Splitting-type variational problems with asymmetrical growth conditions

TL;DR

This paper investigates splitting-type variational problems with asymmetric growth conditions, establishing higher integrability of solutions' gradients under weak hypotheses without assuming symmetry in growth functions.

Contribution

It introduces a novel approach to analyze variational problems with asymmetric growth, proving higher integrability results without symmetry assumptions.

Findings

Higher integrability of the gradient for local minimizers.

Use of a Caccioppoli-type inequality with negative power weights.

Results hold under weak, non-symmetric growth conditions.

Abstract

Splitting-type variational problems \[ \int_\Omega \sum_{i=1}^n f_i(\partial_i w) dx \to \min \] with superlinear growth conditions are studied by assuming \[ h_i(t) \leq f''_i(t) \leq H_i(t) \] with suitable functions $h_i$, $H_i$: $\mathbb{R} \to \mathbb{R}^+$, $i=1$, $\dots$ , $n$, measuring the growth and ellipticity of the energy density. Here, as the main feature, a symmetric behaviour like $h_i(t)\approx h_i(-t)$ and $H_i(t) \approx H_i(-t)$ for large $|t|$ is not supposed. Assuming quite weak hypotheses as above, we establish higher integrability of $|\nabla u|$ for local minimizers $u\in L^\infty(\Omega)$ by using a Caccioppoli-type inequality with some power weights of negative exponent.