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Regularity and boundary variations for the Neumann problem

The formula for the first variation of the Neumann eigenvalues of the Laplace operator under domain perturbation of a smooth submanifold of a Riemannian manifold is calculated. It is shown that this first variation formula remains finite when the smooth submanifold is replaced by a Lipschitz domain in R n . Existence and uniqueness is established for the solution to the inhomogeneous Neumann problem in Lipschitz domains with data in IP Sobolev spaces. Thesis Supervisor: David S. Jerison Title: Professor of Mathematics