Borderline gradient continuity of minima

TL;DR

This paper establishes the continuity of the gradient of local minimizers for certain variational functionals under optimal Lorentz space conditions and Dini-continuity assumptions on the integrand.

Contribution

It proves the borderline gradient continuity of minimizers under the optimal Lorentz space condition $ ext{L}(n,1)$ for the measure data, extending previous regularity results.

Findings

Gradient of local minimizers is continuous under Lorentz space condition $ ext{L}(n,1)$.

Continuity holds when the integrand's dependence on $x$ is Dini-continuous.

Extends regularity theory to borderline measure data cases.

Abstract

The gradient of any local minimiser of functionals of the type $$ w \mapsto \int_\Omega f(x,w,Dw)\,dx+\int_\Omega w\mu\,dx, $$ where $f$ has $p$-growth, $p>1$, and $\Omega \subset \mathbb R^n$, is continuous provided the optimal Lorentz space condition $\mu \in L(n,1)$ is satisfied and $x\to f(x, \cdot)$ is suitably Dini-continuous.