Boundary regularity results for minimisers of convex functionals with $(p,q)$-growth

TL;DR

This paper establishes improved boundary regularity results for minimisers of convex functionals with $(p,q)$-growth, including characterisation of regular boundary points and partial regularity under various boundary conditions.

Contribution

It provides new differentiability results and boundary regularity characterisations for convex functionals with $(p,q)$-growth, extending previous understanding.

Findings

Boundary regularity results for relaxed minimisers

Characterisation of regular boundary points

Partial boundary regularity for smooth domains

Abstract

We prove improved differentiability results for relaxed minimisers of vectorial convex functionals with $(p, q)$-growth, satisfying a H\"older-growth condition in $x$. We consider both Dirichlet and Neumann boundary data. In addition, we obtain a characterisation of regular boundary points for such minimisers. In particular, in case of homogeneous boundary conditions, this allows us to deduce partial boundary regularity of relaxed minimisers on smooth domains for radial integrands. We also obtain some partial boundary regularity results for non-homogeneous Neumann boundary conditions.