Local Lipschitz continuity of the minimizers of nonuniformly convex functionals under the Lower Bounded Slope Condition
Flavia Giannetti, Giulia Treu
TL;DR
This work studies local Lipschitz regularity of minimizers for variational functionals $\mathcal{I}(u)=\int_{\Omega} [f(\nabla u)+g(x)u]\,dx$ when the Lagrangian is not uniformly convex on the whole domain but satisfies a Lower Bounded Slope Condition (LBSC) on the boundary of a $R$-uniformly convex domain. The authors develop barrier constructions and a comparison principle for convex, superlinear Lagrangians under LBSC, coupled with a uniform approximation of $f$ by smooth convex functions, to obtain an a priori gradient bound and pass to the limit. They prove two main results: (i) if $f= f_1+f_2$ with $f_1$ satisfying $(F1)$–$(F3)$ and $f_2$ uniformly convex, every bounded minimizer is locally Lipschitz with a boundary gradient bound; (ii) if $f$ is convex under $(F1)$–$(F3)$, any minimizer is locally Lipschitz. The paper also extends these results to a nonconvex setting by using the convex envelope $f^{**}$ and a pyramidal construction to obtain locally Lipschitz minimizers when the non-coincidence set is suitably structured, with further implications such as Hölder continuity up to the boundary under additional assumptions.
Abstract
We prove the local Lipschitz regularity of the minimizers of functionals of the form \[ \mathcal I(u)=\int_Ωf(\nabla u(x))+g(x)u(x)\,dx\qquad u\inφ+W^{1,1}_0(Ω) \] where $g$ is bounded and $φ$ 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.