Pólya-type estimates for the first Robin eigenvalue of elliptic operators

Abstract

The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p-Laplace operator, namely: λF(β,Ω)=minψ∈W1,p(Ω)\\{0}∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN-1∫Ω|ψ|pdx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lambda _F(\\beta ,\\Omega )= \\min _{\\psi \\in W^{1,p}(\\Omega ){\\setminus }\\{0\\} } \\frac{\\displaystyle \\int _\\Omega F(\ abla \\psi )^p dx +\\beta \\int _{\\partial \\Omega }|\\psi |^p F(\ u _{\\Omega }) d{\\mathcal {H}}^{N-1} }{\\displaystyle \\int _\\Omega |\\psi |^p dx}, \\end{aligned}$$\\end{document}where p∈]1,+∞[,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in ]1,+\\infty [,$$\\end{document}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} is a bounded, convex domain 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}νΩ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _{\\Omega }$$\\end{document} is its Euclidean outward normal, β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} is a real number, and F is a sufficiently smooth norm on 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} We show an upper bound for λF(β,Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _{F}(\\beta ,\\Omega )$$\\end{document} in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} and on the volume and the anisotropic perimeter of 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} in the spirit of the classical estimates of Pólya (J Indian Math Soc (NS) 24:413–419, 1961) for the Euclidean Dirichlet Laplacian. We will also provide a lower bound for the torsional rigidity τp(β,Ω)p-1=maxψ∈W1,p(Ω)\\{0}∫Ω|ψ|dxp∫ΩF(∇ψ)pdx+β∫∂Ω|ψ|pF(νΩ)dHN-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ au _p(\\beta ,\\Omega )^{p-1} = \\max _{\\begin{array}{c} \\psi \\in W^{1,p}(\\Omega ){\\setminus }\\{0\\} \\end{array}} \\dfrac{\\left( \\displaystyle \\int _\\Omega |\\psi | \\, dx\\right) ^p}{\\displaystyle \\int _\\Omega F(\ abla \\psi )^p dx+\\beta \\int _{\\partial \\Omega }|\\psi |^p F(\ u _{\\Omega }) d{\\mathcal {H}}^{N-1} } \\end{aligned}$$\\end{document}when β>0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >0.$$\\end{document} The obtained results are new also in the case of the classical Euclidean Laplacian.