Boundary Value Problems In Domains With Lower Dimensional Boundaries

Abstract

This thesis is focused on the regularity boundary value problems in domains with lower dimensional boundaries. The objective is to solve the regularity problem onLipschitz domains. First, we prove that the solvability of the regularity problem is stable under the Carleson perturbation (joint work with J. Feneuil and S. Mayboroda). This is an analogue of the earlier work of C. E. Kenig and J. Pipher, who originally developed the Carleson perturbation theory for the regularity problem in co-dimension 1. Here we extend it to the higher co-dimensional case. Next, we will solve the regularity problem in L2 on flat domains Rn \ Rdd:= {(x, t) ∈ Rn|x ∈ Rd, t ∈ Rn−d \ {0}} (joint work with J. Feneuil and S. Mayboroda). Unlike the co-dimension 1 case, the number of non-tangential directions is now greater than 1 and their interplay is delicate. We divide them into two categories angular and radial. The key new step is estimating the “angular derivatives”. Once we handle them properly, the remaining argument is close to that for the co-dimension 1. Finally, we combine these two results to obtain the L2-solvability in domains with Lipschitz boundaries (joint work with J. Feneuil and S. Mayboroda).