PDE Real Analysis Seminar

Date, time & place	Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s)	Yoshikazu Giga, Kazuhiro Ishige, Hiroyoshi Mitake, Tsuyoshi Yoneda
URL	https://www.math.sci.hokudai.ac.jp/coe/sympo/pde_ra/index_en.html

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
藤田安啓 (富山大学)
Asymptotic solutions and Aubry sets for Hamilton-Jacobi equations

[ Abstract ]
In this talk, we consider the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation

$u_t + \\alpha x\\cdot Du + H(Du) =f(x)$ in

${\\rm I}\\!{\\rm R}^N \\times (0,\\infty)$ , where

$\\alpha$ is a positive constant and

$H$ is a convex function on

${\\rm I} \\!{\\rm R}^N$ . We show that, under some assumptions,

$u(x,t) - ct - v(x)$ converges to

$0$ locally uniformly in

${\\rm I}\\!{\\rm R}^N$ as

$t \\to \\infty$ , where

$c$ is a constant and

$v$ is a viscosity solution of the Hamilton-Jacobi equation

$c + \\alpha x\\cdot Dv + H(Dv) = f(x)$ in

${\\rm I}\\!{\\rm R}^N$ . A set in

${\\rm I}\\!{\\rm R}^N$ , which is called the {\\it Aubry set}, gives a concrete representation of the viscosity solution

$v$ . We also discuss convergence rates of this asymptotic behavior. This is a joint work with Professors H. Ishii and P. Loreti.

Graduate School of Mathematical Sciences, The University of Tokyo

3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan TEL: 03-5465-7001 FAX: 03-5465-7011