Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth

Abstract: We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. We establish global Lipschitz regularity of solutions under the weakest possible assumption on right-hand side of the equation, which may also include the gradient term with natural growth exponent. The results hold in either convex domains, or domains enjoying minimal integrability assumptions on the curvature of its boundary.