Yamada, Naoki

Kyoto University Research Information Repository

TitleApplication of the Aubry-Mather theory to a system ofHamilton-Jacobi equations with unilateral implicit obstacles(Viscosity Solutions of Differential Equations and RelatedTopics)Author(s)Yamada, NaokiCitation数理解析研究所講究録 (2009), 1651: 1-9Issue Date2009-05URL http://hdl.handle.net/2433/140793RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityApplication of the Aubry-Mather theory to asystem of Hamilton-Jacobi equations withunilateral implicit obstaclesN. Yamada (Fukuoka University)1 IntroductionIn this note we describe a recent development of the theory of viscosity solu-tions to a system of Hamilton-Jacobi equations and a recent results obtainedby applying the method of Aubry-Mather theory.In section 2, we give a brief review of the theory of viscosity solutionsfor systems of Hamilton-Jacobi equations. Section 3 is devoted to the state-ment of a new result by using Aubry-Mather theory. In section 4 we give arepresentation formula for the solution of the obstacle problem for Hamilton-Jacobi equation.2 Brief historySoon after the notion of viscosity solutions are introduced to the Hamilton-Jacobi equations [5], [6], some people interested in applying this notion tothe system of equations.The main focus is to get the component-wise comparison principle forsolutions to such systems.Since the notion of viscosity solution is based on the maximum principle,the applicable system should have some structural conditions.In 1984, I. Capuzzo-Dolcetta and L. C. Evans [4] introduced a system$\max\{\lambda u^{d}-g^{d}\cdot Du^{d}-f^{d}, u^{d}-M^{d}[u]\}=0$ , $d=1,$ $\ldots,$ $m$ (1)1651 2009 1-9 1in the connection with a optimal switching problem for ordinary differentialequations. Here, we set$M^{d}[u](x)= \min_{j\neq d}\{u^{j}(x)+k(j, d)\}$for $k(j, d)>0$ are given constants.In S. M. Lenhart [15], H. Englar and S. M. Lenhart[7], they treatedthe system of Hamilton-Jacobi equations which they called weakly coupledsystem which has the form$H_{i}(x, Du_{i})+ \sum_{\ell=1}^{m}c_{k\ell}(x)u_{\ell}(x)=f_{k}(x)$ $k=1,$ $\ldots,$ $m$ .H. Ishii and S. Koike [13], [14] introduced the notion of monotone systemor quasi-monotone system for$G_{k}(x, u_{k}, Du_{k}, D^{2}u_{k})+ \sum_{j=1}^{m}d_{kj}(x)u_{j}=0$ $(k=1, \ldots, m)$ .Systems for second order equations of the form:$\min\{\max\{G_{k}(x, u_{k}, Du_{k}, D^{2}u_{k}), u_{k}-M_{k}(x, u)\}, u_{k}-N_{k}(x, u)\}=0$$\max\{G_{k}(x, u_{k}, Du_{k}, D^{2}u_{k}), u_{k}-M_{k}(x, u)\}=0$where$M_{k}(x, u)= \min\{u_{j}+g_{kj}(x)|j=1, \ldots, m, j\neq k\}$$N_{k}(x, u)= \min\{u_{j}-h_{kj}(x)|j=1, \ldots, m, j\neq k\}$are treated by the author [16] and H. Ishii[12].In these systems the monotonicity assumptions for each system work anessential role to get the comparison principle. We review for these monotonic-ity assumptions for typical two systems to compare with the main result ofthis paper.First consider the simplest weakly coupled system:$H_{1}(Du_{1})+d_{11}u_{1}+d_{12}u_{2}=f_{1}$$H_{2}(Du_{2})+d_{21}u_{1}+d_{22}u_{2}=f_{2}$2where $c_{ij}$ are some constants satisfying$d_{11}+d_{12}\geqq\delta_{0}>0$ , $d_{12}\leqq 0$$d_{21}+d_{22}\geqq\delta_{0}>0$ , $d_{21}\leqq 0$ .This is the monotonicity assumption for this system.To see how this assumption works in the proof of the uniqueness, we followthe formal argument. Let $u=(u_{1}, u_{2}),$ $v=(v_{1}, v_{2})$ be classical solutions. Wewant to show $u_{i}\leqq v_{i}(i=1,2)$ . By the contrary, we assume$(u_{1}-v_{1})(x_{0})= \max_{x,i}(u_{i}-v_{i})(x)>0$ .Using $Du_{1}(x_{0})=Dv_{1}(x_{0})$ , substitute the equation of $v_{1}$ from that of $v_{1)}$ weget$d_{11}(u_{1}-v_{1})(x_{0})+d_{12}(u_{2}-v_{2})(x_{0})=0$.Hence we have$0=d_{11}(u_{1}-v_{1})(x_{0})+d_{12}(u_{2}-v_{2})(x_{0})$$\geqq d_{11}(u_{1}-v_{1})(x_{0})+d_{12}(u_{1}-v_{1})(x_{0})$ $(by d_{12}\leqq 0)$$=(d_{11}+d_{12})(u_{1}-v_{1})(x_{0})$$\geqq\delta_{0}(u_{1}-v_{1})(x_{0})>0$ .This is the contradiction.Next, we would like to describe the monotonicity condition for the systemarising from the optimal switching. We restrict ourselves to the simplest case.Consider the following system:$\max\{H_{1}(Du_{1})+u_{1}-f_{1}, u_{1}-u_{2}-k_{1}\}=0$$\max\{H_{2}(Du_{2})+u_{2}-f_{1}, u_{2}-u_{1}-k_{2}\}=0$ .Here $k_{1},$ $k_{2}$ are positive constants. Arising $u_{i}$ in each $H_{i}(Du_{i})+u_{i}$ and thepositivity of $k_{i}$ is the monotonicity condition in this case.For the simplicity, let $u=(u_{1}, u_{2}),$ $v=(v_{1}, v_{2})$ be classical solutions andwe want to show $u_{i}\leqq v_{i}(i=1,2)$ . By the contradiction, we assume$(u_{1}-v_{1})(x_{0})= \max_{x,i}(u_{i}-v_{i})(x)>0$for some $x_{0}$ . In the following we argue at $x_{0}$ .3First note that $H_{1}(Du_{1})+u_{1}-f_{1}\leqq 0$ is always true. We have two cases.(a) The case $H_{1}(Dv_{1})+v_{1}-f_{1}=0$ :In this case, substitute two equations by using $Du_{1}(x_{0})=Dv_{1}(x_{0})$ , wehave $(u_{1}-v_{1})(x_{0})\leqq 0$ , which is a contradiction.(b) The case $v_{1}-v_{2}-k_{1}=0$ :If $v_{2}-v_{1}-k_{2}=0$ in the second equation, we get $-(k_{1}+k_{2})=0$ , whichis a contradiction. Then it must be $H_{2}(Dv_{2})+v_{2}-f_{2}=0$ . On the otherhand, combining the relations $v_{1}-v_{2}-k_{1}=0$ and $u_{1}-u_{2}-k_{1}\leqq 0$ , we have$u_{1}-u_{2}\leqq v_{1}-v_{2}$ . This implies $u_{1}-v_{1}\leqq u_{2}-v_{2}$ , which says$(u_{2}-v_{2})(x_{0})= \max_{x,i}(u_{i}-v_{i})(x)>0$ .Hence we concentrate to the second equation, which satisfies$H_{2}(Du_{2})+u_{2}-f_{2}\leqq 0$$H_{2}(Dv_{2})+v_{2}-f_{2}=0$ .From this we can get $(u_{2}-v_{2})(x_{0})\leqq 0$ , which is also a contradiction.3 Comparison results by applying Aubry-Mather theoryFirst, note that in both of above examples we use also the fact that theHamiltonian $H_{i}(p)+r$ is strictly increasing with respect to $r$ .There are lot of papers that argue the uniqueness of Hamilton-Jacobiequations without these increasing property. For the system, H. Ishii and S.Koike [13] introduced the notion of “quasi-monotone system” and prove thecomparison principle.The method using Aubry set is the one which is recently introduced byA. Fathi [9]. The Aubry set is first introduced in the connection of dynamictheory, and the relation with PDE is investigated by A. Fathi and A. Siconolfi$[$ 10$]$ , $[$ 11].We prepare some notations.Consider the equation $H(x, Du)=0$ and assume that $H(x,p)$ is convexand coercive with respect to $p$ .Assume that there exist a functions $\psi\in C^{1}$ and $f(x)\geqq 0$ such that$H(x, D\psi(x))\leqq-f(x)$ . We call the set$\mathcal{A}=\{x|f(x)=0\}$4the Aubry set of $H$ .If $\mathcal{A}=\emptyset$ , then there exists strict sub-solution of $C^{1}$ class, hence we canget the uniqueness. On the other hand, if $\mathcal{A}\neq\emptyset$ , the Aubry set plays arule as inner boundary in some sense, hence we can get information of thesolution from the value on $\mathcal{A}$ .F. Camilli and P. Loreti [2] applied these results to the system of eikonalequations. Their result is as follows: Consider the system$H_{i}(x, Du_{i})+ \sum_{j=1}^{M}c_{\dot{\tau}j}(x)(u_{i}-u_{j})=0$ , $(i=1, \ldots, M)$ .Assume that each $H_{i}$ satisfies the assumption of convexity and coerciveness.Assume also that there exist functions $\psi\in C^{1}$ and $f_{i}(x)\geqq 0$ satisfying$H_{i}(x, D\psi(x))\leqq-f_{i}(x)$ . Note that we assume that there exists common $\psi$for all $H_{i}$ . Let$\mathcal{A}_{i}=\{x|f_{i}(x)=0\}$ .Theorem 1 (F. Camilli and P. Loreti [2]) Assume that one of the fol-lowing assumption is $satisfied_{J}$ then the uniqueness of the viscosity solutionsholds;(i) $c_{ij}\geqq 0$ $(i\neq j)$ , $A=\emptyset$ $(i=1, \ldots, M)$(ii) $c_{\dot{\tau}j}>0$ $(i\neq j)$ , $\bigcap_{i=1}^{M}A=\emptyset$Note that this assumption includes the case $d_{11}=0,$ $d_{22}=0$ in theprevious example.Now we are in the position to state our result. Soon after I learned theresult of Camilli and Loreti, I asked them how about the associated result tothe system of obstacle problem.We started the joint work and obtained the following result.Consider the system$\max\{H_{i}(x, Du_{i}(x)), u_{i}(x)-(M_{i}u)(x)\}=0$ in $D,$ $i=1,$ $\ldots,$ $M$.Here,$(M_{i}u)(x)=mjn\{u_{j}ji+k_{ij}(x)\}$ .We assume that $H_{i}(x,p)$ are convex and coercive, and $k_{ij}>0(i\neq j)$ .5Assume that there exist functions $\psi\in C^{1}$ and $f_{i}(x)\geqq 0$ such that$H_{i}(x, D\psi(x))\leqq-f_{i}(x)$ . We denote the Aubry sets of $H_{i}$ by$\mathcal{A}_{\eta}\cdot=\{x|f_{i}(x)=0\}$ .Theorem 2 If sub- and super-solutions $u$ and $v$ satisfy$u_{i}\leqq v_{i}$ on $\mathcal{A}_{i}\cup\partial D(i=1, \ldots, M)$ ,then we have$u_{i}\leqq v_{i}$ in $D(i=1, \ldots, M)$ .The idea of the proof is a combination of the previous results. Firstconsider$u_{\lambda}=(\lambda u_{1}+(1-\lambda)\psi, \ldots, \lambda u_{M}+(1-\lambda)\psi)$ ,for $\lambda\in(0,1)$ . If we drive $u_{\lambda}\leqq v$ , then we let $\lambdaarrow 0$ to get the theorem.First we note that $u_{\lambda}$ is a sub-solution of the following system:$\max\{H_{i}(x, Du_{i}))u_{i}(x)-(M_{i}u)(x)\}=-f_{\lambda_{1}i}(x)$where$f_{\lambda,i}(x)=(1- \lambda)\min\{f_{i}(x), \min_{j\neq k}\{k_{i,j}(x)\}\}$ .We use the convexity of $H_{i}$ in this part.To prove $u_{\lambda}\leq v$ , assume by contradiction that this is not true. Hencethere exist $i_{0}\in\{1, \ldots , M\},$ $x_{0}\in D$ and $\delta>0$ such that$u_{\lambda,i_{0}}(x_{0})-v_{i_{0}}(x_{0})= \max_{x,i}\{u_{\lambda,i}(x)-v_{i}(x)\}=\delta$.Now we divide into two cases.(i) $v_{i_{0}}(x_{0})-(M_{i_{0}}v)(x_{0})<0$ ,(ii) $v_{i_{0}}(x_{0})-(M_{i_{0}}v)(x_{0})\geq 0$ .In the case (i), we can discuss as same as single equation and get$-f_{\lambda,i_{0}}(x_{0})<0$ .Then we can argue as usual to get a contradiction.In the case (ii), we can argue as described in the section about mono-tonicity conditions.64 A representation formulaIt is known that the Aubry set plays a role in some sense as an inner boundary.Reflecting this property, it is known a representation formula for thesolution of Hamilton-Jacobi equation by using the given value on the Aubryset.Let us introduce some notations.Consider the Hamilton-Jacobi equation with Dirichlet condition$H(x, Du)=0$ in $D\subset \mathbb{R}^{n}$ ,(2)$u(x)=g(x)$ on $\partial D$ .Let$Z(x)=\{p\in \mathbb{R}^{n}|H(x,p)\leqq 0\}$ ,$\sigma(x, q)=\sup\{p\cdot q|p\in Z(x)\}$for $x\in\overline{D},$ $q\in \mathbb{R}^{n}$ and we put$S(x, y)= \inf\{\int_{0}^{1}\sigma(\xi(s),\dot{\xi}(s))ds|\xi\in Lip(0,1),$ $\xi(0)=x,$ $\xi(1)=y\}$ .We list some properties by A. Fathi and A. Siconolfi $[$ 11]:(1) $S(x, y)\geqq 0,$ $S(x, x)=0$ ,$S(x, y)\leqq S(x, z)+S(z, y)$ ,$S(x, y)\leqq$ $M|x-y|$ .(2) $S(x, \cdot)$ is a sub-solution on $D$ .$S(x, \cdot)$ is a super-solution on $D\backslash \{x\}$ .(3) It is equivalent that $v$ is a sub-solution and that $-S(y, x)\leqq v(x)-$$v(y)\leqq S(y, x)$ .This means that $S(x, y)$ has a similar properties with distance function.Assume that continuous functions $g:\partial D\cup \mathcal{A}arrow \mathbb{R}$ satisfy the compati-bility condition $-S(y, x)\leqq g(x)-g(y)\leqq S(y, x)$ , then the solution of (2)isrepresented as$u(x)= \min_{y\in\partial D\cup A}\{g(y)+S(y, x)\}$ .7We can get a similar representation formula for a obstacle problem$\max\{H(x, Du), u-\phi\}=0$ .Theorem 3 Assume that continuous functions $g:\partial D\cup \mathcal{A}arrow \mathbb{R}$ satisfy thecompatibility $condition-S(y, x)\leqq g(x)-g(y)\leqq S(y, x)$ . Then the solutionof$\max\{H(x, Du), u-\phi\}=0$ in $D$ ,$u(x)=g(x)$ on $\partial D$satisfying $u(x)=g(x)$ on $\mathcal{A}$ is unique and is given by$u(x)= \min\{\min_{y\in\partial D\cup \mathcal{A}}\{g(y)+S(y, x), m_{\frac{in}{D}}\{\phi(x)y\in+S(y, x)\}\}$.References[1] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity So-lutions of Hamilton-Jacobi-Bellman Equations, Birkhauser, 1997.[2] F. Camilli and P. Loreti, Comparisoii results for a class of weakly coupledsystems of eikonal equations, Hokkaido J. Math., 37(2008), 301-326.[3] F. Camilli, P. Loreti and N. Yamada, Systems of convex Hamilton-Jacobiequations with implicit obstacles and the obstacle problem, to appearin Comm. Pure Appl. Anal.[4] I. Capuzzo Dolcetta and L. C. Evans, Optimal switching for ordinarydifferential equations, SIAM J. Control and optimization, 22(1984),143-161.[5] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobiequations, Tlirans. Amer. Math. Soc., 277(1983), 1-42.[6] M. G. Crandall, P. L. Lions and L. C. Evans, Some properties of vis-cosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc.,282(1984), 487-502.[7] H. Engler and S. M. Lenhart, Viscosity solutions for weakly coupled sys-tems of Hamilton-Jacobi equations, Proc. London Math. Soc., 63(1991),212-240.8[8] A. Fathi, $The\acute{o}re\grave{m}e$ KAM faible et the\’orie de Mather sur les $syste\grave{m}$eslagrangiens, C. R. Acad. Sci. Paris $Se\acute{r}$ . I Math., 324(1997), 1043-1046.[9] A, Fathi, Weak KAM Theorem in Lagrangian Dynamics, book to ap-pear,[10] A. Fathi and A. Siconolfi, Existence of $C^{1}$ critical subsolutions of theHamilton-Jacobi equation, Invent. Math., 155(2004), 363-388.[11] A. Fathi and A. Siconolfi, PDE aspects and Aubry-Mather theory forquasi-convex Hamiltonians, Calc. Var. 22(2005), 185-228.[12] H. Ishii, Perron’s method for monotone systems of second-order ellipticpartial differential equations, Differential Integral Equations, 5(1992),1-24.[13] H. Ishii and S. Koike, Viscosity solutions of monotone systems of second-order elliptic PDEs, Comm. Partial Differential Equations., 16(1991),1095-1128.[14] H. Ishii and S. Koike, Viscosity solutions of a system of nonlinearsecond-order elliptic PDEs arising in switching games, Funkcial. Ekvac.,34(1991), $143-arrow 155$ .[15] S. M. Lenhart, Viscosity solutions for weakly coupled systems of first-order partial differential equations, J. Math. Anal. Appl., 131(1988),180-193.[16] N. Yamada, Viscosity solutions for a system of elliptic inequalities withbilateral obstacle, Funkcial. Ekvac., 30(1987), 417-425.Naoki YamadaDepartment of Applied MathematicsFukuoka UniversityNanakuma, Fukuoka814-0180 Japane-mail: nyamada@fukuoka-u.ac.jp9

Application of the Aubry-Mather theory to a system of Hamilton-Jacobi equations with unilateral implicit obstacles (Viscosity Solutions of Differential Equations and Related Topics)

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Application of the Aubry-Mather theory to a system of Hamilton-Jacobi equations with unilateral implicit obstacles (Viscosity Solutions of Differential Equations and Related Topics)

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