# 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.

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