Regularity for double phase problems at nearly linear growth

TL;DR

This paper proves local Hölder continuity of the gradient for minimizers of a class of functionals with nearly linear growth, extending regularity results to a broader range of growth conditions.

Contribution

It establishes regularity results for double phase problems with nearly linear growth, specifically when the exponent q is less than 1 plus the Hölder continuity exponent over dimension.

Findings

Gradient of minimizers is locally Hölder continuous.

Regularity holds for growth exponent q in the range (1, 1+α/n).

Extends known regularity results to nearly linear growth functionals.

Abstract

Minima of functionals of the type $$ w\mapsto \int_{\Omega}\left[\snr{Dw}\log(1+\snr{Dw})+a(x)\snr{Dw}^{q}\right] \dx\,, \quad 0\leq a(\cdot) \in C^{0, \alpha}\,,$$ with $\Omega \subset \er^n$, have locally H\"older continuous gradient provided $1 < q < 1+\alpha/n$.