## Colloquium

Organizer(s) Tomohide Terasoma http://www.ms.u-tokyo.ac.jp/seminar/colloquium/index_e.html

Next seminar

### 2018/11/30

15:30-16:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Hiroyoshi Mitake (The University of Tokyo)
The theory of viscosity solutions and Aubry-Mather theory
(日本語)
[ Abstract ]
In this talk, we give two topics of my recent results.

(i) Asymptotic analysis based on the nonlinear adjoint method: Wepresent two results on the large-time behavior for the Cauchy problem, and the vanishing discount problem for degenerate Hamilton-Jacobiequations.
(ii) Rate of convergence in homogenization of Hamilton-Jacobi equations: The convergence appearing in the homogenization was proved in a famous unpublished paper by Lions, Papanicolaou, Varadhan (1987). In this talk, we present some recent progress in obtaining the optimal rate of convergence $O(¥epsilon)$ in periodic homogenization of Hamilton-Jacobi equations. Our method is completely different from previous pure PDE approaches which only provides $O(¥epsilon^{1/3})$. We have discovered a natural connection between the convergence rate and the underlying Hamiltonian system.