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

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 95, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE-2005/95\hfil Comparison principle]
{Comparison principle for parabolic equations in the Heisenberg group} 

\author[T. Bieske \hfil EJDE-2005/95\hfilneg]
{Thomas Bieske}  

\address{Department of Mathematics\\
University of South Florida\\ 
Tampa, FL 33620, USA}
\email{tbieske@math.usf.edu}


\date{}
\thanks{Submitted April 7, 2005. Published September 1, 2005.}

\subjclass[2000]{35K55, 22E25}
\keywords{Heisenberg group; viscosity solutions; parabolic equations}


\begin{abstract}
 We define two notions of viscosity solutions to parabolic equations 
 in the Heisenberg group, depending on whether the test functions 
 concern only the past or both the past and the future. 
 We then exploit the Heisenberg geometry to prove a comparison principle 
 for a class of parabolic equations and show the sufficiency of 
 considering the test functions that concern only the past.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}  
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newcommand{\ip}[2]{\langle #1,#2 \rangle}


\section{Background and Motivation}
In \cite{B:HG}, viscosity solutions to a class of fully nonlinear subelliptic 
equations in the Heisenberg group were introduced and comparison principles 
were proved by heavily exploiting the geometry of the Heisenberg group. 
It is natural to adapt the geometric workings of \cite{B:HG} to parabolic 
equations in the Heisenberg group. For example, such equations have been 
recently used by Bonk and Capogna to study mean curvature \cite{BC:MC}. 

Our objective is to find the Heisenberg analog of the Euclidean comparison 
principle for a class of parabolic equations as found in 
\cite[Section 8]{CIL:UGTVS}. 
Relying on the Heisenberg geometry, we prove such an analog as our main 
theorem,  Theorem \ref{comp}. In addition, we examine a significant 
consequence of this comparison principle.  Namely, we are able to 
prove that for this class of parabolic equations, it is sufficient 
to consider only test functions that refer to the past.  
This was originally proved in the Euclidean case by Juutinen \cite{Ju:P}. 

Before presenting these two results in Section 4, we begin with a brief 
introduction to Heisenberg groups in Section 2 and discuss 
viscosity solutions to parabolic equations in Section 3.  

\section{The Heisenberg Group}

We begin with $\mathbb{R}^{2n+1}$ using the coordinates 
$(x_1,x_2,\dots, x_{2n},z)$ and consider the linearly
independent vector fields $ \{X_i,Z\}, $ where the index $i$ ranges 
from $1$ to $2n$, defined by  
\begin{gather*}
 X_i  =  \begin{cases} 
  \frac{ \partial }{\partial x_i} - 
  \frac{ x_{n+i}} { 2}\frac{ \partial }{\partial z}
  & \text{ if  $1 \leq i\leq n$} \\[3pt]
 \frac{ \partial }{ \partial x_{i}} + 
 \frac{x_{i-n}} { 2}\frac{ \partial }{\partial z} 
  & \text{ if  $n < i \leq 2n$,}
\end{cases}\\
Z  =  \frac{ \partial }{ \partial z}  \,.
\end{gather*} 
These vector fields obey the relations 
\begin{equation*}
 	[X_i,X_j] =  \begin{cases} 
  			Z & \text{if  $j=i+n$} \\ 
			0 & \text{otherwise}
			\end{cases}
\end{equation*}
and for all $i$, 
\begin{equation*}
[X_i,Z] = 0.
\end{equation*}
We then have a Lie Algebra denoted $h_n$ that decomposes as a direct sum 
\begin{equation*}
h_n = V_1 \oplus V_2 
\end{equation*}
where $V_1$ is
spanned by the $X_i$'s and $V_2$ is spanned by $Z$. 
We endow $h_n$  with an inner
product $\ip{\cdot}{\cdot}$ and related norm $\|\cdot\|$ so that this basis is
orthonormal. 
The corresponding Lie Group is called the
general Heisenberg group of dimension $n$ and is denoted by $\mathbb{H}_n$. 
The choice of vector fields and their Lie bracket relations forces the 
exponential map to be the identity and so elements of $h_n$ and 
$\mathbb{H}_n$ can be identified with each other.  Namely,
$$
\sum_{i=1}^{2n}x_iX_i+ zZ \in h_n \leftrightarrow
(x_1, x_2, \dots, x_{2n},z)\in \mathbb{H}_n.
$$
In particular,  for any $p,q$ in $\mathbb{H}_n$, written as 
$p=(x_1, x_2, \dots, x_{2n},z_1)$ 
and $q=(y_1, y_2, \dots, y_{2n},z_2)$ the group multiplication law is given by 
\[
p \cdot q = \big( x_1+y_1, x_2+y_2, \dots, x_{2n}+y_{2n}, z_{1}+z_{2} 
+ \frac{1}{2}\sum_{i=1}^{n} (x_iy_{n+i}-x_{n+i}y_i)\big).
\]
The natural metric on $\mathbb{H}_n$ is the Carnot-Carath\'{e}odory metric  
given by 
$$
d_C(p,q)= \inf_{\Gamma} \int_{0}^{1} \| \gamma '(t) \| dt 
$$
where the set $ \Gamma $
is the set of all curves $ \gamma $ such that $\gamma (0) = p, \gamma (1) = q$ 
and $\gamma'(t) \in V_1$.  By Chow's theorem  any two points
 can be connected by a horizontal curve, which
makes $d_C(p,q)$ a left-invariant metric on $\mathbb{H}_n$. (See, for example,
\cite{BR:SRG}.)
This metric induces a
homogeneous norm on $\mathbb{H}_n$, denoted $|\cdot|$, by 
$$|p|=d_C(0,p)
$$ 
and we have the estimate 
$$
|p| \sim \sum_{i=1}^{2n}|x_i|+|z|^{1/2}.
$$
This estimate leads us to define the left-invariant gauge $\mathcal{N}$ that 
is comparable to the  Carnot-Carath\'{e}odory metric and is given by 
$$
\mathcal{N}(p)=\Big(\big(\sum_{i=1}^{2n}x_i^2\big)^2+16z^2\Big)^{1/4}\,.
$$ We define the 
Carnot-Carath\'{e}odory balls $B(p,r)$ and the gauge balls $B_\mathcal{N}(p,r)$ 
in the obvious way.

Given a smooth function $u:\mathbb{H}_n \mapsto \mathbb{R}$, we define the 
horizontal gradient by
$$
\nabla_0 u = (X_1u,X_2u,\dots, X_{2n}u),
$$ 
the full gradient by 
$$
\nabla u = (X_1u,X_2u,\dots, X_{2n}u,Zu),
$$
and the symmetrized horizontal second
derivative matrix $(D^2u)^{\star}$  by 
\[ 
((D^2u)^{\star})_{ij}=\frac{1}{2}(X_iX_ju + X_jX_iu).
\]
A function $f$ is $\mathcal{C}^1$ if $X_if$ is continuous for all 
$i$ and $f$ is $\mathcal{C}^2$ if $f$ is $\mathcal{C}^1$ and $X_iX_jf$ is 
continuous for all $i$ and $j$.  

For a more complete treatment of the Heisenberg group, the interested
reader is directed to \cite{BR:SRG}, \cite{B:HG}, \cite{F:SE}, \cite{FS:HSHG}
\cite{G:MS}, \cite{H:CCG}, \cite{K:LGHT}, \cite{St:HA} and the references therein.

\section{Parabolic Jets and Solutions to Parabolic Equations}

In this section, we define and compare various notions of solutions to 
parabolic equations in the Heisenberg group, in the spirit of 
 \cite[Section 8]{CIL:UGTVS}. 
We begin by letting $u(p,t)$ be a function in $\mathbb{H}_n \times [0,T]$ 
for some $T>0$.  We consider parabolic equations of the form 
\begin{equation}\label{main}
u_t+F(t,p,u,\nabla u,(D^2u)^{\star})=0
\end{equation}
for continuous and proper $F:[0,T]\times \mathbb{H}_n \times \mathbb{R} 
\times h_n \times S^{2n} \mapsto \mathbb{R}$.  
We recall that $S^{k}$ is the set of $k \times k$ symmetric matrices and 
the derivatives  $\nabla u$ and $(D^2u)^{\star}$ are taken in the space 
variable $p$.    Examples of parabolic equations include the parabolic 
$P$-Laplace equation for $2 \leq P < \infty$ given by
$$
u_t+\Delta_Pu = u_t - \textmd{div}(\|\nabla_0u\|^{P-2}\nabla_0u)=0
$$ 
and the parabolic infinite Laplace equation 
$$
u_t+\Delta_{\infty}u = u_t - \ip{(D^2u)^\star\nabla_0u}{\nabla_0u}=0.
$$

For such equations, we define the parabolic superjet of $u(p,t)$ at the point 
$(p_0,t_0) \in \mathcal{O}_T \equiv \mathcal{O} \times (0,T)$, denoted 
$P^{2,+}u(p_0,t_0)$, by using triples 
$(a,\eta,X) \in \mathbb{R} \times h_n \times S^{2n}$ so that 
$(a,\eta,X) \in P^{2,+}u(p_0,t_0)$ if 
\begin{align*}
u(p,t) & \leq  u(p_0,t_0) + a(t-t_0) + \ip{\eta}{p_0^{-1}\cdot p}
+\frac{1}{2}\ip{X\overline{p_0^{-1}\cdot p}}{\overline{p_0^{-1}\cdot p}}\\
 & \quad + o(|t-t_0|+|p_0^{-1}\cdot p|^2)\quad  \textmd{as } (p,t)\to (p_0,t_0).
\end{align*}
We recall that $\overline{p_0^{-1}\cdot p}$ is the first $2n$ coordinates of 
$p_0^{-1}\cdot p$, given by $(x_1-x_1^0,x_2-x_2^0,\dots,x_{2n}-x^0_{2n})$.  
This definition is analogous to the superjet definition for subelliptic 
equations, as detailed in \cite{B:HG}.  We define the subjet $P^{2,-}u(p_0,t_0)$ 
by 
$$
P^{2,-}u(p_0,t_0)=-P^{2,+}(-u)(p_0,t_0).
$$  
We define the set theoretic closure of the superjet, denoted 
$\overline{P}^{2,+}u(p_0,t_0)$, by requiring 
$(a,\eta,X) \in \overline{P}^{2,+}u(p_0,t_0)$ exactly when there is a 
sequence 
$$
(a_n,p_n,t_n,u(p_n,t_n),\eta_n,X_n)\to (a,p_0,t_0,u(p_0,t_0),\eta,X)
$$ 
with the triple $(a_n,\eta_n,X_n)\in P^{2,+}u(p_n,t_n)$. 
A similar definition holds for the closure of the subjet.

As in the subelliptic case, we may also define jets using the appropriate 
test functions.  Namely, we consider the set $\mathcal{A}u(p_0,t_0)$ by 
\begin{align*}
\mathcal{A}u(p_0,t_0)=\{\phi \in \mathcal{C}^2(\mathcal{O}_T): u(p,t)-\phi(p,t) \leq 
u(p_0,t_0)-\phi(p_0,t_0)=0\}
\end{align*}
consisting of all test functions that touch from above. We define 
the set of all test functions that touch from below, denoted $\mathcal{B}u(p_0,t_0)$,
similarly. The following lemma is proved in the same way as 
the Euclidean version (\cite{C:VS} and \cite{H:H}) except we replace the 
Euclidean distance $|p-p_0|$ with the Heisenberg gauge 
$\mathcal{N}(p_0^{-1}\cdot p)$.

\begin{lemma}
$$P^{2,+}u(p_0,t_0)=\{(\phi_t(p_0,t_0),\nabla \phi(p_0,t_0), 
(D^2\phi(p_0,t_0))^\star): \phi \in \mathcal{A}u(p_0,t_0)\}.
$$ 
\end{lemma}

We may now relate the traditional Euclidean parabolic jets found in 
\cite{CIL:UGTVS} to the Heisenberg parabolic jets via the following lemma.

\begin{lemma} \label{jets}
Let $DL_{p_0}$ be the differential of the left multiplication map at the
point
$p_0$, let $P_{\textmd{eucl}}^{2,+}u(p_0,t_0)$ be the traditional Euclidean 
parabolic superjet
of $u$ at the point $(p_0,t_0)$ and let  $ (a,\eta, X) \in
\mathbb{R}\times \mathbb{R}^{2n+1} \times S^{2n+1}$.  Then, 
$$ 
(a,\eta, X) \in \overline{P}_{\textmd{eucl}}^{2,+}u(p_0,t_0)
$$ 
gives the element  
$$
\Big(a,DL_{p_0}\eta, (DL_{p_0} \; X \; (DL_{p_0})^{T})_{2n} \Big) \in
\overline{P}^{2,+}u(p_0,t_0)
$$ 
with the convention that for any matrix $M$, $M_{m}$ is the $m \times m$ 
principal minor.
\end{lemma}

\begin{proof}
This proof is similar to the corresponding result for subelliptic jets as 
found in \cite{B:HG}. We then highlight the main details.

We may assume that $u(p_0,t_0)=0$.  We first consider the case when $p_0$ 
is the origin.  Let $(a,\eta,X) \in P_{\textmd{eucl}}^{2,+}u(0,t_0)$.  
Then we have 
$$
u(p,t) \leq a(t-t_0)+ \ip{\eta}{p}_{\textmd{eucl}}+\ip{Xp}{p}_{\textmd{eucl}}
+o(|t-t_0|+\|p\|^2_{\textmd{eucl}})
$$ 
for $(p,t)$ near $(0,t_0)$. Suppose that 
$\alpha$ is $o(|t-t_0|+\|p\|^2_{\textmd{eucl}})$.  Then we have 
\begin{align*}
\frac{\alpha}{|t-t_0|+|p|^2} 
& =  \frac{\alpha}{|t-t_0|+\|p\|^2_{\textmd{eucl}}}\times 
\frac{|t-t_0|+\|p\|^2_{\textmd{eucl}}}{|t-t_0|+|p|^2} \\
 &  \leq   \frac{\alpha}{|t-t_0|+\|p\|^2_{\textmd{eucl}}}\times 
 \big(1+\frac{\|p\|^2_{\textmd{eucl}}}{|p|^2}\big)_.
\end{align*} 
We thus conclude that $\alpha$ is $o(|t-t_0|+|p|^2)$.  Using the fact that 
$\ip{\eta}{p}_{\textmd{eucl}}= \ip{\eta}{p}$ at the origin, we obtain
$$
u(p,t) \leq a(t-t_0)+ \ip{\eta}{p}+\ip{Xp}{p}_{\textmd{eucl}}+o(|t-t_0|+|p|^2).
$$
We next observe that 
$$
\ip{Xp}{p}_{\textmd{eucl}}= \ip{(X)_{2n}\overline{p}}{\overline{p}}+ o(|p|^2)
$$
where $(X)_{2n}$ is the $2n \times 2n$ principal minor and $\overline{p}$ 
is as above.  We therefore obtain the inequality
$$
u(p,t) \leq a(t-t_0)+ \ip{\eta}{p}+\ip{(X)_{2n}\overline{p}}{\overline{p}}
+o(|t-t_0|+|p|^2).
$$
The general case follows from left translation of $p_0$. 
\end{proof}

We then use these jets to define subsolutions and supersolutions 
to Equation \eqref{main} in the usual way.

\begin{definition} \rm
Let $(p_0,t_0)\in \mathcal{O}_T$ be as above.  The upper semicontinuous 
function $u$ is a \emph{viscosity subsolution} in $\mathcal{O}_T$ if for all 
$(p_0,t_0) \in \mathcal{O}_T$ we have $(a,\eta,X) \in P^{2,+}u(p_0,t_0)$ 
produces 
$$ 
a+F(t_0,p_0,u(p_0,t_0),\eta,X)\leq 0.
$$
A lower semicontinuous function $u$ is a \emph{viscosity supersolution} in 
$\mathcal{O}_T$ if for all $(p_0,t_0) \in \mathcal{O}_T$ we have 
$(b,\nu,Y) \in P^{2,-}u(p_0,t_0)$ produces 
$$ 
b+F(t_0,p_0,u(p_0,t_0),\nu,Y)\geq 0.
$$
A continuous function $u$ is a \emph{viscosity solution} in $\mathcal{O}_T$ 
if it is both a viscosity subsolution and viscosity supersolution.
\end{definition}

We also wish to define what \cite{Ju:P} refers to as parabolic viscosity 
solutions. We first need to consider the set 
$$
\mathcal{A}^-u(p_0,t_0)=\{\phi \in \mathcal{C}^2(\mathcal{O}_T): u(p,t)
-\phi(p,t) \leq u(p_0,t_0)-\phi(p_0,t_0)=0\  \textmd{for}\ t < t_0\}
$$ 
consisting of all functions that touch from above only when $t<t_0$.  
Note that this set is larger than $\mathcal{A}u$ and corresponds physically 
to the past alone playing a role in determining the present. We define 
$\mathcal{B}^-u(p_0,t_0)$ similarly.
We then have the following definition.

\begin{definition} \rm
An upper semicontinuous function $u$ on $\mathcal{O}_T$ is a 
\emph{parabolic viscosity subsolution} in $\mathcal{O}_T$ if 
$\phi\in \mathcal{A}^-u(p_0,t_0)$ produces 
$$
\phi_t(p_0,t_0)+F(t_0,p_0,u(p_0,t_0),\nabla \phi(p_0,t_0),
(D^2\phi(p_0,t_0))^{\star}) \leq 0.
$$ 
An lower semicontinuous function $u$ on $\mathcal{O}_T$ is a 
\emph{parabolic viscosity supersolution} in $\mathcal{O}_T$ if 
$\phi\in \mathcal{B}^-u(p_0,t_0)$ produces 
$$
\phi_t(p_0,t_0)+F(t_0,p_0,u(p_0,t_0),\nabla \phi(p_0,t_0),(D^2
\phi(p_0,t_0))^{\star}) \geq 0.
$$ 
A continuous function is a \emph{parabolic viscosity solution} if it is both 
a parabolic viscosity supersolution and subsolution.
\end{definition} 
It is easily checked that parabolic viscosity sub(super-)solutions are 
viscosity  sub(super-)solutions.  The reverse implication will be a 
consequence of the comparison principle proved in the next section.


\section{Comparison Principle}

To prove the comparison principle, we need to chose the proper penalty 
function that has the desired properties.  
We consider the function 
$\varphi: \mathbb{H}_n \times \mathbb{H}_n \mapsto \mathbb{R}$ given by 
$$
\varphi(p,q)= \frac{1}{2}\sum_{i=1}^{2n}(x_i-y_i)^{2}
+\frac{1}{2}\Big(z_1-z_2+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i})
\Big)^{2}.
$$ 
We observe that we have 
$$
\varphi(p,q) = \frac{1}{2}\|q^{-1}\cdot p\|^2.
$$
This choice gives us desired properties as detailed in the next lemma.

\begin{lemma}\label{properties}
Let the vector $\eta$ be given by 
\[
\eta = q^{-1} \cdot p = \begin{pmatrix}
(x_1-y_1) \\(x_2-y_2) \\ \vdots \\ (x_{2n}-y_{2n}) \\
z_1-z_2+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i})
\end{pmatrix}\,.
\]
Recall that the differential of left multiplication with respect to $p$, 
denoted $DL_p$, is given by 
\[
\begin{pmatrix}
I_{2n\times 2n} & \mathcal{P} \\
0_{1 \times 2n} & 1
\end{pmatrix}
\]
with the $2n \times 1$ vector $\mathcal{P}$ given by 
\[
(-\frac{1}{2} x_{n+1},
-\frac{1}{2} x_{n+2},
\dots,
-\frac{1}{2} x_{2n},
\frac{1}{2} x_{1}, 
\frac{1}{2} x_{2}, 
\dots,
\frac{1}{2} x_{n})^T
\]
with a similar definition for $DL_q$ using the vector $\mathcal{Q}$.  
Denoting Euclidean differentiation with respect to the point $r$ by $D_r$, 
then have the following properties:
\begin{enumerate}
\item $ D_p \varphi (p,q) = DL_q \eta$
\item $ D_q \varphi (p,q) = -DL_p \eta$
\item $ D_p \eta = DL_q^T $
\item $ D_q \eta = -DL_p^T $
\item $ DL_p DL_q = DL_q DL_p $
\item $DL_p D_p \varphi(p,q)=-DL_q D_q \varphi(p,q) \equiv \Upsilon(p,q)$
\item \begin{align*}
& DL_p(D_{pp}\varphi (p,q)DL_p^T+D_{pq}\varphi(p,q)DL_q^T)\\
&= \frac{1}{2}
(z_1-z_2+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i}))
\begin{pmatrix}
0_{n\times n} & -I_{n \times n} & 0_{n\times 1} \\
I_{n\times n} & 0_{n \times n} & 0_{n\times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix}
\end{align*}
\item 
\begin{align*}
& DL_q(D_{qq}\varphi (p,q)DL_q^T+D_{qp}\varphi(p,q)DL_p^T)\\
&= \frac{1}{2}
(z_1-z_2+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i}))
\begin{pmatrix}
0_{n\times n} & I_{n \times n} & 0_{n\times 1} \\
-I_{n\times n} & 0_{n \times n} & 0_{n\times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix}
\end{align*}
\item Let $v \in h_n$ be a vector.  Then 
\begin{align*}
&\ip{D_{pp}\varphi(p,q)DL_p^Tv}{DL_p^Tv}+
\ip{D_{pq}\varphi(p,q)DL_q^Tv}{DL_p^Tv}\\
&+\ip{D_{qp}\varphi(p,q)DL_p^Tv}{DL_q^Tv}+
\ip{D_{qq}\varphi(p,q)DL_q^Tv}{DL_q^Tv}=0
\end{align*}
\item Let $v \in h_n$ be a vector. We recall that $\overline{v}$ is the 
first $2n$ coordinates of $v$.  Then 
\begin{align*}
& \|\begin{pmatrix}
D_{pp}\varphi(p,q) & D_{pq}\varphi(p,q) \\ 
D_{qp}\varphi(p,q) & D_{qq}\varphi(p,q)\end{pmatrix}
\begin{pmatrix} DL_p^Tv\\ DL_q^Tv\end{pmatrix} \|^2 \\ 
& =\frac{1}{2}\|\overline{v}\|^2(z_1-z_2
+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i}))^2\,. 
\end{align*}
\end{enumerate}
\end{lemma}

\begin{proof}
The first five properties are elementary calculations and left to the reader.  
The sixth follows from the fifth and the first two.  We therefore turn our 
attention to the last four. Let $M_{pq}$ be the left-hand side of $(7)$.  
Then, 
\begin{align*}
M_{pq} & =  DL_p\Big(D_p(DL_q\eta)DL_p^T+D_q(DL_q\eta)DL_q^T\Big) \\
 & =  DL_p\Big(DL_q D_p\eta DL_p^T+D_q(DL_q)\eta DL_q^T+DL_q D_q\eta DL_q^T\Big)\\
 & =  DL_p\Big(DL_q DL_q^T DL_p^T+D_q(DL_q)\eta DL_q^T-DL_q DL_p^T DL_q^T\Big)\\
 & =  DL_p\Big(D_q(DL_q)\eta DL_q^T \Big)
\end{align*}
and so we are left to compute only the derivative of the matrix $DL_q$.  
Knowing the definition of $\eta$ above and the formula for $DL_q$ as given 
above, we see that when $1 \leq i \leq n$, we have $D_{y_i}(DL_q)$ is a 
matrix with every entry $0$ except for the $(i+n,2n+1)$ entry, which 
is $\frac{1}{2}$.  When $n < i < 2n$, we have  
$D_{y_i}(DL_q)$ has all entries $0$ except for the $(i-n,2n+1)$ entry, 
which is $-\frac{1}{2}$. Cleary, $D_{z_2}(DL_q)$ is the $0$ matrix. 
We then compute 
\[
D_q(DL_q)\eta  =  \frac{1}{2}(z_1-z_2+\frac{1}{2}
\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i}))
\begin{pmatrix}
0_{n \times n} & -I_{n \times n} & 0_{n \times 1} \\
I_{n \times n} & 0_{n \times n} & 0_{n \times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix}\,.
\]
We then have 
\begin{align*}
&\begin{pmatrix}
I_{2n\times 2n} &  \mathcal{P} \\
0_{1 \times 2n} &  1
\end{pmatrix}
\begin{pmatrix}
0_{n \times n} & -I_{n \times n} & 0_{n \times 1} \\
I_{n \times n} & 0_{n \times n} & 0_{n \times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix}
\begin{pmatrix}
I_{2n\times 2n} & 0_{2n \times 1}\\
\mathcal{Q}^T & 1
\end{pmatrix}\\
&= \begin{pmatrix}
I_{2n\times 2n} & \mathcal{P} \\
0_{1 \times 2n} & 1
\end{pmatrix}
\begin{pmatrix}
0_{n \times n} & -I_{n \times n} & 0_{n \times 1} \\
I_{n \times n} & 0_{n \times n} & 0_{n \times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix}
=
\begin{pmatrix}
0_{n \times n} & -I_{n \times n} & 0_{n \times 1} \\
I_{n \times n} & 0_{n \times n} & 0_{n \times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix}
\end{align*}
and Property $(7)$ follows.  To prove $(8)$, we let $M_{pq}$ be the 
left-hand side of  $(8)$.  Then 
\begin{align*}
M_{pq} 
 & =  DL_q\Big(-D_q(DL_p\eta)DL_q^T+D_p(-DL_p\eta)DL_p^T\Big) \\
 & =  DL_q\Big(-DL_p D_q\eta DL_q^T-D_p(DL_p)\eta DL_p^T-DL_p D_p\eta DL_p^T\Big)\\
 & =  DL_q\Big(DL_p DL_p^T DL_q^T-D_p(DL_p)\eta DL_p^T-DL_p DL_q^T DL_p^T\Big)\\
 & =  -DL_q\Big(D_p(DL_p)\eta DL_p^T \Big)
\end{align*}
and we compute $D_p(DL_p)\eta$ in the same way as the above computation 
for $D_q(DL_q)\eta$
and arrive at Property $(8)$.

To prove Property $(9)$, we note that the right hand side can be written as 
\begin{align*}
&\ip{DL_p(D_{pp}\varphi (p,q)DL_p^T+D_{pq}\varphi(p,q)DL_q^T)v}{v} \\
&+ \ip{DL_q(D_{qq}\varphi (p,q)DL_q^T+D_{qp}\varphi(p,q)DL_p^T)v}{v}
\end{align*}
Using Properties $(7)$ and $(8)$ we see this is zero. Property $(9)$ then follows.

Using the proofs of Properties $(7)$ and $(8)$, we have 
\begin{align*}
&\begin{pmatrix}
D_{pp}\varphi(p,q) & D_{pq}\varphi(p,q) \\ 
D_{qp}\varphi(p,q) & D_{qq}\varphi(p,q)\end{pmatrix}
\begin{pmatrix} DL_p^Tv\\DL_q^Tv\end{pmatrix}\\
&=
\begin{pmatrix}
(D_{pp}\varphi(p,q)DL_p^T+D_{pq}\varphi(p,q)DL_q^T)v \\
(D_{qp}\varphi(p,q)DL_p^T+D_{qq}\varphi(p,q)DL_q^T)v 
\end{pmatrix}\\
&= 
\frac{1}{2}(z_1-z_2+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i}))
\begin{bmatrix}
\begin{pmatrix}
0_{n \times n} & -I_{n \times n} & 0_{n\times 1} \\
I_{n \times n} & 0_{n \times n} & 0_{n\times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix} v \\ 
\begin{pmatrix}
0_{n \times n} & I_{n \times n} & 0_{n\times 1} \\
-I_{n \times n} & 0_{n \times n} & 0_{n\times 1} \\
0_{1 \times n} & 0_{1 \times n} & 0
\end{pmatrix} v
\end{bmatrix}.
\end{align*}
We then see that 
\begin{align*}
&\big\| \begin{pmatrix}
D_{pp}\varphi(p,q) & D_{pq}\varphi(p,q) \\ 
D_{qp}\varphi(p,q) & D_{qq}\varphi(p,q)\end{pmatrix}
\begin{pmatrix} DL_p^Tv\\ DL_q^Tv\end{pmatrix}\big\|^2   \\ 
& =  \frac{1}{2}\|\overline{v}\|^2(z_1-z_2
+\frac{1}{2}\sum_{i=1}^{n}(x_{n+i}y_i-x_iy_{n+i}))^2
\end{align*}
and Property $(10)$ is proved.
\end{proof}

Using the penalty function $\varphi$ we next need to show the existence 
of Heisenberg jet elements when considering  subsolutions and supersolutions 
in $\mathbb{H}_n$.  This theorem is based on \cite[Thm. 8.2]{CIL:UGTVS}, 
which details the Euclidean case. 


\begin{theorem}
Let $u$ be a viscosity subsolution to Equation \eqref{main} and $v$ be
a viscosity supersolution to Equation \eqref{main} in the bounded parabolic 
set $\Omega \times (0,T)$ where $\Omega$ is a (bounded) domain. Let $\tau$ be 
a positive real parameter and let $\varphi(p,q)$ be as above.
Suppose the local maximum of 
$$
M_\tau(p,q,t) \equiv u(p,t)-v(q,t)-\tau\varphi(p,q)
$$ 
occurs at the interior point $(p_\tau, q_\tau, t_\tau)$ of the parabolic 
set $\Omega\times \Omega \times (0,T)$. Then, for each $\tau>0$, there 
are elements 
$(a,\tau \Upsilon,\mathcal{X}^{\tau}) \in \overline{P}^{2,+}u(p_\tau,t_\tau)$ 
and 
$(a,\tau \Upsilon,\mathcal{Y}^{\tau}) \in \overline{P}^{2,-}v(q_\tau,t_\tau)$ 
so that if 
$$
\lim_{\tau \to \infty}\tau\varphi(p_\tau,q_\tau)=0,
$$ 
then we have 
$\mathcal{X}^{\tau}\leq \mathcal{Y}^{\tau} +\mathcal{R}^{\tau}$ with 
$\mathcal{R}^{\tau} \to 0$ as $\tau \to \infty$.   
\end{theorem}

\begin{proof}
We first need to check that Condition 8.5  in \cite{CIL:UGTVS} is satisfied, 
namely that there exists an $r>0$ so that for each $M$, there exists 
a $C$ so that $b \leq C$ 
when $(b,\eta,X) \in P_{\textmd{eucl}}^{2,+}u(p,t), |p-p_\tau|+|t-t_\tau|<r$, 
and  $|u(p,t)|+\|\eta\|+\|X\|\leq M$ with a similar statement holding for $-v$. 
If this condition is not met, then for each $r>0$, we have an $M$ so that for 
all $C$, $b>C$ when $(b,\eta,X) \in P_{\textmd{eucl}}^{2,+}u(p,t)$.  
By Lemma \ref{jets} we would have 
$$(b,DL_p\eta,(DL_p\;X\;DL_p^T)_{2n})\in P^{2,+}u(p,t)
$$
contradicting the fact that $u$ is a subsolution.  A similar conclusion is 
reached for $-v$ and so we conclude that this condition holds.  
We may then apply Theorem 8.3 of \cite{CIL:UGTVS} and obtain, by our choice 
of $\varphi$, 
\begin{gather*}
(a,\tau D_p\varphi(p_\tau,q_\tau),X^\tau)  \in  
\overline{P}^{2,+}_{\textmd{eucl}}u(p_\tau,t_\tau) \\
(a,-\tau D_q\varphi(p_\tau,q_\tau),Y^\tau)  \in  
\overline{P}^{2,-}_{\textmd{eucl}}v(q_\tau,t_\tau)
\end{gather*}
and by Lemma \ref{jets} we have 
\begin{gather*}
(a,\tau \Upsilon(p_\tau,q_\tau),\mathcal{X}^\tau)  
\in  \overline{P}^{2,+}u(p_\tau,t_\tau) \\
(a,\tau \Upsilon(p_\tau,q_\tau),\mathcal{Y}^\tau)  
\in  \overline{P}^{2,-}v(q_\tau,t_\tau)
\end{gather*}
where 
\begin{gather*}
\mathcal{X}^\tau  =  (DL_{p_\tau} \; X^\tau \; DL_{p_\tau}^{T})_{2n}\\
\mathcal{Y}^\tau  =  (DL_{q_\tau} \; Y^\tau \; DL_{q_\tau}^{T})_{2n}.
\end{gather*}
Given a vector $v=(v_1,v_2,\dots, v_{2n})$, we consider the extension 
$\hat{v}$ to all of $h_n$ by $\hat{v}=(v,0)$. We then have 
\begin{align*}
\ip{\mathcal{X}^\tau v}{v}-\ip{\mathcal{Y}^\tau v}{v} 
& = 
\ip{DL_{p_\tau} \; X^\tau \; DL_{p_\tau}^{T}\hat{v}}{\hat{v}}-\ip{DL_{q_\tau} 
\; Y^\tau \; DL_{q_\tau}^{T}\hat{v}}{\hat{v}} \\
 & \leq  \tau \ip{(A^2+A)\Big(DL_{p_\tau}^{T}\hat{v} \oplus DL_{q_\tau}^{T}
 \hat{v}\Big)}
 {\Big(DL_{p_\tau}^{T}\hat{v} \oplus DL_{q_\tau}^{T}\hat{v}\Big)} 
\end{align*}
where the matrix $A$ is given by 
$$
A=\left(\begin{array}{cc}D_{pp}\varphi(p_\tau,q_\tau) & D_{pq}\varphi(p_\tau,q_\tau) \\ 
D_{qp}\varphi(p_\tau,q_\tau) & D_{qq}\varphi(p_\tau,q_\tau) \end{array}\right)_.
$$  
We may then combine Properties $(9)$ and $(10)$ of Lemma \ref{properties} and 
the fact that $\tau\varphi(p_\tau,q_\tau) \to 0$ as 
$\tau \to \infty$ to obtain the matrix estimate.
\end{proof}

Using the vector $\Upsilon$, we may define a class of parabolic equations to 
which we shall prove a comparison principle. 

\begin{definition} \rm
We say the continuous, proper function 
$$
F:[0,T]\times \overline{\Omega}\times \mathbb{R}\times h_n \times S^{2n} \mapsto \mathbb{R}
$$ 
is \emph{admissable} if 
for each $t \in [0,T]$, there is the same function 
$\omega:[0,\infty] \mapsto [0,\infty]$ with $\omega(0+)=0$ so that $F$ 
satisfies 
\begin{equation}\label{cond}
F(t,q,r,\tau \Upsilon,Y)-F(t,p,r,\tau \Upsilon,X) \leq \omega(d_C(p,q)
+\tau\|\Upsilon(p,q)\|^2+\|X-Y\|).
\end{equation}  
Note that $\|\Upsilon(p,q)\|^2 \sim \varphi(p,q)$ by the calculations above.
\end{definition}

We now formulate the comparison principle for the  problem
\begin{equation}\label{problem}
\begin{alignedat}{2}
 u_t+F(t,p,u,\nabla u, (D^2u)^\star) = 0 \quad & \textmd{in }(0,T)\times \Omega 
 &&\mathrm{(E)}\\
u(p,t)=g(p,t) \quad & p \in \partial \Omega,\; t \in [0,T) \quad &&\mathrm{(BC)}\\
u(p,0) = \psi(p) \quad & p \in \overline{\Omega} &&\mathrm{(IC)} 
\end{alignedat}
\end{equation}
Here, $\psi \in C(\overline{\Omega})$ and 
$g \in C(\overline{\Omega} \times [0,T))$.  Note that this is the 
Heisenberg version of the problem considered in \cite{CIL:UGTVS}. 
We also adopt their definition that a subsolution $u(p,t)$ to 
Problem \eqref{problem} is a viscosity subsolution to (E), 
$u(p,t) \leq g(p,t)$ on $\partial \Omega$ with $0 \leq t < T$ and 
$u(p,0) \leq \psi(p)$ on $\overline{\Omega}$.  Supersolutions and 
solutions are defined in an analogous matter. 
\begin{theorem}\label{comp}
Let $\Omega$ be a bounded domain in $\mathbb{H}_n$.  Let $F$ be admissible.  
If $u$ is a viscosity subsolution and $v$ a viscosity supersolution to 
Problem \eqref{problem} then $u \leq v$ on $[0,T) \times \Omega$.
\end{theorem}

\begin{proof}
Our proof follows that of \cite{CIL:UGTVS}[Thm. 8.2] and so we discuss only 
the main parts. 
For $\varepsilon > 0$, we substitute $\tilde{u}=u-\frac{\varepsilon}{T-t}$ 
for $u$ and prove the theorem for 
\begin{gather*}
u_t+F(t,p,u,\nabla u,(D^2u)^\star) \leq -\frac{\varepsilon}{T^2} < 0 \\ 
\lim_{t \uparrow T}u(p,t) = -\infty \ \  \textmd{uniformly on }\overline{\Omega}
\end{gather*}
and take limits to obtain the desired result.
Assume the maximum occurs at $(p_0,t_0)\in \Omega \times (0,T)$ with 
$$
u(p_0,t_0)-v(p_0,t_0)= \delta >0.
$$
Let 
$$
M_\tau=u(p_\tau,t_\tau)-v(q_\tau,t_\tau)-\tau\varphi(p_\tau,q_\tau)
$$ 
with 
$(p_\tau,q_\tau,t_\tau)$ the maximum point in 
$\overline{\Omega} \times \overline{\Omega} \times [0,T)$ of 
$u(p,t)-v(q,t)-\tau \varphi(p,q)$. 
Using the same proof as  in \cite[Lemma 5.2]{B:HG} we conclude that 
$$
\lim_{\tau\to \infty}\tau\varphi(p_\tau,q_\tau) =0.
$$ 
If $t_\tau=0$, we have 
$$
0 < \delta \leq M_\tau \leq \sup_{\overline{\Omega}\times
\overline{\Omega}}(\psi(p)-\psi(q)-\tau \varphi(p,q))
$$ 
leading to a contradiction for large $\tau$.  We therefore conclude 
$t_\tau >0$ for large $\tau$.  Since $u \leq v$ on
$\partial \Omega \times [0,T)$ by Equation (BC) of Problem \eqref{problem}, 
we conclude that for large $\tau$, we have $(p_\tau,q_\tau,t_\tau)$ is an 
interior point. That is, 
$(p_\tau,q_\tau,t_\tau) \in \Omega \times \Omega \times (0,T)$.
Using the previous theorem, we obtain
\begin{gather*}
(a,\tau \Upsilon(p_\tau,q_\tau), \mathcal{X}^\tau)  \in 
 \overline{P}^{2,+}u(p_\tau,t_\tau) \\
(a,\tau \Upsilon(p_\tau,q_\tau), \mathcal{Y}^\tau)  \in  
\overline{P}^{2,-}v(q_\tau,t_\tau)
\end{gather*}  
that satisfy the equations 
\begin{gather*}
a+F(t_\tau,p_\tau,u(p_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau), \mathcal{X}^\tau) 
 \leq  -\frac{\varepsilon}{T^2} \\
a+F(t_\tau,q_\tau,v(q_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau), \mathcal{Y}^\tau) 
 \geq  0.
\end{gather*}
Using the fact that $F$ is proper and that 
$u(p_\tau,t_\tau)\geq v(q_\tau,t_\tau)$ (otherwise $M_\tau < 0$), we have  
\begin{align*}
0 < \frac{\varepsilon}{T^2}  
& \leq  F(t_\tau,q_\tau,v(q_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau), 
  \mathcal{Y}^\tau)-
F(t_\tau,p_\tau,u(p_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau), \mathcal{X}^\tau)\\
& \leq   \omega (d_C(p_\tau,q_\tau)+C\tau\varphi(p_\tau,q_\tau)+\|\mathcal{R}_\tau\|).
  \end{align*}
We arrive at a contradiction as $\tau \to \infty$.  
\end{proof}
We then have the following corollary, showing the equivalence of parabolic 
viscosity solutions and viscosity solutions.

\begin{corollary}
For admissable $F$, 
we have the parabolic viscosity solutions are exactly the viscosity solutions.  
\end{corollary}

\begin{proof}
We showed above that parabolic viscosity sub(super-)solutions are viscosity 
sub(super-)solutions. To prove the converse, we will follow the proof of the 
subsolution case found in  \cite{Ju:P}, highlighting the main details. 
 Assume that $u$ is not a parabolic viscosity subsolution. 
Let $\phi \in \mathcal{A}^-u(p_0,t_0)$ have the property that 
$$
\phi_t(p_0,t_0)+F(t_0,p_0,\phi(p_0,t_0),\nabla \phi(p_0,t_0),
(D^2\phi(p_0,t_0))^\star) \geq \epsilon > 0
$$ 
for a small parameter $\epsilon$.  We may assume $p_0$ is the origin.
Let $r > 0$ and define $S_r= B_{\mathcal{N}}(r) \times (t_0-r,t_0)$ and let 
$\partial S_r$ be its parabolic boundary. Then the function 
$$
\tilde{\phi}_r(p,t)= \phi(p,t)+(t_0-t)^8-r^8+(\mathcal{N}(p))^8
$$ 
is a classical supersolution for sufficiently small $r$.  We then observe 
that $u \leq \tilde{\phi}_r$ on $\partial S_r$ but 
$u(0,t_0) > \tilde{\phi}(0,t_0)$. Thus, the comparison prinicple, 
Theorem \ref{comp}, does not hold.  Thus, $u$ is not a viscosity subsolution.  
The supersolution case is identical and omitted. 
\end{proof}

\begin{thebibliography}{CIL}
\bibitem{BR:SRG}Bella\" {\i}che, Andr\' {e}. The Tangent Space in
Sub-Riemannian Geometry. In \emph{Sub-Riemannian Geometry};
Bella\" {\i}che, Andr\' {e}., Risler, Jean-Jacques., Eds.; 
Progress in Mathematics;  Birkh\" {a}user: Basel, Switzerland.
1996; Vol. 144, 1--78.

\bibitem{B:HG}Bieske, Thomas. On Infinite Harmonic Functions on
the Heisenberg Group. Comm. in PDE. \textbf{2002}, \emph{27} (3\&4), 727--762.

\bibitem{BC:MC}Bonk, Mario.;Capogna, Luca. Mean Curvature Flow and the 
Isoperimetric Problem in the Heisenberg Group. Preprint.

\bibitem{C:VS}Crandall, Michael. \emph{Viscosity Solutions:  A Primer};
Lecture Notes in Mathematics 1660; Springer-Verlag: Berlin, 1997.

\bibitem{CIL:UGTVS}Crandall, Michael.; Ishii, Hitoshi.; 
Lions, Pierre-Louis.  User's Guide to Viscosity Solutions of Second Order
Partial Differential Equations. Bull. of Amer. Math. Soc. \textbf{1992}, \emph{27} 
(1), 1--67.  

\bibitem{F:SE}Folland, G.B.
Subelliptic Estimates and Function Spaces on Nilpotent Lie Groups.
Ark. Mat. \textbf{1975}, \emph{13}, 161--207.

\bibitem{FS:HSHG}Folland, G.B.; Stein, Elias M.
\emph{Hardy Spaces on Homogeneous Groups};
Princeton University Press: Princeton, NJ. 1982.

\bibitem{G:MS}Gromov, Mikhael.
\emph{Metric Structures for Riemannian and  Non-Riemannian  Spaces};
Birkhäuser Boston Inc: Boston, 1999. 

\bibitem{H:CCG}Heinonen, Juha.
Calculus on Carnot Groups.
In \emph{Fall School in Analysis Report No. 68},
Fall School in Analysis, Jyv\"{a}skyl\"{a}, 1994;  
Univ. Jyv\"{a}skyl\"{a}: Jyv\"{a}skyl\"{a}, Finland. 1995; 1--31. 

\bibitem{HH:QR}Heinonen, Juha.; Holopainen, Ilkka.
Quasiregular Maps on Carnot Groups.
J. of Geo. Anal. \textbf{1997},  \emph{7} (1), 109--148.

\bibitem{H:H}Ishii, Hitoshii. Viscosity Solutions of Nonlinear Partial 
Differential Equations. Sugaku Exp. \textbf{1996}, \emph{9} (2), 135--152.

\bibitem{Ju:P}Juutinen, Petri. On the Definition of Viscosity Solutions for 
Parabolic Equations. Proc. Amer. Math. Soc. \textbf{2001}, \emph{129} (10), 
2907--2911.

\bibitem{K:LGHT}Kaplan, Aroldo.
Lie Groups of Heisenberg Type.
Rend. Sem. Mat. Univ. Politec. Torino 1983 Special Issue \textbf{1984}, 117--130.

\bibitem{St:HA}Stein, Elias M. 
\emph{Harmonic Analysis};
Princeton University Press: Princeton, NJ. 1993.

\end{thebibliography}
\end{document}