Analysis Seminar: "The Dirichlet problem with Sobolev boundary data for the Laplace equation in rough domains"

Abstract: In this talk I will present some recent advances on Boundary Value Problems for the Laplace operator with rough boundary data in a bounded corkscrew domain in R^{n+1} whose boundary is uniformly n-rectifiable. In particular, I will discuss the equivalence between solvability of the Dirichlet problem for the Laplacian with boundary data in L^{p′} and solvability of the regularity problem for the Laplacian with boundary data in an appropriate Sobolev space W^{1,p}, where p is in (1,2+ϵ) and 1/p+1/p′=1. As chord-arc domains satisfy the aforementioned geometric assumptions, our result answers a question posed by Carlos Kenig in 1991. This is joint work with Xavier Tolsa.

Host: Walton Green