An elementary partial regularity proof for vector-valued obstacle problems
We analyze the partial regularity properties of vector-valued Sobolev- functions u which minimize energy-type functionals under a non-convex side-condition of the form Im(u)\(\subset \bar M\) for a bounded smooth domain M in the image space. Although our results are not new, the method outlined here simplifies and unifies the various techniques developed by F. Duzaar and the author (see the references quoted in the paper). The approach presented here is based on a rather elementary blow-up lemma and an extension-theorem for maps defined on spheres with values in \=M to the whole ball. Combining these facts we get an energy-decay estimate near points with small energy. Moreover, the same technique applies to the related p-harmonic obstacle problem for exponents \(p>2\), which could not be handled with our former techniques.
scientific article; zbMATH DE number 3985731