Hölder continuity of weak solutions of p-Laplacian PDEs with VMO coefficients. (August 2019) - Goodrich, Christopher S.
Abstract: We consider solutions u ∈ W 1, p ( Ω ; R N ) of the p -Laplacian PDE ∇ ⋅ ( a ( x ) | D u | p − 2 D u ) = 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 ) | D u | ) D u | D u | = 0, x ∈ Ω as well as minimizers of the functional ∫ Ω g ( a ( x ) | D u | ) d x . In each case, the coefficient map a : Ω → R is only assumed to be of class V M O ( 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, D u ) d x in case f is only asymptotically convex.