Regularity for Double Phase Problems at Nearly Linear Growth

Abstract

Minima of functionals of the type w↦∫Ω|Dw|log(1+|Dw|)+a(x)|Dw|qdx,0≤a(·)∈C0,α,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} w\\mapsto \\int _{\\varOmega }\\left[ |Dw|\\log (1+|Dw|)+a(x)|Dw|^{q}\\right] \\, \ extrm{d}x, \\quad 0\\le a(\\cdot ) \\in C^{0, \\alpha }, \\end{aligned}$$\\end{document}with varOmega subset {mathbb {R}}^n, have locally Hölder continuous gradient provided 1< q < 1+alpha /n.