{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T20:45:28Z","timestamp":1774644328665,"version":"3.50.1"},"reference-count":42,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2012285"],"award-info":[{"award-number":["DMS-2012285"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this paper, we show that the so-called Korn inequality holds for vector fields with a zero normal or tangential trace on a subset (of positive measure) of the boundary of Lipschitz domains.\nWe further show that the validity of this inequality depends on the geometry of this subset of the boundary.\nWe then consider three eigenvalue problems for the Lam\u00e9 operator: we constrain the traction in the tangential direction and the normal component of the displacement, the related problem of constraining the normal component of the traction and the tangential component of the displacement, and a third eigenproblem that considers mixed boundary conditions.\nWe show that eigenpairs for these eigenproblems exist on a broad variety of domains.\nAnalytic solutions for some of these eigenproblems are given on simple domains.<\/jats:p>","DOI":"10.1515\/cmam-2021-0144","type":"journal-article","created":{"date-parts":[[2022,3,19]],"date-time":"2022-03-19T19:40:00Z","timestamp":1647718800000},"page":"821-837","source":"Crossref","is-referenced-by-count":3,"title":["Korn\u2019s Inequality and Eigenproblems for the Lam\u00e9 Operator"],"prefix":"10.1515","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3542-8539","authenticated-orcid":false,"given":"Sebasti\u00e1n A.","family":"Dom\u00ednguez-Rivera","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics , University of Saskatchewan , Saskatoon , Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6739-1179","authenticated-orcid":false,"given":"Nilima","family":"Nigam","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Simon Fraser University , Burnaby , Canada"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1944-2872","authenticated-orcid":false,"given":"Jeffrey S.","family":"Ovall","sequence":"additional","affiliation":[{"name":"Fariborz Maseeh Department of Mathematics and Statistics , Portland State University , Portland , OR 97201 , USA"}]}],"member":"374","published-online":{"date-parts":[[2022,3,17]]},"reference":[{"key":"2023033113381587284_j_cmam-2021-0144_ref_001","doi-asserted-by":"crossref","unstructured":"G. Acosta, R. G. Dur\u00e1n and M. A. Muschietti,\nSolutions of the divergence operator on John domains,\nAdv. Math. 206 (2006), no. 2, 373\u2013401.","DOI":"10.1016\/j.aim.2005.09.004"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_002","doi-asserted-by":"crossref","unstructured":"G. Acosta and I. Ojea,\nKorn\u2019s inequalities for generalized external cusps,\nMath. Methods Appl. Sci. 39 (2016), no. 17, 4935\u20134950.","DOI":"10.1002\/mma.3170"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_003","doi-asserted-by":"crossref","unstructured":"S. Bauer and D. Pauly,\nOn Korn\u2019s first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in \n                  \n                     \n                        \n                           R\n                           N\n                        \n                     \n                     \n                     \\mathbb{R}^{N}\n                  \n               ,\nAnn. Univ. Ferrara Sez. VII Sci. Mat. 62 (2016), no. 2, 173\u2013188.","DOI":"10.1007\/s11565-016-0247-x"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_004","doi-asserted-by":"crossref","unstructured":"S. Bauer and D. Pauly,\nOn Korn\u2019s first inequality for tangential or normal boundary conditions with explicit constants,\nMath. Methods Appl. Sci. 39 (2016), no. 18, 5695\u20135704.","DOI":"10.1002\/mma.3954"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_005","doi-asserted-by":"crossref","unstructured":"E. C\u00e1ceres, J. Guzm\u00e1n and M. Olshanskii,\nNew stability estimates for an unfitted finite element method for two-phase Stokes problem,\nSIAM J. Numer. Anal. 58 (2020), no. 4, 2165\u20132192.","DOI":"10.1137\/19M1266897"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_006","doi-asserted-by":"crossref","unstructured":"M. Chipot,\nOn inequalities of Korn\u2019s type,\nJ. Math. Pures Appl. (9) 148 (2021), 199\u2013220.","DOI":"10.1016\/j.matpur.2020.08.012"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_007","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\nOn Korn\u2019s inequality,\nChin. Ann. Math. Ser. B 31 (2010), no. 5, 607\u2013618.","DOI":"10.1007\/s11401-010-0606-3"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_008","doi-asserted-by":"crossref","unstructured":"S. Conti, D. Faraco and F. Maggi,\nA new approach to counterexamples to \n                  \n                     \n                        \n                           L\n                           1\n                        \n                     \n                     \n                     L^{1}\n                  \n                estimates: Korn\u2019s inequality, geometric rigidity, and regularity for gradients of separately convex functions,\nArch. Ration. Mech. Anal. 175 (2005), no. 2, 287\u2013300.","DOI":"10.1007\/s00205-004-0350-5"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_009","doi-asserted-by":"crossref","unstructured":"A. Damlamian,\nSome remarks on Korn inequalities,\nChin. Ann. Math. Ser. B 39 (2018), no. 2, 335\u2013344.","DOI":"10.1007\/s11401-018-1067-3"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_010","doi-asserted-by":"crossref","unstructured":"F. Demengel and G. Demengel,\nFunctional Spaces for the Theory Of Elliptic Partial Differential Equations,\nUniversitext,\nSpringer, London, 2012.","DOI":"10.1007\/978-1-4471-2807-6"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_011","doi-asserted-by":"crossref","unstructured":"L. Desvillettes and C. Villani,\nOn a variant of Korn\u2019s inequality arising in statistical mechanics,\nESAIM Control Optim. Calc. Var. 8 (2002), 603\u2013619.","DOI":"10.1051\/cocv:2002036"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_012","doi-asserted-by":"crossref","unstructured":"S. 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Berichte 36 (1906), 351\u2013401."},{"key":"2023033113381587284_j_cmam-2021-0144_ref_029","doi-asserted-by":"crossref","unstructured":"A. Korn,\nSolution g\u00e9n\u00e9rale du probl\u00e8me d\u2019\u00e9quilibre dans la th\u00e9orie de l\u2019\u00e9lasticit\u00e9 dans le cas o\u00f9 les efforts sont donn\u00e9s \u00e0 la surface,\nAnn. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2) 10 (1908), 473\u2013473.","DOI":"10.5802\/afst.255"},{"key":"2023033113381587284_j_cmam-2021-0144_ref_030","unstructured":"A. Korn,\n\u00dcber einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen,\nBull. Int. Acad. Sci. de Cracovie 9 (1909), 705\u2013724."},{"key":"2023033113381587284_j_cmam-2021-0144_ref_031","doi-asserted-by":"crossref","unstructured":"C. Le Roux,\nExistence and uniqueness of the flow of second-grade fluids with slip boundary conditions,\nArch. Ration. Mech. 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