Boundary Problems for Harmonic Functions and Norm Estimates for Inverses of Singular Integrals in Two Dimensions

ABSTRACT

In this paper, we establish sharp well-posedness results for tangential derivative problems for the Laplacian with data in L ^p , 1 < p < ∞, on curvilinear polygons. Furthermore, we produce norm estimates/formulas for inverses of singular integral operators relevant for the Dirichlet, Neumann, tangential derivative, and transmission boundary value problems associated with the Laplacian in a distinguished subclass of Lipschitz domains in two dimensions. Our approach relies on Calderón-Zygmund theory and Mellin transform techniques.

Keywords:

• Curvilinear polygons
• Dirichlet problem
• Layer potentials
• Neumann problem
• Tangential derivative problem

Mathematics Subject Classification:

• Primary 45E05, 47A05
• Secondary 35J25, 42B20

ACKNOWLEDGMENTS

This work has been supported in part by the NSF grant DMS 0513173 and University of Virginia FEST grant.