Hölder continuity of weak solutions of p-Laplacian PDEs with VMO coefficients

We consider solutions u∈W 1,p (Ω;R N ) of the p-Laplacian PDE ∇⋅(a(x)|Du| p−2 Du)=0,for x∈Ω⊆R n , where Ω is open and bounded. More generally, we consider solutions of the elliptic system ∇⋅a(x)g ′ (a(x)|Du|)[Formula presented]=0,x∈Ωas well as minimizers of the functional ∫ Ω g(a(x)|Du|)dx.In each case, the coefficient map a:Ω→R is only assumed to be of class VMO(Ω)∩L ∞ (Ω), which means that it may be discontinuous. Without assuming that x↦a(x) has any weak differentiability, we show that u∈C loc 0,α (Ω) for each 0<α<1. The preceding results are, in fact, a corollary of a much more general result, which applies to the functional ∫ Ω f(x,u,Du)dx in case f is only asymptotically convex. © 2019 Elsevier Ltd