I will introduce the free boundary problem for the p-Laplacian with
emphasis on the free boundary condition. Then any uniform sub-
sequential limit is proved to solve the free boundary problem for
the infinity Laplacian.
We introduce a concept of viscosity solutions of Hamilton-Jacobi equations in metric spaces and in some cases relate it to viscosity solutions in the sense of differentials in the Wasserstein space. Our study is motivated physical systems which consist of infinitely many particles in motion (This is a joint work with Andzrej Swiech).
I will present a new PDE approach to obtain large time behavior
of Hamilton-Jacobi equations. This applies to usual Hamilton-Jacobi
equations, as well as the degenerate viscous cases, and weakly coupled
systems. The degenerate viscous case was an open problem in last 15 years.
This is the joint work with Cagnetti, Gomes, and Mitake.
In this talk, I will discuss the existence of a unique global weak solution
to the general Ericksen-Leslie system in $R^2$, which is smooth away from possiblyfinite many singular times, for any initial data. This is a joint work with Jinrui Huang and Fanghua Lin.