Analysis Seminar: The Dirichlet problem as the boundary of the Poisson problem.

Abstract: We will describe a novel approximation result about how solutions to the Dirichlet problem for second-order real elliptic PDEs with boundary data in  Lp on rough domains may be globally approximated, up to the boundary in a precise sense, by a family of solutions to certain corresponding inhomogeneous Poisson problems with null boundary data. This result is new even for the Laplacian on the unit ball, but is shown in high geometric generality, as well as with minimal assumptions on the coefficients.

This approximation result was inspired by our second main result: we fully characterize the dual space to the space of functions whose Kenig-Pipher modified non-tangential maximal function lies in Lp, answering a question of Hytonen and Rosen. We show that the dual space consists of exactly two components, and each of them correspond to the Banach spaces in which the Dirichlet and Poisson problems are solved with control of the Lp norm of the non-tangential maximal function. Moreover, one component is the weak-* boundary of the other component. This relationship at the level of the data in fact "lifts" to the level of solution spaces to the boundary value problems, at least partially, allowing us to make the following statement precise: the Dirichlet problem lies on the boundary of the Poisson problem, in a certain topology.

These results may be thought of as deeper instances of the robust relationship between (singular) boundary value problems and the inhomogeneous problems recently investigated in the literature. We start our talk with a discussion of these results. This is joint work with Mihalis Mourgoglou.

Host: Alan Chang