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

We prove the local Lipschitz regularity of the minimizers of functionals of the form $$\mathcal{I}(u)={\int }_{\mathrm{\Omega }}f(\mathrm{\nabla }u(x))+g(x)u(x)dxu\in \phi +W_0^{1,1}(\mathrm{\Omega })$$ 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.