Lipschitz regularity for solutions of a general class of elliptic equations

Abstract

AbstractWe prove local Lipschitz regularity for local minimisers of $$\begin{aligned} W^{1,1}(\Omega )\ni v\mapsto \int _\Omega F(Dv)\, dx \end{aligned}$$ W 1 , 1 ( Ω ) ∋ v ↦ ∫ Ω F ( D v ) d x where $$\Omega \subseteq {\mathbb {R}}^N$$ Ω ⊆ R N , $$N\ge 2$$ N ≥ 2 and $$F:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ F : R N → R is a quasiuniformly convex integrand in the sense of Kovalev and Maldonado (Ill J Math 49:1039–1060, 2005), i. e. a convex $$C^1$$ C 1 -function such that the ratio between the maximum and minimum eigenvalues of $$D^2F$$ D 2 F is essentially bounded. This class of integrands includes the standard singular/degenerate functions $$F(z)=|z|^p$$ F ( z ) = | z | p for any $$p>1$$ p > 1 and arises as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations.