Analysis Seminar: "Solvability of the Poisson problem with interior data in Lp Carleson spaces and its applications to the regularity problem."

Abstract: On Corkscrew domains with Ahlfors-regular boundary, we prove the equivalence of the classically considered Lp-solvability of the (homogeneous) Dirichlet problem with a new concept which we introduce, of the solvability of the inhomogeneous Poisson problem with interior data in an Lp-Carleson space (with a natural bound on the Lp norm of the non-tangential maximal function of the solution), and we study several applications.  Our main application is towards the Lq Dirichlet-regularity problem for second-order elliptic operators satisfying the Dahlberg-Kenig-Pipher condition (this is, roughly speaking, a Carleson measure condition on the square of the gradient of the coefficients), in the geometric generality of bounded Corkscrew domains with uniformly rectifiable boundaries. This solves an open problem from 2001. Other applications include: several new characterizations of the Lp-solvability of the Dirichlet problem, new non-tangential maximal function estimates for the Green's function, a new local T1-type theorem for the Lp solvability of the Dirichlet problem, new estimates for eigenfunctions, and a bridge to the theory of the landscape function (also known as torsion function). This is joint work with Mihalis Mourgoglou and Xavier Tolsa.

Host: Walton Green