Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition

Abstract

We prove the local Lipschitz regularity of the minimizers of functionals of the form \[ \mathcal I(u)=\int_\Omega f(\nabla u(x))+g(x)u(x)\,dx\qquad u\in\phi+W^{1,1}_0(\Omega) \] where $g$ is bounded and $\phi$ satisfies the Lower Bounded Slope Condition. The function $f$ is assumed to be convex but not uniformly convex everywhere. As byproduct, we also prove the existence of a locally Lipschitz minimizer for a class of functionals of the type above but allowing to the function $f$ to be nonconvex.