Joint BCAM-UPV/EHU Analysis and PDE seminar: Degenerate Boundary Value Problems

In Boundary Value Problems (BVPs) one aims to understand solutions to a differential equation (the problem) under some constraint (value at the boundary).

On euclidean space, BVPs can be attacked with Fourier methods: the Fourier transform provides us with a representation of the solution and a characterisation of the trace space of the solutions. What happens when the boundary of our domain becomes rough and symmetries are lost? Can we still find a way to describe solutions and trace space when Fourier methods break down? Ultimately: how do solutions (and these methods) depend on small perturbations of the boundary? 

In this talk I will introduce the "first order approach" for divergence form equations  -div A ∇u = 0, which relates harmonic extensions from the real line and holomorphic functions. This relation works in higher dimensions as well, and allows us to rewrite our problem in a suitable way so that holomorphic functional calculus can be applied.

We will take a closer look at degenerate BVPs: when the coefficient A(x) of our divergence form equation lacks uniform boundedness and accretivity, and can exhibit singularities. Current state-of-the-art results can handle singularities characterised by scalar Muckenhoupt weights. These results have been recently extended on manifolds satisfying some curvature assumption. But even on flat euclidean space, anisotropic degenerate coefficients have been out of reach, due to the lack of off-diagonal estimates. Is there another way to handle more general matrix-degenerate coefficients? How far can we push the new methods before the theory falls apart?

New results are from a joint work with Andreas Rosén.