Regularity for Double Phase Functionals with Two Modulating Coefficients

In this paper, we establish regularity results for local minimizers of functionals with non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral with two modulating coefficients w↦∫[a(x)|Dw|p+b(x)|Dw|q]dx,1<p<q,a(·),b(·)≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w\mapsto \int [a(x)|Dw|^p+b(x)|Dw|^q] dx, \qquad 1<p<q, \qquad a(\cdot ),b(\cdot )\ge 0, \end{aligned}$$\end{document}with 0<μ≤a(·)+b(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\mu \le a(\cdot )+b(\cdot )$$\end{document}. Here, the coefficient b(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(\cdot )$$\end{document} is assumed to be Hölder continuous and the coefficient a(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(\cdot )$$\end{document} is assumed to be uniformly continuous.