Abstract

In this talk I am going to introduce the Dirichlet boundary value problem with $L^p$ boundary data for elliptic linear second order PDEs and discuss the Carleson condition as sufficient condition for solvability of this problem. In the last few years, the theory involving Carleson type conditions is continuously making progress, for example in related different problems like the real or complex valued Neumann or Regularity problem.

To start with we will motivate the Carleson condition and explain why it arises “naturally” as condition for solvability of the Dirichlet boundary value problem. Motivated by the Carleson condition we will discuss perturbation theory of operators without drift term. This perturbation theory allows us to extent the class of operators which a Carleson type condition can be applied to, and hence we get a wider class of operators for whom the $L^p$ Dirichlet boundary value problem is solvable. The perturbation theory and its application is joint work Martin Dindoš and Erik Sätterqvist.