In recent work with Guy David we introduce the notion of almost
minimizer for a series of functionals previously studied by Alt-Caffarelli
and Alt-Caffarelli-Friedman.

We prove regularity results for these almost minimizers and explore the
structure of the corresponding free boundary. A key ingredient in the study
of the 2-phase problem is the existence of almost monotone quantities. The
goal of this talk is to present these results in a self-contained manner,
emphasizing both the similarities and differences between minimizers and
almost minimizers.

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on joint work with Gui-Qiang Chen.

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.

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.