Best constants in second-order Sobolev inequalities on Riemannian manifolds and applications.
type: Item
instance of: scholarly article
title: Best constants in second-order Sobolev inequalities on Riemannian manifolds and applications.
zbMATH Open document ID: 1043.58010
DOI: 10.1016/S0021-7824(03)00018-7
author: Rodney Josué Biezuner
author: Marcos Montenegro
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-09-25T00:00:00Z
review text: Let \(M\) be a compact Riemannian manifold of dimension \(n\geq 3\) and let \(1<p<n/2\). Set \(p^*=np/(n-2p)\). If \(M\) has no boundary look at the space \(E=H^{2,p}(M)\), if \(M\) has nonempty boundary let \(E=H^{2,p}_0(M)\) or \(E=H^{2,p}(M)\cap H^{1,p}_0(M)\). By the Sobolev embedding theorem the inclusion \(E \subset L^{p^*}(M)\) is continuous. The authors study the problem of best constants for \[ \| u\| ^p_{L^{p^*}(M)} \leq A\| \Delta u\| ^p_{L^{p}(M)} + B \| u\| ^p_{L^{p}(M)}. \] The results are then applied to construct solutions to certain nonlinear elliptic fourth order equations.
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 1985984
zbMATH Keywords: Sobolev inequalities
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: concentration-compactness principle
Wikidata QID: Q115340559
reviewed by: Christian Bär
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1985984