九州関数方程式セミナー 平成22年度前期講演

日時	6月 4日(金) 14:45--15:45
会場	福岡大学 セミナーハウス 2階 セミナー室
講師	曽我 幸平 氏 (早稲田大)
題目	The Aubry-Mather theory to difference Hamilton-Jacobi equations
概要	We consider periodically time-dependent Hamiltonian systems with one degree of freedom, the corresponding Hamilton-Jacobi equations and scalar conservation laws. The relation among them is made clear by Albert Fathi and Weinan E. In this talk we focus our attention on numerical aspects of the issue. First we see results of difference approximation of periodic entropy solutions to the conservation laws and apply them to the computation of the Aubry-Mather sets. Then we translate the approximation into that of viscosity solutions of the Hamilton-Jacobi equations and find Aubry-Mather theory like relation among these approximate objects. The key tool of our arguments is a stochastic and variational representation of difference solutions which corresponds to the variational representation of viscosity solutions by the value function.

日時	6月 4日(金) 16:00--17:00
会場	福岡大学 セミナーハウス 2階 セミナー室
講師	A. Fathi 氏 (Ecole Norm. Sup. Lyon)
題目	Smoother critical subsolutions of the Hamilton-Jacobi Equation
概要	We will be interested in C$^1$ globally defined and periodic functions $u$ which satisfy $H(x,d_xu)\leq c$ everywhere with the smallest possible $c$. These are called critical subsolutions. Usually there does not exist C$^2$ critical subsolutions. We will explain how Denjoy theory puts in two dimensions strong restrictions on the existence of smoother critical subsolutions. We will also show that (in all dimensions) the obstruction to find smoother subsolutions is localized at the neighborhood of the Aubry set.