\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 $$\label{main} u_t+F(t,p,u,\nabla u,(D^2u)^{\star})=0$$ 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 $t0$, 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 $(b,\eta,X) \in P_{\textmd{eucl}}^{2,+}u(p,t), |p-p_\tau|+|t-t_\tau|0$, 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 $$\label{cond} F(t,q,r,\tau \Upsilon,Y)-F(t,p,r,\tau \Upsilon,X) \leq \omega(d_C(p,q) +\tau\|\Upsilon(p,q)\|^2+\|X-Y\|).$$ 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 \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} 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. 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