On the best Sobolev inequality
type: Item
instance of: scholarly article
title: On the best Sobolev inequality
zbMATH Open document ID: 0944.46027
DOI: 10.1016/S0021-7824(99)00012-4
author: Thierry Aubin
author: Yan Yan Li
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-09-20T00:00:00Z
review text: Sobolev embeddings \(W^{1,p}\subset L^{p^*}\), \(1\leq p<n\) and \(p^*= {np\over n-p}\), hold for the first-order Sobolev spaces \(W^{1,p}\) defined on a complete \(n\)-dimensional Riemannian manifold with positive injectivity radius and the Riccati curvature bounded from below. Let \(K= K(n,p)\) be the norm of the inclusion \(W^{1,p}\subset L^{p^*}\) on \(\mathbb{R}^n\). Then for any \(\varepsilon> 0\) there exists a constant \(A_P(\varepsilon)\) such that every \(\varphi\in W^{1,p}\) satisfies \[ \|\varphi\|_{L^{p^*}}\leq(K+ \varepsilon)\|\nabla\varphi\|_{L^p}+ A_p(\varepsilon)\|\varphi\|_{L^p},\tag{1} \] and \(K\) is the smallest constant having this property. The best constant is achieved if there exist \(A_p(0)\). One can expect that the sharp constant exists. In an earlier work T. Aubin made a conjecture that there exist \(A(p)\) such that any \(\varphi\in W^{1,p}\) satisfies \[ \|\varphi\|^p_{L^{p^*}}\leq K^p\|\nabla\varphi\|_{L^p}+ A(p)\|\varphi\|^p_{L^p}\quad\text{if }1\leq p\leq 2,\tag{2} \] and \[ \|\varphi\|^2_{L^{p^*}}\leq K^2\|\nabla\varphi\|^2_{L^p}+ A(p)\|\varphi\|^2_{L^p}\quad\text{if }2< p<n.\tag{3} \] The main theorem of the paper asserts that if \(M\) is a complete \(n\)-dimensional Riemannian manifolds with positive injectivity radius and if the Riemann curvature tensor and its covariant derivative are bounded on \(M\) then the inequality (2) holds for \(1<p\leq 2\) and (3) holds with \(2<p<n\). In particular, the theorem holds if \(M\) is compact. It is proved even more. The exponent in the inequality (3) can be improved if the dimension \(n\geq 3\). The improved version of the inequality (2) is also proved.
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 1344437
zbMATH Keywords: Sobolev embeddings
zbMATH Keywords: first-order Sobolev spaces
zbMATH Keywords: Riccati curvature
zbMATH Keywords: Riemannian manifolds with positive injectivity radius
zbMATH Keywords: Riemann curvature tensor
reviewed by: Leszek Skrzypczak
MaRDI profile type: Publication
cites work: Some remarks on Sobolev type inequalities
cites work: Elliptic equations with nearly critical growth
cites work: Q3657016
cites work: Q4398855
cites work: Q4093975
cites work: Problèmes isoperimetriques et espaces de Sobolev
cites work: Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature
cites work: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality
cites work: Sobolev inequalities with remainder terms
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Q4735483
cites work: Q3756181
cites work: Extremals of functionals with competing symmetries
cites work: Q3338984
cites work: Degenerate parabolic equations
cites work: Optimal Sobolev inequalities of arbitrary order on compact Riemannian manifolds
cites work: Q3818662
cites work: A new proof of local \(C^{1,\alpha}\) regularity for solutions of certain degenerate elliptic P.D.E
cites work: Q4040640
cites work: Quasilinear elliptic equations involving critical Sobolev exponents
cites work: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent
cites work: Degenerate elliptic systems and applications to Ginzburg-Landau type equations. I
cites work: Optimal Sobolev Inequalities on Complete Riemannian Manifolds with Ricci Curvature Bounded Below and Positive Injectivity Radius
cites work: Sobolev spaces on Riemannian manifolds
cites work: Best constants in the Sobolev imbedding theorem
cites work: The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds
cites work: On the smoothness of weak solutions of quasilinear equations in several variables and of variational problems
cites work: Q4342727
cites work: Sharp Sobolev inequalities involving boundary terms
cites work: Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
cites work: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
cites work: Q5614914
cites work: Q3027501
cites work: Best constant in Sobolev inequality
cites work: Regularity for a more general class of quasilinear equations
cites work: Q5547919
cites work: Some general forms of sharp Sobolev inequalities
full work available at URL: s0021-7824(99)00012-4
OpenAlex ID: W2058303982
description: scientific article; zbMATH DE number 1344437