Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

Abstract

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.