The minimal regularity Dirichlet problem for elliptic PDEs beyond symmetric coefficients - Note Thursday!
Analysis and Geometry Seminar
8th November 2018, 3:00 pm – 4:00 pm
Howard House, 4th Floor Seminar Room
We will begin with an overview of the classical construction of harmonic measure followed by the relationship between the $A_\infty$-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 nonsymmetric coefficients on Lipschitz domains is solvable when the boundary data is in $L^p$ for some $p2$ 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 $A_\infty$ with respect to a weighted Lebesgue measure on the domain boundary. The Carleson measure estimate allows us to avoid applying the method of $\epsilon$-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients.
This is joint work with Steve Hofmann and Phi Le.
Note: Thursday and on 4th floor