Regularity for double-phase functionals with nearly linear growth and two modulating coefficients

Abstract

We deal with non-uniformly elliptic integral functionals w↦∫c(x)∣Dw∣log(1+∣Dw∣)+a(x)(∣Dw∣2+s2)q2+1dx,w\mapsto \int \left[{\mathfrak{c}}\left(x)| Dw| \log \left(1+| Dw| )+a\left(x){\left({| Dw| }^{2}+{s}^{2})}^{\tfrac{q}{2}}+1\right]{\rm{d}}x, with s∈[0,1]s\in \left[0,1], q>1q\gt 1, 0≤c(⋅)≤Λ,a(⋅)≥00\le {\mathfrak{c}}\left(\cdot )\le \Lambda ,a\left(\cdot )\ge 0, and a(x)+c(x)≥1Λa\left(x)+{\mathfrak{c}}\left(x)\ge \frac{1}{\Lambda } for some Λ>0\Lambda \gt 0. In this article, we establish that the gradient of a local minimizer for the aforementioned functionals is locally Hölder continuous.

Keywords

• nearly linear growth
• double-phase problem
• schauder estimate
• minimizer
• nonuniform ellipticity
• primary 35b65
• secondary 35a15
• 35j70
• 49n60