Abstract
Geometric variational problems are among the most fascinating problems of the calculus of variations. A lot among the most interesting problems of differential geometry are variational in nature. The solution to such problems describes equilibrium shapes of physical systems or give selected representative belonging to particular class of homology and homotopy. To study existence multiplicity and geometric characterization of such solutions is of fundamental in pure and applied mathematics. This project have as a primary goal do improve our understanding about them developing new methods and approaches. Among the problems treated in this project we have the isoperimetric problem, the oriented Plateau's problem, fase transition problems with emphasis in the Cahn-Hilliard equation. All these problems will be treated in smooth and non-smooth metric measures spaces using the most recent technics of non-smooth analysis in metric measure spaces combined with the sophisticated theory of regularity for $Q$-valued functions of Almgren. great relevance will be given to the theory of existence and multiplicity of geometric variational problems with lack of compactness in non compact ambient spaces with bounded geometry. In these spaces we will give a quite satisfactory general answer. (AU)
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