Variational problems with non-convex obstacles and an integral constraint for vector-valued functions
We minimize the Dirichlet-integral (or related quadratic functionals) in a class of vector-valued functions \(u: {\mathbb{R}}^ n\supset D\to {\mathbb{R}}^ N\) which is described by an integral constraint and a side condition of the form \(Im(u)\subset \bar M\) and show partial regularity of weak minima up to a set of vanishing \({\mathbb{H}}^{n-2}\)-measure, provided M is a bounded open set with certain smoothness properties. In contrast to the results of Hildebrandt, Tomi and others our theorem is not restricted to special dimensions n or requires convexity of M. The method consists in a suitable linearisation of the problem giving an Euler-inequality with quadratic growth in Du. In combination with certain reverse Hölder-inequalities the techniques of Giaquinta and Giusti imply the partial regularity result. Moreover, their arguments allow to reduce the dimension of the singular set S to \({\mathbb{H}}-\dim (S)\leq n-3\).
scientific article; zbMATH DE number 3907217
scientific article