Gradient regularity in mixed local and nonlocal problems

Abstract

Minimizers of functionals of the type w↦∫Ω[|Dw|p-fw]dx+∫Rn∫Rn|w(x)-w(y)|γ|x-y|n+sγdxdy\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} w\\mapsto \\int _{\\Omega }[|Dw|^{p}-fw]\\,\ extrm{d}x+\\int _{{\\mathbb {R}}^{n}}\\int _{{\\mathbb {R}}^{n}}\\frac{|w(x)-w(y)|^{\\gamma }}{|x-y|^{n+s\\gamma }}\\,\ extrm{d}x\\,\ extrm{d}y\\end{aligned}$$\\end{document}with p,γ>1>s>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p, \\gamma>1>s >0$$\\end{document} and p>sγ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p> s\\gamma $$\\end{document}, are locally C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1, \\alpha }$$\\end{document}-regular in Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} and globally Hölder continuous.