Coercive estimates for the Laplace-Beltrami operator
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"The layout of the thesis is as follows. In Chapter 1 we review some notation and basic definitions which are relevant to our work, and we prove a partition of unity result. Chapter 2 deals with the theory of integration on surfaces. Here we also show how to define a unit normal TV to a surface which is locally given by graphs of C^k functions, k > 2. For such surfaces we also have a local description of the unit normal 7 to the boundary of a surface. In Chapter 3, we discuss first-order tangential differential operators. Here we also prove that tangential operators annihilate functions which are constant on a surface. We begin Chapter 4 by proving the existence of a distinguished extension to a neighborhood of the surface of the unit normal to a surface. We then introduce the Gauss curvature for a surface and prove that it is actually the divergence of our extension of the unit normal. We finish the chapter by proving some useful properties of a particular family of tangential operators and defining the n x n matrix-valued function R which appears in the identities on surfaces we prove later in the thesis. Chapter 5 focuses on two integration by parts results. In Chapter 6 we prove identity (1.17). This is done using the formalism and properties of a particular family of operators (Dj)^nj = 1 defined here as well as the work done in previous chapters. We finish with Chapter 7 in which we prove Sobolev norm estimates on surfaces for solutions to the Poisson problem for the Laplace-Beltrami operator on surfaces with homogeneous Dirichlet, Neumann, and mixed boundary conditions."--Page 7.
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M.S.