A note on removable singularities for minima of certain vector-valued obstacle problems
We analyze the regularity properties of vector-valued functions \(u: {\mathbb{R}}^ n\supset D\to {\mathbb{R}}^ N\) minimizing a quadratic variational integral \[ F(u):=\sum ^{N}_{i=1}\sum ^{n}_{\alpha,\beta =1}\int _{D}a_{\alpha \beta}(\cdot,u)D_{\alpha}u^ iD_{\beta}u^ idx \] under an additional side-condition of the form Im(u)\(\subset \bar M\) for a smooth region M in \({\mathbb{R}}^ N\). We try to exhibit reasonable geometric conditions on the set M excluding singular points. For example we show that the singular set is empty if M is a star-like domain of sufficiently small diameter. A counter-example demonstrates the sharpness of our result.
scientific article; zbMATH DE number 3983973
scientific article