In this talk we will present various results on the size of the nodal (zero) set for solutions of partial differential equations of elliptic and parabolic type. In particular, we will establish a sharp upper bound for the (n-1)-dimensional Hausdorff measure of the nodal sets of the eigenfunctions of regular analytic elliptic problems. We will also show certain more recent results concerning semilinear equations (e.g. Navier-Stokes equations) and equations with non-analytic coefficients.
In recent work with Guy David we introduce the notion of almost
minimizer for a series of functionals previously studied by Alt-Caffarelli
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
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.
The theory of Hamilton-Jacobi equations in Hilbert and some
Banach spaces is relatively well developed. Much less is known about equations in spaces of measures, and more general metric spaces. We will present a notion of metric viscosity solution which applies to a class of Hamilton-Jacobi equations in geodesic metric spaces and gives well posedness for such equations. We will also discuss other approaches to Hamilton-Jacobi equations in metric spaces, in particular in the Wasserstein space, and discuss some applications of such equations.