Partial Hölder continuity for discontinuous elliptic problems with VMO-coefficients
Verena Bögelein, Frank Duzaar, Jens Habermann, Christoph Scheven · Proceedings of the London Mathematical Society · 2011
We establish partial Hölder continuity for vector-valued solutions u : Ω → ℝN to elliptic systems of the type div a ( x , u , D u ) = 0 in Ω , as well as for minimizers u : Ω → ℝN of quasi-convex functionals F [ u ] : = ∫ Ω f ( x , u , D u ) d x , where the structure function a, respectively, the integrand f is possibly discontinuous with respect to x. More precisely, we merely impose a uniform VMO-condition with respect to the x-dependence and continuity with respect to the u-dependence and prove Hölder continuity of the solutions, respectively, the minimizers outside of a negligible set.