The minimal regularity Dirichlet problem for elliptic PDEs beyond symmetric coefficients

Andrew Morris (Birmingham). Part of the analysis and applications seminar series

We will begin with an overview of the classical construction of harmonic measure followed by the relationship between the Α-property of this measure and solvability of the Dirichlet problem. We will then discuss a recent proof that the Dirichlet problem for degenerate elliptic equations with non-symmetric coefficients on Lipschitz domains is solvable when the boundary data is in Lp for some p<∞. The result is achieved without requiring any structure on the coefficient matrix, thus allowing for coefficients that are not symmetric, in which case p>2 becomes necessary.

More specifically, the coefficients are only assumed to be measurable, real-valued and independent of the transversal direction to the boundary, with a degenerate bound and ellipticity controlled by a Muckenhoupt weight. The result is achieved by obtaining a Carleson measure estimate on bounded solutions. This is combined with an oscillation estimate in order to deduce that the degenerate harmonic measure is in Α with respect to a weighted Lebesgue measure on the domain boundary. The Carleson measure estimate allows us to avoid applying the method of ε-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients.

This is joint work with Steve Hofmann and Phi Le.