{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:54:29Z","timestamp":1787324069170,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Math. Anal."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>A G\u00e5rding-type inequality is proved for a quadratic form associated to $\\mathcal{A}$-quasiconvex functions. This quadratic form appears as the relative entropy in the theory of conservation laws and it is related to the Weierstrass excess function in the calculus of variations. The former provides weak-strong uniqueness results, whereas the latter has been used to provide sufficiency theorems for local minimizers. Using this new G\u00e5rding inequality we provide an extension of these results to PDE constrained problems in dynamics and statics under $\\mathcal{A}$-quasiconvexity assumptions. The application in statics improves existing results by proving uniqueness of $L^p$-local minimizers in the classical $\\mathcal{A}={curl}$ case.<\/jats:p>","DOI":"10.1137\/20m1345128","type":"journal-article","created":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T10:36:31Z","timestamp":1627554991000},"page":"4178-4211","source":"Crossref","is-referenced-by-count":6,"title":["$\\mathcal{A}$-Quasiconvexity, G\u00e5rding Inequalities, and Applications in PDE Constrained Problems in Dynamics and Statics"],"prefix":"10.1137","volume":"53","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9760-6839","authenticated-orcid":true,"given":"Konstantinos","family":"Koumatos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas P.","family":"Vikelis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,7,29]]},"reference":[{"key":"atypb1","first-page":"53","volume":"19","author":"Agranovich M. S.","year":"1964","journal-title":"Uspekhi Mat. Nauk"},{"key":"atypb2","volume-title":"Characterization of Generalized Young Measures Generated by $\\mathcal{A}$-Free Measures, preprint, arXiv:1908.03186","author":"Arroyo-Rabasa A.","year":"2019"},{"key":"atypb3","volume-title":"On the Fine Properties of Elliptic Operators, preprint, arXiv:1911.08474","author":"Arroyo-Rabasa A.","year":"2019"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.2140\/apde.2020.13.559"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-003-0291-4"},{"key":"atypb6","volume-title":"DPhil. dissertation","author":"Campos-Cordero J.","year":"2014"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2017.02.027"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1051\/cocv\/2019019"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/BF01587765"},{"key":"atypb10","first-page":"627","volume":"9","author":"Dacorogna B.","year":"1982","journal-title":"Ann. Sc. Norm. Super. Pisa Cl. Sci."},{"key":"atypb11","series-title":"Lecture Notes in Math. 922","volume-title":"Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals","author":"Dacorogna B.","year":"2006"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/BF00280911"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-29089-3"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2012.0344"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2016.184.3.10"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/1007108"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141098339885"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10364"},{"key":"atypb19","volume-title":"Ann. Math. Stud. 105","author":"Giaquinta M.","year":"1983"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-018-01346-5"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jfa.2019.108278"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-08-04786-7"},{"key":"atypb23","volume-title":"OxPDE technical report","author":"Guerra A.","year":"2020"},{"key":"atypb24","volume-title":"Null Lagrangians, and Hardy Space Integrability under Constant Rank Constraints, preprint, arXiv:1909.03923","author":"Guerra A.","year":"2019"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-016-0967-1"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21801"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21917"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1515\/acv-2015-0009"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1007\/s002080050277"},{"key":"atypb30","volume-title":"private communication","author":"Kristensen J.","year":"2019"},{"key":"atypb31","volume-title":"Oscillation and Concentration in Sequences of PDE Constrained Measures, preprint, arXiv:1912.09190","author":"Kristensen J.","year":"2019"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0092670"},{"key":"atypb33","first-page":"489","volume":"5","author":"Murat F.","year":"1978","journal-title":"Ann. Sc. Norm. Super. Pisa Cl. Sci."},{"key":"atypb34","volume-title":"Masson","author":"Necas J.","year":"1967"},{"key":"atypb35","volume-title":"Progr. Nonlinear Differential Equations Appl. 30","author":"Pedregal P.","year":"2012"},{"key":"atypb36","first-page":"58","author":"Rai\u0163\u0103 B.","year":"2019","journal-title":"Calc. Var. Partial Differential Equations"},{"key":"atypb37","volume-title":"Continuity and Canceling Operators of Order $n$ on $\\mathbb{R}^n$, preprint, arXiv:1903.03574","author":"Rai\u0163\u0103 B.","year":"2019"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-77637-8"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-003-0303-4"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1007\/s00526-017-1269-7"},{"key":"atypb41","volume-title":"Lower Semi-Continuity for $\\mathcal{A}$-quasiconvex Functionals Under Convex Restrictions, preprint, arXiv:1909.11543","author":"Skipper J. W.","year":"2019"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0061808"},{"key":"atypb43","first-page":"136","volume-title":"Nonlinear Analysis and Mechanics: Heriot-Watt Symposium","volume":"4","author":"Tartar L.","year":"1979"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1992-1037211-6"}],"container-title":["SIAM Journal on Mathematical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1345128","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:18:12Z","timestamp":1787321892000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1345128"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":44,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1345128"],"URL":"https:\/\/doi.org\/10.1137\/20m1345128","relation":{},"ISSN":["0036-1410","1095-7154"],"issn-type":[{"value":"0036-1410","type":"print"},{"value":"1095-7154","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}