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The Dirichlet Problem of the cmc Equation

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Constant Mean Curvature Surfaces with Boundary

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

We shall study the Dirichlet problem for the cmc equation considering that the domain is bounded. Thanks to the topological apparatus of a priori estimates of Ladyzhenskaya and Uraltseva, the existence of the Dirichlet problem reduces to the question of obtaining a priori estimates for the solution and its gradient. The purpose of the present chapter is to study the geometry that lies behind the Dirichlet problem, giving a geometric approach to the usual techniques that appear in the PDE theory. Under assumptions on the length or the curvature of the boundary curve, we shall use known surfaces such as pieces of Delaunay surfaces as barriers to give C 1-estimates of solutions of the Dirichlet problem. If these barriers are spherical caps or cylinders, we shall assume that the domain is convex, and if we employ nodoids, we suppose that the domain satisfies a certain exterior circle condition. We finish the chapter with a discussion of the corresponding Dirichlet problem of the cmc equation in domains of the unit sphere whose solutions represent radial graphs.

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  1. Department of Geometry and Topology, University of Granada, Granada, Spain

    Rafael López

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López, R. (2013). The Dirichlet Problem of the cmc Equation. In: Constant Mean Curvature Surfaces with Boundary. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39626-7_8

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