Lipschitz minimizers for a class of integral functionals under the bounded slope condition

TL;DR

This paper investigates the minimizers of a specific convex integral functional with a boundary condition, proving the existence, uniqueness, and Lipschitz regularity of solutions under certain conditions.

Contribution

It establishes the existence and uniqueness of Lipschitz continuous minimizers for a class of integral functionals with a bounded slope boundary condition.

Findings

Existence and uniqueness of minimizers under bounded slope condition

Minimizers are Lipschitz continuous

Results apply to a class of convex functionals with specific boundary conditions

Abstract

We consider the functional $\int_\Omega g(\nabla u+\textbf X^\ast)d\mathscr L^{2n}$ where $g$ is convex and $\textbf X^\ast(x,y)=2(-y,x)$ and we study the minimizers in $BV(\Omega)$ of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on $g$, there exists a unique minimizer which is also Lipschitz continuous.