{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T14:29:31Z","timestamp":1772288971944,"version":"3.50.1"},"reference-count":11,"publisher":"MathDoc\/Centre Mersenne","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"\n Let\n N<\/jats:italic>\n \u2a7e5,\n a<\/jats:italic>\n >0,\n \n \u03a9<\/mml:mi>\n <\/mml:math>\n be a smooth bounded domain in\n \n \n \u211d<\/mml:mi>\n N<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n ,\n \n \n \n 2<\/mml:mn>\n *<\/mml:mo>\n <\/mml:msup>\n =<\/mml:mo>\n \n \n 2<\/mml:mn>\n N<\/mml:mi>\n <\/mml:mrow>\n \n N<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math>\n ,\n \n \n \n 2<\/mml:mn>\n #<\/mml:mo>\n <\/mml:msup>\n =<\/mml:mo>\n \n \n 2<\/mml:mn>\n (<\/mml:mo>\n N<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \n N<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math>\n and \u2016\n u<\/jats:italic>\n \u2016\n 2<\/jats:sup>\n =|\u2207\n u<\/jats:italic>\n |\n 2<\/jats:sub>\n 2<\/jats:sup>\n +\n a<\/jats:italic>\n |\n u<\/jats:italic>\n |\n 2<\/jats:sub>\n 2<\/jats:sup>\n . We prove there exists an\n \u03b1<\/jats:italic>\n 0<\/jats:sub>\n >0 such that, for all\n \n \n u<\/mml:mi>\n \u2208<\/mml:mo>\n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n \n (<\/mml:mo>\n \u03a9<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u29f9<\/mml:mi>\n \n {<\/mml:mo>\n 0<\/mml:mn>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n ,\n \n \n \n S<\/mml:mi>\n \n 2<\/mml:mn>\n \n 2<\/mml:mn>\n \/<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mfrac>\n \u2a7d<\/mml:mi>\n \n \n \n \u2225<\/mml:mo>\n u<\/mml:mi>\n \u2225<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \n |<\/mml:mo>\n u<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n \n 2<\/mml:mn>\n *<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msubsup>\n <\/mml:mfrac>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n \n \u03b1<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \n \n \n |<\/mml:mo>\n u<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n \n 2<\/mml:mn>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n #<\/mml:mo>\n <\/mml:msup>\n <\/mml:msubsup>\n \n \n \u2225<\/mml:mo>\n u<\/mml:mi>\n \u2225<\/mml:mo>\n \u00b7<\/mml:mi>\n |<\/mml:mo>\n u<\/mml:mi>\n |<\/mml:mo>\n <\/mml:mrow>\n \n \n 2<\/mml:mn>\n *<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n \n \n 2<\/mml:mn>\n *<\/mml:mo>\n <\/mml:msup>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mfrac>\n <\/mml:mfenced>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n This inequality implies Cherrier's inequality.\n <\/jats:p>","DOI":"10.1016\/s1631-073x(02)02215-x","type":"journal-article","created":{"date-parts":[[2002,7,25]],"date-time":"2002-07-25T23:41:31Z","timestamp":1027640491000},"page":"105-108","source":"Crossref","is-referenced-by-count":4,"title":["A sharp inequality for Sobolev functions"],"prefix":"10.5802","volume":"334","author":[{"given":"Pedro M.","family":"Gir\u00e3o","sequence":"first","affiliation":[]}],"member":"3842","published-online":{"date-parts":[[2002,1,1]]},"reference":[{"key":"key2025102009563789378_1","first-page":"9","article-title":"The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi","author":"Adimurthi","year":"1991","unstructured":"[1] Adimurthi; Mancini, G. The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi, Scuola Norm. Sup. Pisa (1991), pp. 9-25","journal-title":"Scuola Norm. Sup. Pisa"},{"key":"key2025102009563789378_2","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1006\/jfan.1993.1053","article-title":"Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity","volume":"113","author":"Adimurthi","year":"1993","unstructured":"[2] Adimurthi; Pacella, F.; Yadava, S.L. Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., Volume 113 (1993), pp. 318-350","journal-title":"J. Funct. Anal."},{"key":"key2025102009563789378_3","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0022-1236(85)90020-5","article-title":"Sobolev inequalities with remainder terms","volume":"62","author":"Brezis, H.","year":"1985","unstructured":"[3] Brezis, H.; Lieb, E. Sobolev inequalities with remainder terms, J. Funct. Anal., Volume 62 (1985), pp. 73-86","journal-title":"J. Funct. Anal."},{"key":"key2025102009563789378_4","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1002\/cpa.3160360405","article-title":"Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents","volume":"36","author":"Brezis, H.","year":"1983","unstructured":"[4] Brezis, H.; Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., Volume 36 (1983), pp. 437-477","journal-title":"Comm. Pure Appl. 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Iberoamericana"},{"key":"key2025102009563789378_9","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/0022-0396(91)90014-Z","article-title":"Neumann problems of semilinear elliptic equations involving critical Sobolev exponents","volume":"93","author":"Wang, X.J.","year":"1991","unstructured":"[9] Wang, X.J. Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, Volume 93 (1991), pp. 283-310","journal-title":"J. Differential Equations"},{"key":"key2025102009563789378_10","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s005260050119","article-title":"Existence and nonexistence of G<\/span>-least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains","volume":"8","author":"Wang, Z.Q.","year":"1999","unstructured":"[10] Wang, Z.Q. Existence and nonexistence of G-least energy solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Calc. Var., Volume 8 (1999), pp. 109-122","journal-title":"Calc. Var."},{"key":"key2025102009563789378_11","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s005260050115","article-title":"Sharp Sobolev inequalities with interior norms","volume":"8","author":"Zhu, M.","year":"1999","unstructured":"[11] Zhu, M. Sharp Sobolev inequalities with interior norms, Calc. Var., Volume 8 (1999), pp. 27-43","journal-title":"Calc. Var."}],"container-title":["Comptes Rendus. Math\u00e9matique"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/comptes-rendus.academie-sciences.fr\/mathematique\/item\/10.1016\/S1631-073X(02)02215-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T07:57:02Z","timestamp":1760947022000},"score":1,"resource":{"primary":{"URL":"https:\/\/comptes-rendus.academie-sciences.fr\/mathematique\/articles\/10.1016\/S1631-073X(02)02215-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1,1]]},"references-count":11,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2002]]}},"alternative-id":["10.1016\/S1631-073X(02)02215-X"],"URL":"https:\/\/doi.org\/10.1016\/s1631-073x(02)02215-x","relation":{},"ISSN":["1631-073X","1778-3569"],"issn-type":[{"value":"1631-073X","type":"print"},{"value":"1778-3569","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1,1]]}}}