,p,q) =u(t,p)-\tau |q^{-1}\cdot p|^{4}-\varphi(t,p).
$$
Then $\Phi(t,p,q) = \limsup^{*}_{\tau \to \infty}\Phi^{\tau}(t,p,q) = -\infty$
when $p\neq q$, and
$\Phi(t,p,p)=\limsup^{*}_{\tau \to \infty}\Phi^{\tau}(t,p,p)
= u(t,p)-\varphi(t,p)$.
By the convergence of maximum points \cite[Lemma 2.2.5]{G}),
there exists a sequence $(t^{\tau},p^{\tau},q^{\tau})$ converging to
$(\hat{t}, \hat{p},\hat{p})$ such that $\Phi^{\tau}$ attains a maximum
at $(t^{\tau},p^{\tau},q^{\tau})$. Moreover
\begin{equation}\label{lim}
\lim_{\tau \to \infty}\Phi^{\tau}(t^{\tau},p^{\tau},q^{\tau})
= \Phi(\hat{t},\hat{p},\hat{p}).
\end{equation}
Hence, since $\varphi(t^{\tau},p^{\tau}) \to \varphi(\hat{t},\hat{p})$
as $\tau \to \infty$, we derive from \eqref{lim} and the upper
semicontinuity of $u$ that
\begin{equation}
\begin{split}
u(\hat{t},\hat{p})
&= \liminf_{\tau \to \infty}\big(u(t^{\tau},p^{\tau})
 -\tau |(q^{\tau})^{-1}\cdot p^{\tau}|^{4}\big) \\
& \leq \liminf_{\tau \to \infty}u(t^{\tau},p^{\tau})\\
&\leq   \limsup_{\tau \to \infty}u(t^{\tau},p^{\tau})
\leq u(\hat{t},\hat{p}).
\end{split}
\end{equation}
Concluding that
\begin{equation}\label{lim u}
\lim_{\tau \to \infty}u(t^{\tau},p^{\tau}) = u(\hat{t},\hat{p}).
\end{equation}
With \eqref{lim u} in mind, we may proceed with the proof as in \cite{FLM1}.
\end{proof}


Finally, we proceed as in \cite{CIL,Bi2} and we introduce the definition
of viscosity solution to the problem \eqref{boundaryVP2}.


\begin{definition} \label{visc.sol} \rm
A subsolution $u(t,p)$ to problem \eqref{boundaryVP2} is a viscosity
subsolution to (E), $u(t,p)\leq g(t,p)$ on $\partial \Omega$,
$t\in [0,T)$, and $u(0,p) \leq h(p)$ in $\overline{\Omega}$.
Supersolutions and solutions are defined in an analogous manner.
\end{definition}

\section{Comparison principle}\label{comparisonprinciple8}

Let $G:\mathbb{R}^{2} \to \mathbb{R}$ be a smooth function verifying
that there is no point $(p_1,p_2) \in \mathbb{R}^{2}\setminus \{0\}$ such that
\begin{equation}\label{assumption-on-g}
p_1 \frac{\partial G}{\partial p_1}(p_1,p_2)
+ p_2\frac{\partial G}{\partial p_2}(p_1,p_2)=0.
\end{equation}
In particular, observe that the Euclidean gradient
$ (\partial G/\partial p_1 (p_1,p_2), \partial G/\partial p_2(p_1,p_2))$
is not zero for all $(p_1,p_2) \neq (0,0)$.
We are interested in a comparison principle for the problem \eqref{boundaryVP2}
in the case of sub or supersolutions $u$ which are symmetric with respect to
the surface $p_3=G(p_1,p_2)$:
$$
u(t, p_1,p_2,p_3)=u(t, \hat{p}_1,\hat{p}_2,p_3), \quad\text{when }
G(p_1,p_2)=G(\hat{p}_1,\hat{p}_2).
$$
This is the content of our main result.


\begin{theorem}\label{thm4.1}
Let $u$ and $v$ be respectively an upper semicontinuous subsolution and
a lower semicontinuous supersolution to \eqref{boundaryVP2}.
Assume that either $u$ or $v$ are symmetric with respect to the surface
$p_3=G(p_1,p_2)$. Then:
$$
u\leq v \quad \text{in }  [0,T)\times\overline{\Omega} .
$$
\end{theorem}

\begin{proof}
Let us assume that $u$ is symmetric with respect to the surface
$p_3=G(p_1,p_2)$. To obtain a contradiction, we assume   that there exists
a point $(\overline{t}, \overline{p}) \in  (0,T) \times\Omega$ so that
$$
(u-v)(\overline{t},\overline{p}) >0.
$$
Then, we are able to find a positive number $\delta >0$ satisfying
$$
u(\overline{t},\overline{p}) - v(\overline{t},\overline{p})
-\frac{\delta}{T-\overline{t}} >0.
$$
Let $(\hat{t}, \hat{p}) \in   [0,T) \times \overline{\Omega}$ so that
\begin{equation}\label{145}
M=u(\hat{t},\hat{p}) -v(\hat{t},\hat{p}) -\frac{\delta}{T-\hat{t}}
= \max_{  [0,T) \times \overline{\Omega}}\Big(u(t,p)-v(t,p)
-\frac{\delta}{T-t}\Big) > 0.
\end{equation}
As usual in the proof of comparison principles, we double the variables
and  proceed with the penalizing process defining the function  $M^{\tau}$ by
$$
M^{\tau}(t,p,s,q) = u(t,p)-v(s,q)-\tau g^{2}(p,q)
-\frac{\tau}{2}(t-s)^{2}-\frac{\delta}{T-t},
$$
where $g(p,q)=|p\cdot q^{-1}|^{4}_\mathcal{H}$.
This is the same penalizing process as in \cite{FLM1}.
We  take maximizers
$(t^{\tau},p^{\tau},s^{\tau},q^{\tau})\in (  [0,T) \times \overline{\Omega})^{2}$
 of $M^{\tau}$. In view of \eqref{145} and the boundedness from above
of the functions $u$ and $-v$, the points $t^{\tau}$  lie in a compact
subset of $[0,T)$ for $\tau$ large. Moreover, the inequality
\begin{equation}\label{inequality}
M^{\tau}(t^{\tau},p^{\tau},s^{\tau},q^{\tau})
\geq M^{\tau}(\hat{t},\hat{p},\hat{t},\hat{p})
\end{equation}
implies that
\begin{equation}\label{same-convergence}
|p^{\tau}\cdot (q^{\tau})^{-1}|_\mathcal{H}\to 0
\quad \text{and}\quad |t^{\tau}-s^{\tau}|\to 0,
\end{equation}
as $\tau \to \infty.$ This fact, together with the compactness of
 the set $\overline{\Omega}$ yield the existence of a point
$(t_0, p_0) \in   [0,T) \times \overline{\Omega}$ such that
$p^{\tau}, q^{\tau} \to p_0$ and $t^{\tau}, s^{\tau} \to t_0$.
It also follows from \eqref{inequality} and the above convergences, that
$$
0 \leq \limsup_{\tau \to \infty} \Big(\tau g^{2}(p^{\tau}, q^{\tau})
+ \frac{\tau}{2}(t^{\tau}-s^{\tau})^{2}\Big) \leq 0.
$$
Concluding that
$$
\tau g^{2}(p^{\tau}, q^{\tau})+ \frac{\tau}{2}(t^{\tau}-s^{\tau})^{2}
\to 0 \quad \text{as }\tau \to \infty.
$$
In addition, the fact $u(0, p)\leq v(0,p)$ implies that $t_0 \neq 0$.
Indeed, if $t_0 =0$, then
$$
0 < M = M^{\tau}(\hat{t},\hat{p},\hat{t},\hat{p})
\leq \lim_{\tau \to \infty}M^{\tau}(t^{\tau},p^{\tau},s^{\tau},q^{\tau})
= u(0,p_0)-v(0,p_0) -\frac{\delta}{T}\leq 0
$$
which is a contradiction. Hence, $t_0 \in (0,T)$. On the other hand,
we also need to check that  $p_0 \in \Omega$. Observe that
\begin{equation}\label{A}
M \leq \lim_{\tau \to \infty}M^{\tau}(t^{\tau},p^{\tau},s^{\tau},q^{\tau})
= u(t_0,p_0)-v(t_0,p_0)-\frac{\delta}{T-t_0}.
\end{equation}
Hence, If $p_0 \in \partial \Omega$,  the property
$u(t,p)\leq v(t,p)$ on $ [0,T) \times\partial \Omega $
says that the right-hand side above is negative, which contradicts
that $M > 0$. Therefore, $p_0$ should be an interior point.
Thus, we may apply the parabolic Euclidean Crandall-Ishii
lemma \cite[Theorem 8.3]{CIL}) to the functions
$$
(t,p) \to u(t,p) -v(s^{\tau}, q^{\tau}) -\tau g^{2}(p,q^{\tau})
-\frac{\tau}{2}(t-s^{\tau})^{2} - \frac{\delta}{T-t}
$$
and
$$
(s,q) \to u(t^{\tau},p^{\tau}) -v(s, q) -\tau g^{2}(p^{\tau},q)
-\frac{\tau}{2}(t^{\tau}-s)^{2} - \frac{\delta}{T-t^{\tau}}
$$
which have, respectively, a maximum at the points $(t^{\tau}, p^{\tau})$
and $(s^{\tau}, q^{\tau})$. To do so, we need to check
\cite[Condition 8.5]{CIL},
namely, that there is an $r > 0$ such that for
every $M > 0$ there is a $C$ such that  for all
$(b, \beta,X) \in P_{Eucl}^{2,+}u(t, p)$:
 if  $|u(p,t)|+ |\beta|+\|X\| \leq M$ and  $|t-t^{\tau}|+\|p-p^{\tau}\|_E < r$
hold, then $b \leq C$, with an analogous statement for $-v$.
Indeed, if this is not true, for all $r>0$, there is an $M>0$ such that
for all $C$ there is $(b, \beta,X)\in P_{Eucl}^{2,+}u(p, t)$ so that
$|u(t,p)|+ |\beta|+\|X\| \leq M$ and $|t-t^{\tau}|+\|p-p^{\tau}\|_E < r$
 but $b> C$. This would imply that
$$
\big(b, (DL_p\beta,DL_pXDL_p^{T})_{2 \times 2}\big) \in P^{2,+}u(t, p)
$$
which contradicts the fact that $u$ is a subsolution for large $C$
in view of the local boundedness of the function $F_*$.
A similar argument applies to $-v$. Hence, we may apply
\cite[Theorem 8.3]{CIL} to obtain matrices
$X^{\tau}, Y^{\tau} \in S^{3}(\mathbb{R})$ such that
\begin{equation}\label{123}
\Big(\frac{\delta}{(T-t^{\tau})^{2}}+ \tau(t^{\tau}-s^{\tau}),
\tau \nabla_p g^{2}(p^{\tau},q^{\tau}), X^{\tau}\Big)
\in \overline{P}_{Eucl}^{2,+}u(t^{\tau}, p^{\tau})
\end{equation}
$$
\Big(\tau(t^{\tau}-s^{\tau}), -\tau \nabla_q g^{2}(p^{\tau},q^{\tau}),
Y^{\tau}\Big) \in \overline{P}_{Eucl}^{2,-}v(s^{\tau}, q^{\tau}),
$$
with the property that
\begin{equation}\label{matrix-inequality}
\begin{pmatrix}
X^{\tau} & 0 \\
0 & -Y^{\tau}\end{pmatrix}
 \leq \tau \big[ \nabla^{2}_{p,q}g^{2}(p^{\tau}, q^{\tau})
+ (\nabla^{2}_{p,q}g^{2}(p^{\tau}, q^{\tau}))^{2}\big].
\end{equation}
It is also clear that
\begin{equation}\label{1234}
\Big(\frac{\delta}{(T-t^{\tau})^{2}}+ \tau(t^{\tau}-s^{\tau}),
\tau DL_{p^{\tau}} \nabla_p g^{2}(p^{\tau},q^{\tau}),
(DL_{p^{\tau}}X^{\tau}DL_{p^{\tau}}^{T})_{2 \times 2}\Big)
\in \overline{P}^{ 2, +}u(t^{\tau}, p^{\tau})
\end{equation}
and
$$
\Big(\tau(t^{\tau}-s^{\tau}), -\tau DL_{q^{\tau}}
\nabla_q g^{2}(p^{\tau},q^{\tau}), (DL_{q^{\tau}}Y^{\tau}DL_{q^{\tau}}^{T}\Big)
 \in \overline{P}^{2,-}v(s^{\tau}, q^{\tau}).
$$
As in \cite{Bi} and \cite{FLM1}, let
\begin{gather*}
w_{p^{\tau}} = (DL_{p^{\tau}})^{T}(w_1,w_2,0)^{T}
= \Big(w_1,w_2, \frac{1}{2}(p_1^{\tau}w_2 - p_2^{\tau}w_1)\Big),\\
w_{q^{\tau}}=(DL_{q^{\tau}})^{T}(w_1,w_2,0)^{T}
= \Big(w_1,w_2,\frac{1}{2}(q_1^{\tau}w_2-q_2^{\tau}w_1)\Big).
\end{gather*}
Then, defining the matrices
$\mathcal{X}^{\tau}, \mathcal{Y}^{\tau} \in S^{2}(\mathbb{R})$
as
\begin{gather*}
\mathcal{X}^{\tau} = (DL_{p^{\tau}}X^{\tau}DL_{p^{\tau}}^{T})_{2 \times 2},\\
\mathcal{Y}^{\tau} = (DL_{p^{\tau}}Y^{\tau}DL_{p^{\tau}}^{T})_{2 \times 2},
\end{gather*}
we deduce from \eqref{matrix-inequality} that
\begin{equation}\label{inequality2}
\begin{split}
&\langle \mathcal{X}^{\tau}w, w\rangle - \langle \mathcal{Y}^{\tau}w,w\rangle \\
&= \langle X^{\tau}w_{p^{\tau}},w_{p^{\tau}}\rangle
 - \langle Y^{\tau}w_{q^{\tau}}, w_{q^{\tau}}\rangle \\
& \leq  \tau \Big\langle \big[ \nabla^{2}_{p,q}g^{2}(p^{\tau}, q^{\tau})
+ (\nabla^{2}_{p,q}g^{2}(p^{\tau}, q^{\tau}))^{2}\big] (w_{p^{\tau}}
\oplus w_{q^{\tau}}), w_{p^{\tau}}\oplus w_{q^{\tau}}\Big \rangle = o(1),
\end{split}
\end{equation}
locally uniformly in $\| w \|$.

Because of the singularity of $F$ at $\eta = 0$, we have to consider two cases.
Firstly, assume that
$$
\eta^{\tau} = \tau \nabla_\mathcal{H}^{p}g^{2}(p^{\tau}, q^{\tau}) 
= - \tau  \nabla_\mathcal{H}^{q}g^{2}(p^{\tau}, q^{\tau})\neq 0
$$
for all  large $\tau$. Using that $u$ is a subsolution and $v$ is a 
supersolution to equation (E), we obtain
\begin{gather}\label{1}
\frac{\delta}{(T-t^{\tau})^{2}}+ \tau(t^{\tau}-s^{\tau}) + F(t^{\tau}, 
p^{\tau}, u(t^{\tau},p^{\tau}), \eta^{\tau}, \mathcal{X}^{\tau}) \leq 0, \\
\label{2}
\tau(t^{\tau}-s^{\tau}) + F( s^{\tau}, q^{\tau},v(s^{\tau}, q^{\tau}), 
\eta^{\tau}, \mathcal{Y}^{\tau}) \geq 0,
\end{gather}
Subtracting \eqref{2} from \eqref{1}, we have
\begin{equation}
\frac{\delta}{(T-t^{\tau})^{2}} +F( t^{\tau}, p^{\tau},u( t^{\tau},p^{\tau}), 
\eta^{\tau}, \mathcal{X}^{\tau})  -F(s^{\tau}, q^{\tau}, v(s^{\tau}, q^{\tau}), 
\eta^{\tau}, \mathcal{Y}^{\tau}) \leq 0.
\end{equation}
Whence
$$
0 < \frac{\delta}{(T-t^{\tau})^{2}}
\leq F(s^{\tau}, q^{\tau}, v(s^{\tau}, q^{\tau}), \eta^{\tau},
 \mathcal{Y}^{\tau}) - F( t^{\tau}, p^{\tau},u( t^{\tau},p^{\tau}), 
\eta^{\tau}, \mathcal{X}^{\tau}).
$$
We now estimate the difference on the right-hand side. 
From assumption (1) on $F$, we obtain
\begin{equation}\label{est.1}
\begin{split}
& F(s^{\tau}, q^{\tau}, v(s^{\tau}, q^{\tau}), \eta^{\tau}, \mathcal{Y}^{\tau}) 
- F( t^{\tau}, p^{\tau},u( t^{\tau},p^{\tau}), \eta^{\tau}, \mathcal{X}^{\tau}) \\
&\leq \omega\big(|t^{\tau} - s^{\tau}| + d_C(p^{\tau}, q^{\tau})\big)
+ F(t^{\tau}, p^{\tau}, v(s^{\tau}, q^{\tau}), \eta^{\tau}, \mathcal{Y}^{\tau})\\
&\quad -F( t^{\tau}, p^{\tau},u( t^{\tau},p^{\tau}), \eta^{\tau}, \mathcal{X}^{\tau}).
\end{split}
\end{equation}
By \eqref{A}, we have
$$
u(t^{\tau}, p^{\tau}) - v(s^{\tau}, q^{\tau}) > 0,
$$
for large enough $\tau$. Hence, by assumption (2), \eqref{inequality2} 
and assumption (4), the inequality  \eqref{est.1} becomes
\begin{equation}
 F(s^{\tau}, q^{\tau}, v(s^{\tau}, q^{\tau}), \eta^{\tau}, \mathcal{Y}^{\tau}) 
- F( t^{\tau}, p^{\tau},u( t^{\tau},p^{\tau}), \eta^{\tau}, 
\mathcal{X}^{\tau}) \leq o(1)  \text{ as }\tau \to \infty.
\end{equation} 
Then we obtain the contradiction
$$
0 < \frac{\delta}{(T-t_0)^{2}}  \leq o(1).
$$
Secondly, suppose that $\eta^{\tau_j} = 0$ for a subsequence 
$\tau_j \to \infty$. One has to distinguish two subcases:
\smallskip

\noindent\textbf{Subcase 1:}
 If $g(p^{\tau_j}, q^{\tau_j}) = 0$, then reasoning as in \cite{FLM1},
 we obtain the contradiction
$$
\frac{\delta}{(T-t_0)^{2}} \leq 0.
$$
\smallskip

\noindent\textbf{Subcase 2:} Suppose $g(p^{\tau_j}, q^{\tau_j}) \neq 0$, then it follows that
 $\nabla_\mathcal{H}^{p}g(p^{\tau_j},q^{\tau_j}) = 0$. 
We first prove that $p^{\tau_j}_1=p^{\tau_j}_2=0$. Assume to get a contradiction 
that $(p_1^{\tau_j})^{2}+ (p_2^{\tau_j})^{2}\neq 0$. Observe that the 
function $p \to  u(p,q^{\tau_{j}},t^{\tau_j})- \tau g^{2}(p, q^{\tau_j})$ 
attains a maximum at $p^{\tau_j}$, which is an interior point of $\Omega$ for 
$\tau_j$ large enough. For all point 
$\hat{p}=(\hat{p}_1,\hat{p}_2, \hat{p}_3) \neq 0$ closed to $p^{\tau_j}$ such that
$$
\hat{p}_3= G(\hat{p}_1, \hat{p}_2) = G(p_1^{\tau_j}, p_2^{\tau_j})= p_3^{\tau_j},
$$
we have by the assumed symmetry of $u$ that
$$
u(\hat{p}, t^{\tau_j}) -\tau g^{2}(\hat{p}, q^{\tau_j}) 
\leq u(p^{\tau_j}, t^{\tau})-\tau g^{2}(p^{\tau_j}, q^{\tau_j})
$$
which yields: 
$$
g(\hat{p}, q^{\tau_j}) \geq g(p^{\tau_j}, q^{\tau_j}).
$$
The method of Lagrange multipliers says that there exists a constant 
$\lambda \in \mathbb{R}$ such that
\begin{equation}\label{lagrange}
\begin{gathered}
\frac{\partial g}{\partial p_1}(p^{\tau_j},q^{\tau_j}) 
= \lambda \frac{\partial G}{\partial p_1}(p^{\tau_j})\\
\frac{\partial g}{\partial p_2}(p^{\tau_j},q^{\tau_j}) 
= \lambda \frac{\partial G}{\partial p_2}(p^{\tau_j}).
\end{gathered}
\end{equation}
From the assumption $\eta^{\tau_j} = 0$ and \eqref{lagrange}, we obtain
\begin{equation}\label{12}
\begin{gathered}
\lambda \frac{\partial G}{\partial p_1}(p^{\tau_j})-\frac{p_2^{\tau_j}}{2}
\frac{\partial g}{\partial p_3}(p^{\tau_j},q^{\tau_j}) = 0\\
\lambda \frac{\partial G}{\partial p_2}(p^{\tau_j})
+\frac{p_1^{\tau_j}}{2}\frac{\partial g}{\partial p_3}(p^{\tau_j},q^{\tau_j}) = 0.
\end{gathered}
\end{equation}
If $\lambda \neq 0$, then
\begin{equation}
\begin{gathered}
p_1^{\tau_j} \frac{\partial G}{\partial p_1}(p^{\tau_j})
= \frac{p_1^{\tau_j}p_2^{\tau_j}}{2 \lambda}
 \frac{\partial g}{\partial p_3}(p^{\tau_j},q^{\tau_j}) \\
p_2^{\tau_j} \frac{\partial G}{\partial p_2}(p^{\tau_j})
=-\frac{p_1^{\tau_j}p_2^{\tau_j}}{2 \lambda}
 \frac{\partial g}{\partial p_3}(p^{\tau_j},q^{\tau_j}) .
\end{gathered}
\end{equation}
Adding these equations gives a contradiction to \eqref{assumption-on-g}. 
Hence, $\lambda = 0$. Thus, equations \eqref{lagrange} and \eqref{12} take the form
\begin{equation}
\begin{gathered}
0=\frac{\partial g}{\partial p_1}(p^{\tau_j}, q^{\tau_j}) 
= 4\big[(p_1^{\tau_j} - q_1^{\tau_j})^{2} +(p_2^{\tau_j}-q_2^{\tau_j})^{2}  
\big](p_1^{\tau_j} - q_1^{\tau_j})\\
0=\frac{\partial g}{\partial p_2}(p^{\tau_j}, q^{\tau_j}) 
= 4\big[(p_1^{\tau_j} - q_1^{\tau_j})^{2} +(p_2^{\tau_j}
 -q_2^{\tau_j})^{2}  \big](p_2^{\tau_j} - q_2^{\tau_j})\\
0 = \frac{\partial g}{\partial p_3}(p^{\tau_j}, q^{\tau_j}) 
=2\Big(p_3^{\tau_j} -q_3^{\tau_j} +\frac{1}{2}p_2^{\tau_j}q_1^{\tau_j} 
+ \frac{1}{2}p_1^{\tau_j}q_2^{\tau_j}\Big),
\end{gathered}
\end{equation}
which yields $p^{\tau_j}=q^{\tau_j}$, contradicting the assumption 
$g(p^{\tau_j}, q^{\tau_j}) \neq 0$.

Therefore, the claim is proved. As in \cite{FLM1}, it also follows that 
$q_1^{\tau_j}=q_2^{\tau_j}$ which implies
$$
(\nabla_\mathcal{H}^{p,2}g^{2})^{*}(p^{\tau_j},q^{\tau_j}) 
= (\nabla_\mathcal{H}^{q,2}g^{2})^{*}(p^{\tau_j},q^{\tau_j}) =0.
$$
An application of Definition \ref{second-def}, gives the contradiction:
$$
\frac{\delta}{(T-t^{\tau_j})^{2}}+\tau_j(t^{\tau_j}-s^{\tau_j}) \leq 0
$$
and
$$
\tau_j(t^{\tau_j}-s^{\tau_j}) \geq 0.$$Hence, the proof is complete.
\end{proof}

\begin{remark} \label{rmk} \rm
Another way of proving Theorem \ref{thm4.1} is the following: 
for each $\delta > 0$, the function
$$
\tilde{u}=u -\frac{\delta}{T-t}
$$
is also a subsolution. Indeed, if $\varphi$ is a smooth function touching 
$\tilde{u}$ from above at some point $(t_0,p_0) \in (0,T) \times \Omega$ 
such that
$$
\max_{(0,T)\times \Omega}(\tilde{u}-\varphi) = (\tilde{u}-\varphi)(t_0,p_0)
$$
then it clear that
$$
\max_{(0,T)\times \Omega}(u-\tilde{\varphi}) 
= (u-\tilde{\varphi})(t_0,p_0),
$$
where
$$
\tilde{\varphi}(t,p) = \varphi(t,p) + \frac{\delta}{T-t}.
$$
Observe that $\nabla_\mathcal{H}\tilde{\varphi} = \nabla_\mathcal{H}\varphi$ 
and $(\nabla_\mathcal{H}\tilde{\varphi})^{*}= (\nabla_\mathcal{H}\varphi)^{*}$. 
Hence, if $\nabla_\mathcal{H}\varphi \neq 0$, we have by Definition 
\ref{second-def} and assumption (2) on $F$ that
\begin{equation}
\begin{split}
\varphi_t +F(t,p,\tilde{u}, \nabla_\mathcal{H}\varphi, 
(\nabla_\mathcal{H}^{2}\varphi)^{*})  
&\leq  \tilde{\varphi}_t + F(t,p,u, \nabla_\mathcal{H}\tilde{\varphi}, 
(\nabla_\mathcal{H}^{2}\tilde{\varphi})^{*})  -\frac{\delta}{(T-t)^{2}}  \\
&\leq -\frac{\delta}{(T-t)^{2}} < 0.
\end{split}
\end{equation}
If $\nabla_\mathcal{H}\varphi =0$ and $(\nabla_\mathcal{H}^{2}\varphi)^{*}=0$, 
then by Definition \ref{second-def},
\begin{equation}
\varphi_t  =  \tilde{\varphi}_t  -\frac{\delta}{(T-t)^{2}}  
\leq -\frac{\delta}{(T-t)^{2}} < 0.
\end{equation}
Therefore, without loss of generality, we may assume that the subsolution $u$ 
satisfies
$$
u_t +F(t,p,u, \nabla_\mathcal{H}u, (\nabla_\mathcal{H}^{2}u)^{*}) 
\leq -\frac{\delta}{T^{2}} < 0, 
$$
in the sense of viscosity subsolution; that is, $u$ is a subsolution of
$$
u_t +\overline{F}(t,p,u, \nabla_\mathcal{H}u, (\nabla_\mathcal{H}^{2}u)^{*}) \leq 0
$$
where
\[
\overline{F}(t,p,r, \eta, \mathcal{X} )
= F(t,p,r, \eta, \mathcal{X} ) + \frac{\delta}{T^{2}}.
\]
Also  we may assume that the subsolution $u$ satisfies
\begin{equation} \label{assump2}
\lim_{t \to T}u(t,p) = -\infty, \text{ uniformly in }\overline{\Omega}.
\end{equation}
Then we  take the limit $\delta \to 0$ to obtain the desired result for
 any subsolution $u$. Proceeding as above, we assume
$u(\bar{t}, \bar{p}) - v(\bar{t}, \bar{p}) > 0$ for some point
$(\bar{t}, \bar{p}) \in \Omega\times [0,T)$.
The function $M^{\tau}$ is redefined as follows
$$
M^{\tau}(t,p,s,q)= u(t,p)-v(s,q) -\tau g^{2}(p,q) - \frac{\tau}{2}(t - s)^{2}.
$$
As above, we derive
$$
p^{\tau}, q^{\tau} \to p_0,\quad
t^{\tau}, s^{\tau} \to t_0, \quad \text{as } \tau \to \infty.
$$
Observe that by assumption \eqref{assump2} on $u$, the points $t^{\tau}$
lie in a compact subset of $[0, T)$. Moreover,
$(p_0, t_0) \in \Omega \times (0, T)$, and we may apply the parabolic
Crandall-Ishii lemma as before. Recall that the term $\delta/(T-t^{\tau})^{2}$
does not appear now in the expressions \eqref{123} and \eqref{1234}.
 The proof proceed as above, getting the contradictions:
$$
0 < \frac{\delta}{T^{2}} \leq F(s^{\tau},q^{\tau},v(s^{\tau},q^{\tau}),
\eta^{\tau},\mathcal{Y}^{\tau})-F(t^{\tau},p^{\tau},u(t^{\tau},p^{\tau}),
\eta^{\tau},\mathcal{X}^{\tau}) \leq o(1),
$$
when $\eta^{\tau} \neq 0$ for all large $\tau$, and
$$
\frac{\delta}{T^{2}} \leq 0
$$
in the two subcases of the case $\eta^{\tau_j} \neq 0$ for a subsequence
$\tau_j \to \infty$.
\end{remark}

\section{Examples}\label{examples section}

\subsection{Parabolic infinite Laplacian} 
We consider the following parabolic equation in the Heisenberg group $\mathcal{H}$:
\begin{equation}\label{parabolic infinite laplacian equation}
u_t - \Delta_{\infty,\mathcal{H}}^{N}u = 0, \quad\text{in }
\mathcal{H} \times (0, T).
\end{equation}
Here, the operator $-\Delta_{\infty,\mathcal{H}}^{N}$ denotes the normalized 
$\infty$-Laplacian in the Heisenberg group, and it is defined, for all 
$u$ such that $\nabla_\mathcal{H} u \neq 0$, as follows:
\begin{equation}
\begin{split}
-\Delta_{\infty,\mathcal{H}}^{N}u 
&= -\frac{1}{|\nabla_\mathcal{H} u|^{2}}
\big\langle(\nabla_\mathcal{H}^{2}u)^{*}\nabla_\mathcal{H} u;
 \nabla_\mathcal{H} u \big\rangle\\ 
& = -\frac{1}{|\nabla_\mathcal{H} u|^{2}} \sum_{i, j = 1}^{2}X_iuX_juX_iX_ju.
\end{split}
\end{equation} 
For a vector $\eta \in \mathbb{R}^{2}\setminus \{0\}$, and 
$\mathcal{X} \in S^{2}(\mathbb{R})$, we introduce the function
\begin{equation}\label{function infinite laplacian}
F_\infty(\eta, \mathcal{X}) 
= - \sum_{i, j = 1}^{2}\frac{\eta_i \eta_j}{|\eta|^{2}}\mathcal{X}_{ij}.
\end{equation}
Hence, equation \eqref{parabolic infinite laplacian equation} 
can be written whenever $\nabla_\mathcal{H} u \neq 0$ as
\begin{equation}\label{discontinuous case}
u_t + F_\infty(\Delta_{\mathcal{H}}u, (\Delta_{\mathcal{H}}^{2}u)^{*}) = 0,
\quad \text{in }\mathcal{H} \times (0, T).
\end{equation}
Moreover, observe that
$$
F_\infty^{*}(0, 0) = F_{\infty, *} (0, 0)= 0,
$$
and that, for all pairs 
$(\eta, \mathcal{X}) \in (\mathbb{R}^{2}\setminus \{0\}) \times S^{2}(\mathbb{R})$,
$$
F^{*}_\infty(\eta, \mathcal{X}) = F_{\infty, *}(\eta, \mathcal{X}) 
= F_\infty(\eta, \mathcal{X}).
$$
Finally, it is clear that $F$ also satisfies assumption (4) in 
Section \ref{assumptions}. Therefore, Theorem \ref{thm4.1} tells us that 
there exists a unique symmetric viscosity solution to the problem
\begin{equation}\label{boundaryVP-2}
\begin{gathered} 
u_t - \Delta^{N}_{\infty,\mathcal{H}}u = 0,  \quad\text{in }\Omega \times (0, T)\\
u (t,p)= g(t,p)   \quad p\in \partial \Omega, \; t\in [0,T)\\ 
u(0,p) = h(0,p) \quad p \in \overline{\Omega} 
\end{gathered}
\end{equation} 
for $T> 0$, $g \in \mathcal{C}([0,T) \times \overline{\Omega})$ and 
$h \in \mathcal{C}(\overline{\Omega})$ given.

 \subsection{Mean curvature flow equation}
We consider now the following problem involving the mean curvature flow equation:
 \begin{equation}\label{boundaryVP mean curvature}
\begin{gathered} u_t - \operatorname{tr}\Big[\Big(I-\frac{\nabla_\mathcal{H}u
\otimes \nabla_\mathcal{H}u}{|\nabla_\mathcal{H}u|^{2}}\Big)
(\nabla_\mathcal{H}^{2}u)^{*} \Big)\Big]= 0,  \quad\text{in }\Omega \times (0, T)\\ 
u (t,p)= u_0(t,p)   \quad p\in \partial \Omega, \; t\in [0,T)\\ 
 u(0,p) = u_0(0,p) \quad  p \in \overline{\Omega} 
\end{gathered}
\end{equation}
A derivation and interpretation of the problem \eqref{boundaryVP mean curvature} 
in the Euclidean setting may be seen in \cite{G} and \cite{CGG}, and 
in \cite{CD} and \cite{FLM} for the analogue in the Heisenberg group.

Let $F: \mathbb{R}^{2}\setminus \{0\} \times S^{2}(\mathbb{R}) \to \mathbb{R}$ 
be given by
$$
F(\eta,\mathcal{X}) = -\operatorname{tr}
\Big[ \Big(I-\frac{\eta \otimes \eta}{|\eta|^{2}}\Big)\mathcal{X}\Big].
$$
Observe that $F$ satisfies all the assumptions (1)--(4) from Section 
\ref{assumptions}. 
By Theorem \ref{thm4.1}, the boundary value problem \eqref{boundaryVP mean curvature} 
admits a unique viscosity solution which is symmetric in the sense specified 
in Theorem \ref{thm4.1}.

\subsection{Homogeneous diffusions in $\mathcal{H}$}
Consider the following one parameter family of Cauchy problems in 
the Heisenberg group:
 \begin{equation}\label{boundaryVP mean curvature and laplacian}
\begin{gathered} 
u_t + C_{p}\Delta_{p,\mathcal{H}}^{1}u= 0,  \quad\text{in }\Omega \times (0, T)\\ 
u (t,p)= u_0(t,p)  \quad p\in \partial \Omega, \; t\in [0,T)\\ 
 u(0,p) = u_0(0,p) \quad p \in \overline{\Omega} 
\end{gathered}
\end{equation}
where
 $$
C_p=\frac{p}{p+1},
$$
and the $1$-homogeneous $p$-Laplacian $\Delta_p^{1}$ is defined, 
for $1\leq p \leq \infty$, by
 \begin{equation*}
\Delta_{p,\mathcal{H}}^{1}u =  \begin{cases}
\big(1 - \frac{1}{p}\big)F_1((\nabla_\mathcal{H}^{2}u)^{*}) 
 + \big(\frac{2}{p}-1\big) F\big(\nabla_\mathcal{H}u, (\nabla_\mathcal{H}^{2}u)^{*}\big), 
& \text{if } 1 \leq p \leq 2\\[4pt] 
\frac{1}{p}F_1((\nabla_\mathcal{H}^{2}u)^{*})
 + \big(1-\frac{2}{p}\big)F_\infty(\nabla_\mathcal{H}u,
  (\nabla_\mathcal{H}^{2}u)^{*}), & \text{if } p > 2.
\end{cases} 
 \end{equation*}
Here $F_1: S^{2}(\mathbb{R}) \to \mathbb{R}$ is given by
 $$
F_1(\mathcal{X}) = -\operatorname{tr}\mathcal{X}.
$$
Our result Theorem \ref{thm4.1} indicates that the problem 
\eqref{boundaryVP mean curvature and laplacian} has a unique symmetric 
(with respect to a surface $p_3 =G(p_1,p_2)$) viscosity solution.

\begin{thebibliography}{00}
\bibitem{BC} M. Bardi, I. Capuzzo-Dolcetta;
\emph{Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations}, 
Birkhauser, Basel, 1997.

\bibitem{BS} G. Barles, P. E. Souganidis;
\emph{Convergence of approximation schemes for fully non-linear second order 
equations}, Asymptotic Analysis \textbf{4} (1991), 271-283.

\bibitem{Bi} T. Bieske;
\emph{On $\infty$-harmonic functions on the Heisenberg group}, 
Comm. in PDE \textbf{27} no. 3 (2002), 727-761.

\bibitem{Bi2} T. Bieske;
\emph{Comparison principle for parabolic equations in the Heisenberg group}, 
Electronic Journal of Differential Equations no. 95 (2005), 1-11.

\bibitem{Ov} O. Calin,  D. Chang, P. Greiner;
\emph{Geometric analysis on the Heisenberg group and its generalizations}, 
Studies in advanced Mathematics, Vol. 40 AMS, 2007.

\bibitem{CD} L. Capogna, D. Danielli, S. Pauls, J. Tyson;
\emph{An introduction to the Heisenberg group and the sub-Riemannian Isoperimetric 
problem}, Progress in Mathematics, Vol. 259, Birkhauser-Verlag, Basel, 2007.

\bibitem{CGG} Y. G. Chen, Y. Giga, S. Goto;
\emph{Uniqueness and existence of viscosity solutions of generalized mean curvature 
flow equations}, J. of Differential Geometry \textbf{33} (1991), 749-786.

 \bibitem{CL} M. G.  Crandall, P. Lions;
\emph{Viscosity solutions of Hamilton-Jacobi equations}, 
Trans. Amer. Math. Soc. \textbf{277} (1983), 1-42.

\bibitem{CEL} M. G.  Crandall, L. C. Evans, P.L. Lions;
\emph{Some properties of viscosity solutions of Hamilton-Jacobi equations},
 Transactions of the American Mathematical Society \textbf{282} no. 
2 (1984), 487-502.

\bibitem{C} M. Crandall;
\emph{Viscosity solutions: a Primer}, lecture notes in Mathematics 1660,
Springer-Verlag, 1997.


\bibitem{CIL} M. Crandall, H. Ishii, P-L. Lions;
\emph{User's guide to viscosity solutions of second order partial differential 
equations}, Bull. of Amer. Soc. \textbf{27} no. 1 (1992), 1-67.

\bibitem{E} L. C. Evans;
\emph{On solving certain nonlinear partial differential equations by accretive 
operator method}, Israel Journal of Mathematics  \textbf{36}, (1981) 225-247.

\bibitem{FLM1} F. Ferrari, Q. Liu, J. J. Manfredi;
\emph{On the horizontal mean curvature flow for axisymmetric surfaces in the 
Heisenberg group}, Communications in Contemporary Mathematics \textbf{16} 3 (2014), 
1350027 (41 pages).

\bibitem{FLM} F. Ferrari, Q. Liu, J. J. Manfredi;
\emph{On the characterization of $p$-Harmonic functions on the Heisenberg group 
by mean value properties}, Discrete and Continuous Dynamical Systems \textbf{34} 
7 (2014), 2779-2793.

\bibitem{G} Y. Giga;
\emph{Surface evolution equations: a level set method},
 Monographs in Mathematics 99, Birkhauser Verlag, Basel, 2006.

\bibitem{M} J. J. Manfredi;
\emph{Non-linear subelliptic equations on Carnot groups: Analysis and geometry 
in metric spaces}, Notes of a course given at the Third School on Analysis 
and Geometry in Metric Spaces, Trento, (2003), available at
http://www.pitt.edu/manfredi/

\end{thebibliography}

\end{document}