\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 59, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/59\hfil Hopf-type estimates for solutions] {Hopf-type estimates for solutions to Hamilton-Jacobi equations with concave-convex initial data} \author[ N. H. Tho \& T. D. Van \hfil EJDE--2003/59\hfilneg] { Nguyen Huu Tho \& Tran Duc Van} % in alphabetical order \address{Nguyen Huu Tho\hfill\break Bureau of Education and Training of Hatay, Vietnam} \address{Tran Duc Van\hfill\break Hanoi Institute of Mathematics\\ P.O. Box 631, BoHo, Hanoi, Vietnam} \email{tdvan@thevinh.ac.vn} \date{} \thanks{Submitted July 3, 2002. Published May 21, 2003.} \thanks{Partially supported by the National Council on Natural Science, Vietnam.} \subjclass[2000]{35A05, 35F20, 35F25} \keywords{Hamilton-Jacobi equations, Hopf-type formula, \hfill\break\indent global Lipschitz solutions, viscosity solutions} \begin{abstract} We consider the Cauchy problem for the Hamilton-Jacobi equations with concave-convex initial data. A Hopf-type formula for global Lipschitz solutions and estimates for viscosity solutions of this problem are obtained using techniques of multifunctions and convex analysis. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} This paper is a continuation of the works [10] and [8], where the explicit solutions via Hopf-type formulas of the Cauchy problem to the Hamilton-Jacobi equations with concave-convex hamiltonians were considered. Namely, we consider the Cauchy problem for the Hamilton-Jacobi equation \begin{gather} \frac{\partial u}{\partial t} + H(t,\frac{\partial u}{\partial x}) = 0\quad \text{in } U:=\{t>0,\; x\in\mathbb{R}^n\} \label{e1} \\ u(0,x)= \phi(x)\quad\text{on } \{t = 0,\; x\in\mathbb{R}^n\}. \label{e2} \end{gather} Here $\partial /\partial x =(\partial /\partial x_1,\dots, \partial /\partial x_n )$, the Hamiltonian $H = H(t,p)$ and $\phi = \phi (x)$ are given functions, and $u = u(t,x)$ is unknown. In this paper we shall assume that $n = n_1 + n_2$ and that the variable $x\in\mathbb{R}^n$ is separated as $x = (x', x'')$ with $x'\in\mathbb{R}^{n1}$, $x''\in\mathbb{R}^{n2}$, similarly for $p, q,\dots\in\mathbb{R}^n$. In particular, the zero-vector in $\mathbb{R}^n$ will be $0 = (0',0'')$, where $0'$ and $0''$ stand for the zero-vectors in $\mathbb{R}^{n1}$ and $\mathbb{R}^{n2}$, respectively. \noindent{\bf Definition.} A function $g = g(x', x'')$ is called concave-convex if it is concave in $x'\in\mathbb{R}^{n1}$ for each $x''\in\mathbb{R}^{n2}$ and convex in $x''\in\mathbb{R}^{n2}$ for each $x'\in\mathbb{R}^{n1}$. \noindent For results on the concave-convex functions the reader is referred to [7], [8], [10]. In [10, Chapter 10], Van, Tsuji and Thai Son proposed to examine a class of concave-convex functions in a more general framework where the discussion of the global Legendre transformation still make sense. \noindent Bardi and Faggian [2] found explicit pointwise upper and lower bounds of Hopf-type for the viscosity solutions under the following hypotheses: $H$ depends only on $p$ and is a concave-convex function given by the difference of convex functions, $$ H(p',p''):= H_1(p')-H_2(p''), $$ and $\phi$ is uniformly continuous. Also if $H\in C(\mathbb{R}^n)$ and $\phi = \phi(x)$ is concave-convex function given by special representation $\phi(x) = \phi_1(x) - \phi_2(x)$, where $\phi_1, \phi_2$ are convex and Lipschitz continuous. \noindent Barron, Jensen and Liu [3] and Van, Thanh [11] found Hopf-type estimates for viscosity solutions to the corresponding Cauchy problem when the Hamiltonian $H(\gamma ,p)$, $(\gamma ,p)\in\mathbb{R}\times\mathbb{R}^n$, is a D. C. function in $p$, i.e., $$ H(\gamma ,p) = H_1(\gamma ,p)-H_2(\gamma ,p),\quad (\gamma ,p)\in \mathbb{R}\times\mathbb{R}^n, $$ where $ H_i( \gamma,p)$, $i=1,2$, is a convex function in $p$. \noindent Ngoan [6], Thai Son [8], Van, Tsuji and Thai Son [10] obtained explicit global Lipschitz solutions and upper and lower bounds of viscosity solutions to the Hamilton-Jacobi equations with concave-convex hamiltonians via Hopf-type formulas. The aim of this paper is to look for explicit global Lipschitz solution of the Cauchy problem \eqref{e1}--\eqref{e2} and to establish pointwise upper and lower bounds of Hopf-type for viscosity solutions when the initial function $\phi = \phi (x) =\phi (x', x'')$ is concave-convex on $\mathbb{R}^{n1}\times\mathbb{R}^{n2}$. \noindent{\bf Definition.} A function $u = u(t,x)$ in $\mathop{\rm Lip}(\bar{U})$ will be called a global Lipschitz solution of the Cauchy problem \eqref{e1}--\eqref{e2} if it satisfies \eqref{e1} almost everywhere (a. e.) in $U$, with $u(0,x) = \phi (x)$ for all $x\in\mathbb{R}^n$. \section{Hopf-type formula for global Lipschitz solutions} We consider the Cauchy problem for the Hamilton-Jacobi equation \begin{gather} u_t + H(t,Du) = 0\quad \text{in } U:=\{t>0,\; x\in\mathbb{R}^n\} \label{e3} \\ u(0,x)= \phi(x)\quad \text{on } \{t = 0,\; x\in\mathbb{R}^n\}, \label{e4} \end{gather} where the Hamiltonian $H$ depends on the variable $t$ and the spatial derivatives $Du$. We note that Van, Tsuji, Hoang and Thai Son [9], [10] have obtained a Hopf-type formula with the initial function $\phi = \phi (x)$ nonconvex and $H$ merely continuous. Moreover, a global Lipschitz solution of \eqref{e3}--\eqref{e4} is given by an explicit Hopf-type formula in the following case (see Chap. 9, [10]): The Hamiltonian (depends explicitly on $t$) $H = H(t,p)$ is continuous in $U_G := \{ (t,p) : t\in (0, +\infty) \backslash G,\ p\in\mathbb{R}^n \}$ where $G$ is closed subset of $\mathbb{R}$ with Lebesgue measure zero; and, for each $N\in (0, +\infty)$ corresponds a function $g_N := g_N (t)\in L^{\infty}_{\rm loc} (\mathbb{R})$ so that $$\sup_{|p|\le N} |H(t,p)|\leq g_N (t)\quad \text{for almost } t\in (0, +\infty); $$ while the initial function $\phi = \phi (x)$ satisfies one of the following two conditions: \begin{enumerate} \item $\phi = \phi_1 - \phi_2$, where $ \phi_1, \phi_2$ are convex functions; \item $\phi$ is minimum of a family of convex functions. \end{enumerate} In this section, we look for explicit global Lipschitz solutions of problem \eqref{e3}--\eqref{e4}, where $x\in\mathbb{R}^n$, $n = n_1 + n_2,\ x = (x',x'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ and the initial-valued function $\phi = \phi (x):= \phi (x',x'')$ is a strictly concave-convex function on $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ satisfying the following conditions: \begin{gather} \lim_{| x''|\rightarrow +\infty}\frac{\phi (x',x'')}{| x''|} = +\infty\ \text{for each}\ x'\in\mathbb{R}^{n_1}, \label{e5}\\ \lim_{| x'|\rightarrow +\infty}\frac{\phi (x',x'')}{| x'|} = -\infty\ \text{for each}\ x''\in\mathbb{R}^{n_2}. \label{e6} \end{gather} We now consider the Cauchy problem \eqref{e3}--\eqref{e4} with the following hypotheses: \begin{itemize} \item[(M1)] The Hamiltonian $H = H(t,p)$ is continuous in $$ U_G:= \{(t,p): t\in (0,+\infty)\backslash G, p\in\mathbb{R}^n\} $$ with $G$ be a closed subset of $\mathbb {R}$ with Lebesgue measure 0. Moreover, for each $N\in (0,+\infty)$ there corresponds a function $g_N := g_N(t)\in L_{\rm loc}^{\infty}(\mathbb{R})$ so that $$\sup_{| p|\leq N}| H(t,p)| \leq g_N (t)\quad \text{for almost } t\in (0,+\infty); $$ \item[(M2)] The equality $$ \sup_{p''\in\mathbb{R}^{n_2}} \inf_{p'\in\mathbb{R}^{n_1}} \varphi (t,x,p) = \inf_{p'\in\mathbb{R}^{n_1}}\sup_{p''\in\mathbb{R}^{n_2}}\varphi (t,x,p) $$ is satisfied in $U$, where \begin{equation} \varphi (t,x,p) := \langle p,x\rangle - \phi^*(p) - \int_0^t H(\tau,p)d\tau \end{equation} for $(t,x) = (t,x',x'')\in U$, $p = (p',p'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$. Here, $\phi^*$ denotes the conjugate of $\phi$ which is defined as in Section 3 later. \item[(M3)] To each bounded subset $V$ of $U$ there corresponds a positive number $N(V)$ so that \begin{gather*} \underset{ q''\in\mathbb{R}^{n_2}}{\underset{| q''|\leq N(V)}\max} \inf_{q'\in\mathbb{R}^{n_1}} \varphi (t,x,q',q'') > \inf_{q'\in\mathbb{R}^{n_1}} \varphi (t,x,q',p''), \\ \underset{ q'\in\mathbb{R}^{n_1}}{\underset{| q'|\leq N(V)}\min} \sup_{q''\in\mathbb{R}^{n_2}} \varphi (t,x,q',q'') < \sup_{q''\in\mathbb{R}^{n_2}} \varphi (t,x,p',q''), \end{gather*} whenever $(t,x)\in V$, $p = (p',p'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ and $\min\{| p'|, | p''|\} > N(V)$. \end{itemize} The main result of this Section is as follows. \begin{theorem} \label{thm2.1} Let $\phi$ be a strictly concave-convex function on $\mathbb{R}^n$ with \eqref{e5}--\eqref{e6} and assume M1--M3. Then the formula \begin{equation} \label{e8} u(t,x) := \sup_{p''\in\mathbb{R}^{n_2}} \inf_{p'\in\mathbb{R}^{n_1}} \varphi (t,x,p) = \inf_{p'\in\mathbb{R}^{n_1}} \sup_{p''\in\mathbb{R}^{n_2}} \varphi (t,x,p), \end{equation} for $(t,x)\in U$, determines a global Lipschitz solution of the Cauchy problem \eqref{e3}--\eqref{e4}. \end{theorem} To prove this theorem, we need the following lemmas, which are similar to the lemmas 10.5 and 10.6 in [10]. \begin{lemma} \label{lm2.2} Let $\mathcal{O}$ be an open subset of $\mathbb{R}^m$, and $ \eta = \eta(\xi,p) = \eta(\xi,p',p'')$ be a continuous function on $\mathcal{O}\times\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ with the following properties: \begin{enumerate} \item The equality $$\sup_{p''\in\mathbb{R}^{n_2}} \inf_{p'\in\mathbb{R}^{n_1}} \eta(\xi,p) = \inf_{p'\in\mathbb{R}^{n_1}} \sup_{p''\in\mathbb{R}^{n_2}} \eta (\xi,p) $$ is satisfied in $\mathcal{O}$; \item There is a nonempty subset $E\subset\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ such that $\eta(\xi,p)$ is finite on $\mathcal{O}\times E$ and $\eta(\xi,p) \equiv -\infty$ on $\mathcal{O}\times E^c$, where $E^c = \mathbb{R}^n\setminus E$. Moreover, for each bounded subset $V$ of $\mathcal{O}$, corresponds a positive number $N(V)$ such that $$ \underset{ q''\in\mathbb{R}^{n_2}}{\underset{| q''|\leq N(V)}\max} \inf_{q'\in\mathbb{R}^{n_1}} \eta (\xi,q',q'') > \inf_{q'\in\mathbb{R}^{n_1}} \eta (\xi,q',p''), $$ and $$\underset{ q'\in\mathbb{R}^{n_1}}{\underset{| q'|\leq N(V)}\min} \sup_{q''\in\mathbb{R}^{n_2}} \eta (\xi,q',q'') < \sup_{q''\in\mathbb{R}^{n_2}}\eta (\xi,p',q''), $$ whenever $\xi\in V$, $p = (p',p'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ and $\text{min}\{| p'|,\, | p''|\} > N(V)$; \item For each fixed $p$ of $E$, $\eta = \eta (\xi,p)$ is differentiable in $\xi\in\mathcal{O}$ with continuous gradient $$ \partial\eta /\partial\xi =\ \partial\eta (\xi,p)/\partial\xi $$ on $\mathcal{O}\times\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$. \end{enumerate} Then we have: %\begin{description} \indent i. The function $$\psi = \psi(\xi) := \sup_{p''\in\mathbb{R}^{n_2}} \inf_{p'\in\mathbb{R}^{n_1}} \eta(\xi,p) = \inf_{p'\in\mathbb{R}^{n_1}}\sup_{p''\in\mathbb{R}^{n_2}} \eta (\xi,p) $$ \indent\quad is a locally Lipschitz continuous on $\mathcal{O}$. \indent ii. $\psi = \psi(\xi)$ is directionally differentiable in $\mathcal{O}$ with \begin{align*} \partial_e\psi(\xi) & = \max_{p''\in L''(\xi)} \min_{p'\in L'(\xi)} \langle\partial\eta(\xi,p',p'')/\partial\xi, e\rangle\\ & = \min_{p'\in L'(\xi)} \max_{p''\in L''(\xi)} \langle \partial\eta(\xi,p',p'')/\partial\xi, e\rangle,\quad \xi\in\mathcal{O},\; e\in\mathbb{R}^m \end{align*} \indent\quad where \begin{gather} L'(\xi) := \{p'\in\mathbb{R}^{n_1} :\sup_{p''\in\mathbb{R}^{n_2}}\eta (\xi,p',p'') = \psi(\xi)\} \label{e9}\\ L''(\xi) := \{p''\in\mathbb{R}^{n_2} :\inf_{p'\in\mathbb{R}^{n_1}}\eta (\xi,p',p'') = \psi(\xi)\}. \label{e10} \end{gather} %\end{description} \end{lemma} \vskip0.3cm \begin{lemma} Suppose that the conditions 1--2 in Lemma \ref{lm2.2} are satisfied for a continuous function $\eta = \eta(\xi,p',p'')$ on $\mathcal{O}\times\mathbb{R}^{n_1}\times\mathbb{R}^{n_2} $. Then \eqref{e9}--\eqref{e10} determines the non-empty valued, closed, locally bounded multifunction $L = L(\xi) := L'(\xi)\times L''(\xi)$, $\xi\in\mathcal{O}$. \end{lemma} \begin{proof}[Proof of Theorem \ref{thm2.1}]. We can verify that the function $$ \eta = \eta (\xi,p) := \varphi (t,x,p) $$ satisfies all the assumptions of Lemma \ref{lm2.2}, where $$ E := \mathop{\rm dom}\phi^* \neq \emptyset ,\quad m:= 1+n = 1+n_1 +n_2 ,\quad \xi:= (t,x). $$ Here we put $\mathcal{O} :=\bar U$ and conclude that $$ L(t,x) = L'(t,x)\times L''(t,x) = \{p\in E :\varphi (t,x,p) = u(t,x)\} $$ determines a nonempty-valued, locally bounded, closed multifunction $L = L(t,x)$ of $(t,x)\in\bar U$. Take arbitrary an $r\in (0,+\infty)$ and denote $$ V_r = \{(t,x)\in\bar U : t+| x| < r\},\quad N_r = N(V_r). $$ Let $g_{N_r} = g_{N_r}(t)$ as be in the condition M1. Then for any two points $(t^1,x^1)$ and $(t^2,x^2)$ are in $V_r$, we may choose an element $p = (p^{'1},p^{''2})\in L'(t^1,x^1)\times L''(t^2,x^2)$ of the nonempty set $$ L'(t^1,x^1)\times L''(t^2,x^2)\subset \bar{B}^{n_1}(0',N_r)\times \bar{B}^{n_2}(0'',N_r) $$ and get \begin{align*} u(t^2,x^2) - u(t^1,x^1) &=\inf_{p'\in\mathbb{R}^{n_1}} \varphi (t^2,x^2,p',p^{''2}) - \sup_{p''\in\mathbb{R}^{n_2}} \varphi (t^1,x^1,p^{'1},p'')\\ &\leq \varphi(t^2,x^2,p^{'1},p^{''2}) - \varphi (t^1,x^1,p^{'1},p^{''2})\\ & = \varphi(t^2,x^2,p) - \varphi (t^1,x^1,p)\\ & = \langle p, x^2 - x^1 \rangle + \int_{t_2}^{t_1} H(\tau,p) d\tau\\ &\leq N_r\cdot| x^2 - x^1| + s_r\cdot| t^2 - t^1| \end{align*} where $s_r = \mathop{\rm ess\,sup}{}_{t\in (0,r)} g_{N_r}(t)$. Dually, $$ u(t^1,x^1) - u(t^2,x^2)\leq N_r\cdot| x^2 - x^1| + s_r\cdot| t^2 - t^1|. $$ Hence, $u = u(t,x)$ is a locally Lipschitz continuous in $\bar U$ and thus it be long to $Lip (\bar U)$. Next, let $e^o := (1,0,0,\dots,0,0)$, $e^1 := (0,1,0,\dots,0,0)$, \dots, $e^n := (0,0,0,\dots,0,1)\in\mathbb{R}^{n+1}$. We now replace in Lemma \ref{lm2.2} the set $\mathcal{O} := U_G$. From this lemma we see that $u = u(t,x)$ is directionally differentiable in $U_G$ with \begin{align*} \partial_{e^o} u(t,x) &=\max_{p''\in L''(t,x)} \min_{p'\in L'(t,x)}\{- H(t,p),\, p\in L(t,x)\}\\ &= \min_{p'\in L'(t,x)}\max_{p''\in L''(t,x)} \{- H(t,p),\, p\in L(t,x)\},\\ \partial_{- e^o} u(t,x) &= \max_{p''\in L''(t,x)} \min_{p'\in L'(t,x)}\{ H(t,p),\, p\in L(t,x)\}\\ &=\min_{p'\in L'(t,x)}\max_{p''\in L''(t,x)}\{ H(t,p),\, p\in L(t,x)\}; \end{align*} and for $1\leq i\leq n$: \begin{equation} \label{e11} \begin{aligned} \partial_{e^i} u(t,x) &= \max_{p''\in L''(t,x)} \min_{p'\in L'(t,x)} \{p_i,\ p\in L(t,x)\} \\ &=\min_{p'\in L'(t,x)}\max_{p''\in L''(t,x)}\{p_i,\ p\in L(t,x)\},\\ \partial_{- e^i} u(t,x) &=\max_{p''\in L''(t,x)} \min_{p'\in L'(t,x)}\{ - p_i,\ p\in L(t,x)\}\\ &=\min_{p'\in L'(t,x)}\max_{p''\in L''(t,x)}\{ - p_i,\ p\in L(t,x)\}. \end{aligned} \end{equation} Since $u = u(t,x)$ is locally Lipschitz continuous in $\bar U$, according to Rademacher's Theorem, there exists a set $\mathcal{Q}\subset U$ of ($(n+1)$ dimensional) Lebesgue measure $0$ such that $u = u(t,x)$ is differentiable with \[ \frac{\partial u(t,x)}{\partial t} = \partial_{e^o} u(t,x) = - \partial_{-e^o} u(t,x), \] \begin{equation} \label{e12} \frac{\partial u(t,x)}{\partial x_i} = \partial_{e^i} u(t,x) = - \partial_{-e^i} u(t,x) \end{equation} at any point $(t,x)\in U\setminus\mathcal{Q}$. Hence, \eqref{e11}--\eqref{e12} show that the equalities for $1\leq i\leq n$, \begin{align*} \frac{\partial u(t,x)}{\partial x_i} &=\max_{p''\in L''(t,x)} \min_{p'\in L'(t,x)}\{p_i,\ p\in L(t,x)\}\\ &=\min_{p'\in L'(t,x)} \max_{p''\in L''(t,x)}\{p_i,\ p\in L(t,x)\}\\ &=\min_{p''\in L''(t,x)} \max_{p'\in L'(t,x)}\{p_i,\ p\in L(t,x)\}\\ &=\max_{p'\in L'(t,x)} \min_{p''\in L''(t,x)}\{p_i,\ p\in L(t,x)\} \end{align*} hold for all $(t,x)\in U\setminus\{\mathcal{P} := (G\times\mathbb{R}^n)\cup\mathcal{Q}\} =: U_{\mathcal{P}}$, this implies $$ L(t,x) = \big\{\frac{\partial u(t,x)}{\partial x}\big\},\quad (t,x)\in U_{\mathcal{P}};$$ and we obtain $$ \frac{\partial u(t,x)}{\partial t} = \{- H(t,p),\ p\in L(t,x)\}. $$ Thus, $$\frac{\partial u(t,x)}{\partial t} + H(t,\frac{\partial u(t,x)}{\partial x}) = - H(t,\frac{\partial u(t,x)}{\partial x}) + H(t,\frac{\partial u(t,x)}{\partial x}) = 0$$ hold almost everywhere in $U$. Furthermore \begin{align*} u(0,x) &= u(0,x',x'') \\ &= \sup_{p''\in\mathbb{R}^{n_2}} \inf_{p'\in\mathbb{R}^{n_1}}\{\langle p',x'\rangle + \langle p'',x''\rangle - \phi^*(p',p'')\}\\ &= \inf_{p'\in\mathbb{R}^{n_1}} \sup_{p''\in\mathbb{R}^{n_2}}\{\langle p',x'\rangle + \langle p'',x''\rangle - \phi^*(p',p'')\}\\ &= \bigl(\phi^*(p',p'')\bigr)^* = \phi (x',x'') = \phi (x) \end{align*} for all $x = (x',x'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$. From what has already been proved, we conclude that $u = u(t,x)$ is a global Lipschitz solution of the Cauchy problem \eqref{e3}--\eqref{e4}. \end{proof} \begin{remark} \label{rmk1} \rm If $n_2 = 0$, we obtain the Hopf-type formulas of the Cauchy problem for the convex initial data as in Chapter 8 [10]. \end{remark} \begin{remark} \label{rmk2} \rm Assume (M1), (M2). Then (M3) is satisfied if $$ \underset{p'\in\mathbb{R}^{n_1}}{\inf}\varphi (t,x,p',p'')\rightarrow -\infty\quad \text{locally uniformly in $(t,x)\in\bar{U}$ as $| p''|\rightarrow +\infty$} $$ and $$ \underset{p''\in\mathbb{R}^{n_2}}{\sup}\varphi (t,x,p',p'')\rightarrow +\infty\quad \text{locally uniformly in $(t,x)\in\bar{U}$ as $| p'|\rightarrow +\infty$} $$ i.e, if the following statement holds:\\ For any $\lambda$ and $\mu\in \mathbb{R}$ and any bounded subset $V$ of $\bar U$, there exists positive numbers $N(\lambda,V)$ and $N(\mu,V)$, respectively, so that $$ \inf_{q'\in\mathbb{R}^{n_1}}\varphi (t,x,q',p'') < \lambda\quad \text{whenever } (t,x)\in V,\; | p''| >\ N(\lambda,V) $$ and $$ \sup_{q''\in\mathbb{R}^{n_2}}\varphi (t,x,p',q'') > \mu\quad \text{whenever } (t,x)\in V,\; | p'| >\ N(\mu,V). $$ Indeed, fix an arbitrary $q^0 = (q^{0'},q^{0''})$ in the domain of $\phi^*$, which is not empty. Since the finite function $\bar{U}\ni (t,x)\mapsto\varphi (t,x,q^0)$ is continuous, it follows that: for any bounded subset $V$ of $\bar U$, \begin{gather*} \lambda_V:= \inf_{(t,x)\in V}\varphi (t,x,q^0) > -\infty,\\ \mu_V:= \sup_{(t,x)\in V} \varphi (t,x,q^0) <+\infty. \end{gather*} Under the hypothesis above, we certainly find a number $N(\lambda,V) \geq\ | q^{0''}|$ (for each such $V$) so that $$ \inf_{q'\in\mathbb{R}^{n_1}}\varphi (t,x,q',p'') < \lambda_V = \inf_{(t,x)\in V}\varphi (t,x,q^{0'},q^{0''}) $$ when $(t,x)\in V$ and $| p''|> N(\lambda,V)$, $$\inf_{q'\in\mathbb{R}^{n_1}} \varphi (t,x,q',p'') <\varphi (t,x,q^{0'},q^{0''}) $$ when $(t,x)\in V$, $| p''| >\ N(\lambda,V)$, $$\inf_{q'\in\mathbb{R}^{n_1}}\varphi (t,x,q',p'') <\inf_{q'\in\mathbb{R}^{n_1}} \varphi (t,x,q',q^{0''}) $$ when $(t,x)\in V$, $| p''| > N(\lambda,V)$, $$\inf_{q'\in\mathbb{R}^{n_1}} \varphi (t,x,q',p'') < \underset{ q''\in\mathbb{R}^{n_2}}{\underset{| q''|\leq N(\lambda,V)}{\max}} \inf_{q'\in\mathbb{R}^{n_1}} \varphi (t,x,q',q'') $$ when $(t,x)\in V$, $| p''|\ >\ N(\lambda,V)$. Analogously, we also obtain $$ \sup_{q''\in\mathbb{R}^{n_2}}\varphi (t,x,p',q'') > \underset{ q'\in\mathbb{R}^{n_1}}{\underset{| q'|\leq N(\mu,V)}{\min}} \sup_{q''\in\mathbb{R}^{n_2}} \varphi (t,x,q',q'') $$ when $(t,x)\in V$, $| p'| > N(\mu,V)$, where $N(\mu,V)\geq| q^{0'}|$. Hence (M3) is satisfied. \end{remark} \section{Hopf-type estimates for viscosity solutions} Consider the Cauchy problem for the Hamilton-Jacobi equation \begin{gather} \frac{\partial u}{\partial t} + H(\frac{\partial u}{\partial x}) = 0\quad \text{in } U:=\{t>0,\, x\in\mathbb{R}^n\} \label{e13} \\ u(0,x)= \phi(x)\quad \text{on } \{t = 0,\, x\in\mathbb{R}^n\}. \label{e14} \end{gather} When $H = H(p)$ is continuous and $\phi = \phi (x)$ is uniformly continuous, the Cauchy problem \eqref{e13}--\eqref{e14} has a unique viscosity solution $u = u(t,x)$ which is in the space of continuous functions that are uniformly continuous in $x$ uniformly in $t$, $UC_x([0, +\infty)\times\mathbb{R}^n)$ (see [5]). We also refer the readers to [4,5] for the definition and properties of viscosity solutions. In the case of Lipschitz continuous and convex (or concave) initial data $\phi$ and merely continuous Hamiltonian $H$, or for convex $\phi$ and Lipschitz continuous $H$, the formula $$ u(t,x) = \sup_{p\in\mathbb{R}^n} \{ \langle p,x\rangle - \phi^*(p) - tH(p)\} $$ determines a (unique) viscosity solution $u = u(t,x)\in UC_x([0, +\infty)\times\mathbb{R}^n)$ of the problem \eqref{e13}--\eqref{e14}. Here $\phi^*$ denotes the Legendre transform of $\phi$ (see, [1,2]). In this section we are interested in giving explicit pointwise upper and lower bounds for viscosity solutions where the initial function $\phi = \phi (x',x'')$ is concave-convex. First, we rewrite some main results on the conjugate of the concave-convex functions (for the details, see [10, Chapter 10]). Let $\phi = \phi (x',x'')$ is a concave-convex function on $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$. Then \begin{gather*} \phi^{*1}(p',x'') = \inf_{x'\in\mathbb{R}^{n_1}}\{\langle x',p'\rangle - \phi (x',x'')\}\\ \bigl(\text{resp. } \phi^{*2}(x',p'') = \sup_{x''\in\mathbb{R}^{n_2}} \{\langle x'',p''\rangle - \phi (x',x'')\}\bigr) \end{gather*} is the Fenchel conjugate of $x'$-concave (resp. $x''$-convex) function $\phi (x',x'')$. If $\phi = \phi (x',x'')$ is concave-convex function with conditions \eqref{e5}--\eqref{e6}, then $\phi^{*1}(p',x'')$ (resp. $\phi^{*2}(x',p'')$) is concave (resp. convex) not only in $p'\in\mathbb{R}^{n_1}$ (resp. $p''\in\mathbb{R}^{n_2}$) but also in the whole variable $(p',x'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ (resp. $(x',p'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$) and $$ \lim_{| p'|\rightarrow +\infty}\frac{\phi^{*1}(p',x'')}{| p'|} = -\infty\quad (\text{resp. } \lim_{| p''|\rightarrow +\infty}\frac{\phi^{*2}(x',p'')}{| p''|} = +\infty) $$ locally uniformly in $x''\in\mathbb{R}^{n_2}$ (resp. $x'\in\mathbb{R}^{n_2}$). Besides the Fenchel {``partial conjugate''} $\phi^{*1}$ and $\phi^{*2}$, we consider two ``total conjugate'' of $\phi$: \begin{align*} \bar{\phi^*}(p',p'')&=\inf_{x'\in\mathbb{R}^{n_1}}\{\langle x',p'\rangle + \phi^{*2}(x',p'')\}\\ &=\inf_{x',\in\mathbb{R}^{n_1}}\sup_{x''\in\mathbb{R}^{n_2}}\{\langle x',p'\rangle + \langle x'',p''\rangle - \phi (x',x'')\} \end{align*} and \begin{align*} {\underline\phi}^*(p',p'')&=\sup_{x''\in\mathbb{R}^{n_2}}\{\langle x'',p''\rangle + \phi^{*1}(p',x'')\}\\ &=\sup_{x''\in\mathbb{R}^{n_2}}\inf_{x'\in\mathbb{R}^{n_1}}\{\langle x',p'\rangle + \langle x'',p''\rangle - \phi (x',x'')\}. \end{align*} Therefore, the functions ${\bar\phi}^*$ and ${\underline\phi}^*$ are usually called the upper and lower conjugate, respectively, of $\phi$. Note that $$ {\underline\phi}^*\leq{\bar\phi}^*. $$ These functions are also concave-convex, and with \eqref{e5}--\eqref{e6} they coincide. In this situation, the Fenchel conjugate $$ \phi^* := {\bar\phi}^* = {\underline\phi}^* $$ of $\phi$ will simultaneously have the properties \begin{gather*} \lim_{| p''|\rightarrow +\infty}\frac{\phi^*(p',p'')}{| p''|} = +\infty\quad \text{for each } p'\in\mathbb{R}^{n_1}\\ \lim_{| p'|\rightarrow +\infty}\frac{\phi^*(p',p'')}{| p'|} = -\infty\quad \text{for each } p''\in\mathbb{R}^{n_2}. \end{gather*} If \eqref{e5}--\eqref{e6} are not assumed, the partial conjugates $\phi^{*1}$ and $\phi^{*2}$ are still concave and convex, respectively, but might be infinite somewhere, then the lower and upper conjugates ${\underline\phi}^*$ and ${\bar\phi}^*$ might not coincide. One can claim only that \begin{gather*} \phi^{*1}(p',x'') < +\infty,\quad \forall (p',x'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2},\\ \phi^{*2}(x',p'') > -\infty,\quad \forall(x',p'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}. \end{gather*} Now let \begin{gather*} D_1 := \{p'\in\mathbb{R}^{n_1} : \phi^{*1}(p',x'') > -\infty\; \forall x''\in\mathbb{R}^{n_2}\}, \\ D_2 := \{p''\in\mathbb{R}^{n_2} : \phi^{*2}(x',p'') < +\infty\; \forall x'\in\mathbb{R}^{n_1}\}, \end{gather*} hence for all $x''\in\mathbb{R}^{n_2}$, $\phi^{*1}(p',x'')$ is finite on $D_1$, and for all $x'\in\mathbb{R}^{n_1}$, $\phi^{*2}(x',p'')$ is finite on $D_2$. We now consider the Cauchy problem \eqref{e13}--\eqref{e14} with the hypothesis: \begin{itemize} \item[(M4)] The Hamiltonian $H=H(p)$ is continuous and the initial function $\phi = \phi (x',x'')$ is concave-convex and Lipschitz continuous (without \eqref{e5}--\eqref{e6}). \end{itemize} For $(t,x)\in U$, we set \begin{gather} u_{-}(t,x) := \sup_{p''\in D_2}\inf_{p'\in\mathbb{R}^{n_1}} \{ \langle p,x\rangle - {\bar\phi}^*(p) - tH(p)\} \label{e15}\\ u_{+}(t,x) := \inf_{p'\in D_1} \sup_{p''\in\mathbb{R}^{n_2}} \{ \langle p,x\rangle - {\underline\phi}^*(p) - tH(p)\}. \label{e16} \end{gather} \begin{remark} \label{rmk3} The concave-convex function $\phi = \phi (x',x'')$ is Lipschitz continuous in the sense: $\phi (x',x'')$ is Lipschitz continuous in $x'\in\mathbb{R}^{n_1}$ for each $x''\in\mathbb{R}^{n_2}$ and in $x''\in\mathbb{R}^{n_2}$ for each $x'\in\mathbb{R}^{n_1}$. \end{remark} Our estimates for viscosity solutions in this section read as follows: \begin{theorem} \label{thm3.1} Assume (M4). Then the unique viscosity solution $u = u(t,x)\in UC_x\bigl([0, +\infty)\times\mathbb{R}^n\bigr)$ of the Cauchy problem \eqref{e13}--\eqref{e14} satisfies on $\bar{U}$ the inequalities $$ u_{-}(t,x)\leq\ u(t,x)\leq\ u_{+}(t,x), $$ where $u_{-} $ and $u_{+}$ are defined by \eqref{e15} and \eqref{e16} respectively. \end{theorem} \begin{proof} For each ${\underline{p}}'\in D_1$, let \begin{align*} \Phi (x;{\underline{p}}')&= \Phi (x',x'';{\underline{p}}') := \langle x',{\underline{p}}'\rangle - \phi^{*1}({\underline{p}}',x'')\\ &=\langle x',{\underline{p}}'\rangle -\inf_{x'\in\mathbb{R}^{n_1}} \bigl\{ \langle x',{\underline{p}}'\rangle - \phi (x',x'')\bigr\}\\ &\geq\phi (x',x'')\quad \text{for all } (x',x'')\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}. \end{align*} Since $\phi^{*1}({\underline{p}}',.)$ is a concave and finite, so $-\phi^{*1}({\underline{p}}',.)$ is convex and finite, it is convex and Lipschitz continuous function; therefore , $\Phi (x;{\underline{p}}')$ is convex and Lipschitz continuous with its Fenchel conjugate given by \begin{align*} \Phi^*(p ;{\underline{p}}') & = \Phi^* (p',p'';{\underline{p}}') =\sup_{x\in\mathbb{R}^n}\bigl\{ \langle x,p\rangle - \Phi (x, {\underline{p}}')\bigr\}\\ &= \sup_{x\in\mathbb{R}^n}\bigl\{ \langle x',p'\rangle + \langle x'',p''\rangle - \langle x',{\underline{p}}'\rangle + \phi^{*1}({\underline{p}}',x'')\bigr\}\\ &=\begin{cases} +\infty & \text{if }\ (p',p'')\neq ({\underline{p}}',p'')\\ {\underline\phi}^*({\underline{p}}',p'')& \text{if } (p',p'') = ({\underline{p}}',p''). \end{cases} \end{align*} Next, consider the Cauchy problem \begin{gather*} \frac{\partial v}{\partial t} + H(\frac{\partial v}{\partial x} ) =0\quad \text{in } U =\{ t >0,\, x\in\mathbb{R}^n\},\\ v(0,x) = \Phi (x ;{\underline p}')\quad \text{on }\{ t = 0,\, x\in\mathbb{R}^n\}. \end{gather*} This is the Cauchy problem with the continuous Hamiltonian $H = H(p)$ and the convex and Lipschitz continuous initial function $\Phi = \Phi (x ;{\underline p}')$ for each ${\underline p}'\in D_1$, its unique viscosity solution $v = v(t,x)\in UC_x\bigl([0,+\infty)\times\mathbb{R}^n\bigr)$ is given by \begin{align*} v(t,x)&= \sup_{p\in\mathbb{R}^n}\{ \langle p,x \rangle - \Phi^*(p ;{\underline p}') - tH(p)\}\\ &= \sup_{p''\in\mathbb{R}^{n_2}}\{\langle {\underline p}',x'\rangle + \langle p'',x''\rangle - {\underline\phi}^*({\underline p}',p'') - tH({\underline p}',p'')\} \end{align*} with the initial condition $$ v(0,x) = \Phi (x ;{\underline p}')\geq \phi(x) = u(0,x) $$ for each ${\underline p}'\in D_1$ (see [1]). Hence, for each ${\underline p}'\in D_1,\ v = v(t,x)$ is a (continuous) supersolution of the problem \eqref{e13}--\eqref{e14} (according to a standard comparison theorem for unbounded viscosity solutions (see [5])), that means $$ u(t,x) \leq v(t,x)\quad \text{for each } {\underline p}'\in D_1, $$ and then \begin{gather*} u(t,x)\leq\underset{p'\in D_1}{\inf} \sup_{p''\in\mathbb{R}^{n_2}}\{ \langle p,x\rangle - {\underline\phi}^*(p) - tH(p)\}\\ u(t,x)\leq u_{+}(t,x) \quad \text{on }\bar{U}. \end{gather*} Dually, we also abtain $u(t,x)\geq u_{-}(t,x)$ on $\bar{U}$. Therefore, Theorem \ref{thm3.1} has been proved. \end{proof} \begin{corollary} \label{coro3.2} Assume (M1), (M2) for the case when $H(t,p)$ is not depending on $t$. Moreover, assume that $\phi = \phi (x',x'')$ is concave-convex and Lipschitz continuous function on $\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}$ and satisfies the conditions \eqref{e5}--\eqref{e6}. Then \eqref{e8} determines the unique viscosity solution $u(t,x)\in UC_x\bigl([0,+\infty)\times\mathbb{R}^n\bigr)$ of the Cauchy problem \eqref{e13}--\eqref{e14}. \end{corollary} \begin{proof} Since $\phi = \phi (x',x'')$ is a concave-convex and Lipschitz continuous function so dom$\phi^*$ is a bounded and nonempty set. Independently of $(t,x)\in\bar U$, it follows that \begin{gather*} \varphi (t,x,p',p'')\to -\infty \quad \text{whenever $| p''|$ is large enough}\\ \varphi (t,x,p',p'')\to +\infty \quad \text{whenever $| p'|$ is large enough}. \end{gather*} From Remark \ref{rmk2} implies that hypothesis (M3) hold. Then the conclusion folows from Theorem \ref{thm3.1}. \end{proof} \begin{thebibliography}{00} \bibitem{ba-ev} Bardi, M. and Evans, L.C., \ \emph{On Hopf's formulas for solutions of Hamilton-Jacobi equations,} Nonlinear Anal., {\bf 8} (1984), 1373 - 1381. \bibitem{ba-f} Bardi, M. and Faggian, S., \emph{ Hopf-type estimates and formulas for non-convex, non-concave Hamilton-Jacobi equations,} SIAM J. Math. Anal., {\bf 29} (1998), 1067 - 1086. \bibitem{baj1} Barron, E. N., Jensen, R., and Liu, W., \emph{Applications of the Hopf-Lax formula for $u_t + H(u, Du) = 0$,} SIAM J. Math. Anal.,{\bf 29} (1998), 1022 - 1039. \bibitem{C-E-L]} Crandall, M. G., Evans, L. C., and Lions, P. L., \emph{ Some properties of viscosity solutions of Hamilton-Jacobi equations,} Trans. Amer. Math. Soc., {\bf 282} (1984), 487-502. \bibitem{ish} Ishii, H., \emph{ Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations,} Indiana Univ. Math. J., {\bf 33} (1984), 721-748. \bibitem{ng} Ngoan, H. T., \emph{ Hopf's formula for Lipschitz solutions of Hamilton-Jacobi equations with concave-convex Hamitonian,} Acta Math. Vietn.,{\bf 23} (1998), 269 - 294. \bibitem{roc} Rockafellar, R. I., Convex analysis, \emph{ Princeton University Press, Princeton, New Jerssey,} 1970. \bibitem{thas} Thai Son, N. D., \emph{ Hopf-type estimates for viscosity solutions to concave-convex Hamilton-Jacobi equation,} Tokyo J. Math., {\bf 24}. No. 1 (2001), 231 - 243. \bibitem{vht} Van, T. D., Hoang, N., and Tsuji, M., \emph{ On Hopf's formula for Lipschitz solutions of the Cauchy problem for Hamilton-Jacobi equations,} Nonlinear Analysis, Theory, Methods\,\&\, Applications, {\bf 29} (1997), 1145-1159. \bibitem{vtt} Van, T. D., Tsuji, M., and Thai Son, N. D., \emph{The Characteristic method and its generalizations for first-order nonlinear partial differential equations}, Chapman \& Hall, CRC Press, 2000. \bibitem{vt}Van, T. D., and Thanh, M. D., \emph{On explicit viscosity solutions to nonconvex-nonconcave Hamilton-Jacobi equations,} Acta Math. Vietn., {\bf 26} (2001), 395-405. \end{thebibliography} \end{document}