Existencia de soluciones débiles en problemas de superficies mínimas con aplicaciones a problemas de elasticidad

Abstract

In this work, minimal hypersurfaces are studied using tools from the calculus of variations. Minimal hypersurfaces arise as critical points of the generalized area functional and are characterized by having zero mean curvature. In this context, tools from functional analysis, particularly Sobolev spaces, are employed to formulate the problem within an appropriate framework. Following the development presented in Struwe, M., Variational Methods [p. 7], we formulate the corresponding variational problem in order to prove the existence of weak solutions of minimal energy using the direct method of the calculus of variations. As an illustration of the theory, we consider a model problem of harmonic maps with values in the sphere ( S^2 ), whose equilibrium profile is computed numerically by means of a shooting method with RK45 integration.