On the existence and regularity of mass-minimizing currents with an elastic boundary

Research Report MRR97-021

On the existence and regularity of mass-minimizing currents with an elastic boundary

Felicia Bernatzki

Abstract: We study the following variational problem. For a compact manifold $S_{0}$ embedded in the Euclidean space we consider deformations of $S_{0}$ . They are represented by Lipschitz continuous homeomorphisms of $S_{0}$ whose images are embedded manifolds. We introduce an energy of a deformation $\varphi$ which depends on the first derivative of $\varphi$ , the curvature of $\varphi(S_{0})$ and the mass of a mass minimizing current which is bounded by $\varphi(S_{0})$ . In this paper it is shown that an energy minimizing deformation $\varphi^{*}$ of $S_{0}$ exists. Moreover, in the case that $S_{0}$ has codimension 1, $\varphi^{*}(S_{0})$ is an embedded $C^{3, \alpha}$ -submanifold, if $\varphi^{*}$ is of the class $C^{2,1}$ .

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