Convexity for free boundaries with singular term (nonlinear elliptic case)

Abstract

We consider a free boundary problem in an exterior domain Lu=g(u)inΩ\\K,u=1on∂K,|∇u|=0on∂Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} Lu=g(u)&{}\ ext {in }\\Omega \\setminus K,\\\\ u=1 &{} \ ext {on }\\partial K,\\\\ |\ abla u|=0 &{}\ ext {on }\\partial \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where K is a (given) convex and compact set in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^n$$\\end{document} (n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\ge 2$$\\end{document}), Ω={u>0}⊃K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega =\\{u>0\\}\\supset K$$\\end{document} is an unknown set, and L is either a fully nonlinear or the p-Laplace operator. Under suitable assumptions on K and g, we prove the existence of a nonnegative quasi-concave solution to the above problem. We also consider the cases when the set K is contained in {xn=0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{x_n=0\\}$$\\end{document}, and obtain similar results.