Global Lipschitz regularity for anisotropic elliptic problems with natural gradient growth

Homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic equations of p-Laplace type are considered. 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. Global Lipschitz regularity of solutions is established under the weakest possible integrability assumptions on the right-hand side, which can also include lower order terms with natural growth in the gradient. The results hold in either convex domains, or domains enjoying minimal integrability assumptions on the curvature of their boundary.