The 19th : PDE Real Analysis Seminar

Organizers :: H.Arai (University of Tokyo) Y.Giga (University of Tokyo/Hokkaido University)
Board Members :: H.Ishii (Waseda.U) T.Kawazoe (Keio.U) N.Kenmochi (Chiba.U) M.Sakai (Tokyo Metropolitan.U) Y.Shibata (Waseda.U) K.Mochizuki (Chuo.U) A.Miyachi(Tokyo Woman's Christian.U) M.Yamazaki (Waseda.U)

Place :: Graduate School of Mathematical Sciences the University of Tokyo, Room #056

Programme :

Professor Yasuhiro Fujita (Toyama University): "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.

The 19th PDE Real Analysis Seminar