PDE Real Analysis Seminar

Seminar information archive ~04/21Next seminarFuture seminars 04/22~

Date, time & place Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.)


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.
[ Reference URL ]