Wiener's criterion and obstacle problems for vector valued functions (Q1081831)

The author studies regularity properties of solutions of obstacle problems for vector valued functions. Let \(\Omega ={\mathbb{R}}^ n\) be bounded and open, with \(n\geq 3\), let \(F\subset {\mathbb{R}}^ N\) be closed and convex, and let \(E\subset \Omega\) be closed. The constraint is then of the form (u-\(\psi)\)(x)\(\in F\) for \(x\in E.\) Let u be a solution to the variational inequality \(\int A^{\alpha \beta}D_{\alpha}uD_{\beta}(v-u)dx\geq \int f\cdot (v-u)dx\) which also satisfies the constraint. If \(x_ 0\in E\) and if a Wiener criterion measuring the thickness of E at \(x_ 0\) is satisfied, then u is pointwise continuous at \(x_ 0.\) An estimate on the modulus of continuity is used to establish Hölder continuity of u at \(x_ 0\) when E is ''sufficiently thick'' at \(x_ 0\). Local regularity of u in \(\Omega\) is also treated.