Jose Maria Martell (ICMAT)
Title: Uniform rectifiability and elliptic operators satisfying a Carleson measure condition.
Abstract: In this talk I will study the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. Our setting is that of domains having an Ahlfors regular boundary and satisfying the so-called interior Corkscrew and Harnack chain conditions (these are respectively scale-invariant/quantitative versions of openness and path-connectivity) and we show that for the class of Kenig-Pipher uniformly elliptic operators (operators whose coefficients have controlled oscillation in terms of a Carleson measure condition) the solvability of the $L^p$-Dirichlet problem with some finite $p$ is equivalent to the quantitative openness of the exterior domains or to the uniform rectifiablity of the boundary. Joint work with S. Hofmann, S. Mayboroda, T. Toro, and Z. Zhao.
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