Singular Integrals and Elliptic Boundary Problems on Rough Domains

These notes discuss results on layer potential methods for elliptic boundary problems, with emphasis on the Dirichlet problem for the Laplace operator. They start by reviewing results for domains with moderately smooth boundary, then for Lipschitz domains, and proceed to discuss results in [HMT], obtained with S. Hofmann and M. Mitrea, for a class of domains we call regular Semmes-KenigToro (SKT) domains, often called chord-arc domains with vanishing constant, and for ε-regular SKT domains, often called chord-arc domains with small constant.