Partial Hölder continuity for discontinuous elliptic problems with VMO‐coefficients

We establish partial Hölder continuity for vector‐valued solutions u : Ω → ℝ N to elliptic systems of the type div   a ( x , u , D u ) = 0           in   Ω , as well as for minimizers u : Ω → ℝ N 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.