{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T03:23:19Z","timestamp":1777605799554,"version":"3.51.4"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,5,14]],"date-time":"2021-05-14T00:00:00Z","timestamp":1620950400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2021,5,14]],"date-time":"2021-05-14T00:00:00Z","timestamp":1620950400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["RTG-DMS 1840314"],"award-info":[{"award-number":["RTG-DMS 1840314"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007710","name":"University of Connecticut","doi-asserted-by":"publisher","award":["Research Excellence Grant"],"award-info":[{"award-number":["Research Excellence Grant"]}],"id":[{"id":"10.13039\/100007710","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2021,6]]},"DOI":"10.1007\/s00332-021-09717-1","type":"journal-article","created":{"date-parts":[[2021,5,14]],"date-time":"2021-05-14T13:02:33Z","timestamp":1620997353000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Compactness and Sharp Lower Bound for a 2D Smectics Model"],"prefix":"10.1007","volume":"31","author":[{"given":"Michael","family":"Novack","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaodong","family":"Yan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,5,14]]},"reference":[{"key":"9717_CR1","doi-asserted-by":"publisher","first-page":"257802","DOI":"10.1103\/PhysRevLett.104.257802","volume":"104","author":"GP Alexander","year":"2010","unstructured":"Alexander, G.P., Chen, B.G.-G., Matsumoto, E.A., Kamien, R.D.: Power of the Poincar\u00e9 group: elucidating the hidden symmetries in focal conic domains. Phys. Rev. Lett. 104, 257802 (2010)","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"9717_CR2","doi-asserted-by":"publisher","first-page":"047802","DOI":"10.1103\/PhysRevLett.108.047802","volume":"108","author":"G Alexander","year":"2012","unstructured":"Alexander, G., Kamien, R., Santangelo, C.: Developed smectics: when exact solutions agree. Phys. Rev. Lett. 108(4), 047802 (2012a)","journal-title":"Phys. Rev. Lett."},{"key":"9717_CR3","doi-asserted-by":"publisher","first-page":"050701","DOI":"10.1103\/PhysRevE.85.050701","volume":"85","author":"GP Alexander","year":"2012","unstructured":"Alexander, G.P., Kamien, R.D., Mosna, R.A.: Conformal smectics and their many metrics. Phys. Rev. E 85, 050701 (2012b)","journal-title":"Phys. Rev. E"},{"key":"9717_CR4","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1090\/S0002-9939-1990-0969514-3","volume":"108","author":"L Ambrosio","year":"1990","unstructured":"Ambrosio, L., Dal Maso, G.: A general chain rule for distributional derivatives. Proc. Am. Math. Soc. 108, 691\u2013702 (1990)","journal-title":"Proc. Am. Math. Soc."},{"key":"9717_CR5","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s005260050144","volume":"9","author":"L Ambrosio","year":"1999","unstructured":"Ambrosio, L., De Lellis, C., Mantegazza, C.: Line energies for gradient vector fields in the plane. Calc. Var. Partial Differ. Equ. 9, 327\u2013355 (1999)","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"9717_CR6","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198502456.001.0001","volume-title":"Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs","author":"L Ambrosio","year":"2000","unstructured":"Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)"},{"key":"9717_CR7","unstructured":"Aviles, P., Giga, Y.: A mathematical problem related to the physical theory of liquid crystal configurations. In: Miniconference on Geometry and Partial Differential Equations, 2 (Canberra, 1986), Volume 12 of Proceedings of The Centre for Mathematical Analysis, Australian National University, Canberra, pp. 1\u201316 (1987)"},{"key":"9717_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0308210500027438","volume":"129","author":"P Aviles","year":"1999","unstructured":"Aviles, P., Giga, Y.: On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg\u2013Landau type energy for gradient fields. Proc. R. Soc. Edinb. Sect. A 129, 1\u201317 (1999)","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"key":"9717_CR9","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1209\/epl\/i2002-00109-0","volume":"59","author":"I Bluestein","year":"2002","unstructured":"Bluestein, I., Kamien, R.D.: Nonlinear effects in the TGB a phase. Europhys. Lett. (EPL) 59, 68\u201374 (2002)","journal-title":"Europhys. Lett. (EPL)"},{"key":"9717_CR10","unstructured":"Bogomol\u2019nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys. (Engl. Transl.); (United States), 24 (1976)"},{"key":"9717_CR11","doi-asserted-by":"publisher","first-page":"R4752","DOI":"10.1103\/PhysRevE.59.R4752","volume":"59","author":"EA Brener","year":"1999","unstructured":"Brener, E.A., Marchenko, V.I.: Nonlinear theory of dislocations in smectic crystals: an exact solution. Phys. Rev. E 59, R4752\u2013R4753 (1999)","journal-title":"Phys. Rev. E"},{"key":"9717_CR12","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1007\/s00332-006-0805-y","volume":"17","author":"R Cantero-\u00c1lvarez","year":"2007","unstructured":"Cantero-\u00c1lvarez, R., Otto, F., Steiner, J.: The concertina pattern: a bifurcation in ferromagnetic thin films. J. Nonlinear Sci. 17, 221\u2013281 (2007)","journal-title":"J. Nonlinear Sci."},{"key":"9717_CR13","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511813467","volume-title":"Principles of Condensed Matter Physics","author":"PM Chaikin","year":"1995","unstructured":"Chaikin, P.M., Lubensky, T.C.: Principles of Condensed Matter Physics. Cambridge University Press, Cambridge (1995)"},{"key":"9717_CR14","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/s00208-006-0070-2","volume":"338","author":"S Conti","year":"2007","unstructured":"Conti, S., De Lellis, C.: Sharp upper bounds for a variational problem with singular perturbation. Math. Ann. 338, 119\u2013146 (2007)","journal-title":"Math. Ann."},{"key":"9717_CR15","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1017\/S030821050000113X","volume":"131","author":"A DeSimone","year":"2001","unstructured":"DeSimone, A., M\u00fcller, S., Kohn, R.V., Otto, F.: A compactness result in the gradient theory of phase transitions. Proc. R. Soc. Edinb. Sect. A Math. 131, 833\u2013844 (2001)","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"key":"9717_CR16","doi-asserted-by":"publisher","first-page":"215504","DOI":"10.1103\/PhysRevLett.89.215504","volume":"89","author":"BA DiDonna","year":"2002","unstructured":"DiDonna, B.A., Kamien, R.D.: Smectic phases with cubic symmetry: the splay analog of the blue phase. Phys. Rev. Lett. 89, 215504 (2002)","journal-title":"Phys. Rev. Lett."},{"key":"9717_CR17","doi-asserted-by":"publisher","first-page":"041703, 11","DOI":"10.1103\/PhysRevE.68.041703","volume":"68","author":"BA DiDonna","year":"2003","unstructured":"DiDonna, B.A., Kamien, R.D.: Smectic blue phases: layered systems with high intrinsic curvature. Phys. Rev. E (3) 68, 041703, 11 (2003)","journal-title":"Phys. Rev. E (3)"},{"key":"9717_CR18","volume-title":"The Physics of Liquid Crystals, International Series of Monographs on Physics (Oxford, England)","author":"PD Gennes","year":"1993","unstructured":"Gennes, P.D., Prost, J.: The Physics of Liquid Crystals, International Series of Monographs on Physics (Oxford, England), vol. 83, 2nd edn. Clarendon Press, Oxford University Press, New York, Oxford (1993)","edition":"2"},{"key":"9717_CR19","doi-asserted-by":"publisher","first-page":"1739","DOI":"10.1007\/s00205-020-01501-x","volume":"236","author":"D Golovaty","year":"2020","unstructured":"Golovaty, D., Novack, M., Sternberg, P., Venkatraman, R.: A model problem for nematic\u2013isotropic transitions with highly disparate elastic constants. Arch. Ration. Mech. Anal. 236, 1739\u20131805 (2020)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9717_CR20","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/s00205-010-0325-7","volume":"199","author":"R Ignat","year":"2011","unstructured":"Ignat, R., Merlet, B.: Lower bound for the energy of Bloch walls in micromagnetics. Arch. Ration. Mech. Anal. 199, 369\u2013406 (2011)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9717_CR21","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1007\/s00526-011-0438-3","volume":"44","author":"R Ignat","year":"2012","unstructured":"Ignat, R., Merlet, B.: Entropy method for line-energies. Calc. Var. Partial Differ. Equ. 44, 375\u2013418 (2012)","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"9717_CR22","doi-asserted-by":"publisher","first-page":"771","DOI":"10.1002\/cpa.21867","volume":"73","author":"R Ignat","year":"2020","unstructured":"Ignat, R., Monteil, A.: A De Giorgi-type conjecture for minimal solutions to a nonlinear Stokes equation. Commun. Pure Appl. Math. 73, 771\u2013854 (2020)","journal-title":"Commun. Pure Appl. Math."},{"key":"9717_CR23","unstructured":"Ignat, R., Otto, F., Reid, T., Tsatsoulis, P.: Variational methods for a singular SPDE yielding the universality of the magnetization ripple. arxiv.org\/pdf\/2010.13123"},{"key":"9717_CR24","doi-asserted-by":"publisher","first-page":"R5037","DOI":"10.1103\/PhysRevE.60.R5037","volume":"60","author":"T Ishikawa","year":"1999","unstructured":"Ishikawa, T., Lavrentovich, O.D.: Dislocation profile in cholesteric finger texture. Phys. Rev. E 60, R5037\u2013R5039 (1999)","journal-title":"Phys. Rev. E"},{"key":"9717_CR25","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/s003329910014","volume":"10","author":"W Jin","year":"2000","unstructured":"Jin, W., Kohn, R.V.: Singular perturbation and the energy of folds. J. Nonlinear Sci. 10, 355\u2013390 (2000)","journal-title":"J. Nonlinear Sci."},{"key":"9717_CR26","doi-asserted-by":"publisher","first-page":"2892","DOI":"10.1103\/PhysRevLett.82.2892","volume":"82","author":"RD Kamien","year":"1999","unstructured":"Kamien, R.D., Lubensky, T.C.: Minimal surfaces, screw dislocations, and twist grain boundaries. Phys. Rev. Lett. 82, 2892\u20132895 (1999)","journal-title":"Phys. Rev. Lett."},{"key":"9717_CR27","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s10711-006-9075-y","volume":"120","author":"RD Kamien","year":"2006","unstructured":"Kamien, R.D., Santangelo, C.D.: Smectic liquid crystals: materials with one-dimensional, periodic order. Geom. Dedic. 120, 229\u2013240 (2006)","journal-title":"Geom. Dedic."},{"key":"9717_CR28","volume-title":"Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media, Oxford Mathematical Monographs","author":"M Kl\u00e9man","year":"1983","unstructured":"Kl\u00e9man, M.: Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media, Oxford Mathematical Monographs. Wiley, New York (1983)"},{"key":"9717_CR29","doi-asserted-by":"crossref","unstructured":"Kohn, R.V.: Energy-driven pattern formation. In: International Congress of Mathematicians, vol. I, pp. 359\u2013383. European Mathematical Society, Z\u00fcrich (2007)","DOI":"10.4171\/022-1\/15"},{"key":"9717_CR30","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1098\/rsfs.2011.0095","volume":"2","author":"EA Matsumoto","year":"2012","unstructured":"Matsumoto, E.A., Santangelo, C.D., Kamien, R.D.: Smectic pores and defect cores. J. Interface Focus 2, 617\u2013622 (2012)","journal-title":"J. Interface Focus"},{"issue":"14","key":"9717_CR31","first-page":"285","volume":"5","author":"L Modica","year":"1977","unstructured":"Modica, L., Mortola, S.: Un esempio di $$\\Gamma ^{-}$$-convergenza. Boll. Un. Mat. Ital. B 5(14), 285\u2013299 (1977)","journal-title":"Boll. Un. Mat. Ital. B"},{"issue":"5","key":"9717_CR32","first-page":"489","volume":"4","author":"F Murat","year":"1978","unstructured":"Murat, F.: Compacit\u00e9 par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(5), 489\u2013507 (1978)","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci."},{"issue":"8","key":"9717_CR33","first-page":"69","volume":"4","author":"F Murat","year":"1981","unstructured":"Murat, F.: Compacit\u00e9 par compensation: condition n\u00e9cessaire et suffisante de continuit\u00e9 faible sous une hypoth\u00e8se de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(8), 69\u2013102 (1981a)","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci."},{"issue":"60","key":"9717_CR34","first-page":"309","volume":"9","author":"F Murat","year":"1981","unstructured":"Murat, F.: L\u2019injection du c\u00f4ne positif de $$H^{-1}$$ dans $$W^{-1,\\, q}$$ est compacte pour tout $$q<2$$. J. Math. Pures Appl. 9(60), 309\u2013322 (1981b)","journal-title":"J. Math. Pures Appl."},{"key":"9717_CR35","unstructured":"Novack, M., Yan, X.: Nonlinear approximation of 3D smectic liquid crystals: compactness and sharp lower bound (xxx). (In preparation)"},{"key":"9717_CR36","unstructured":"Novack, M., Yan, X.: A smectic liquid crystal model in periodic setting (xxxx). (In preparation)"},{"key":"9717_CR37","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/s00526-009-0305-7","volume":"39","author":"F Otto","year":"2010","unstructured":"Otto, F., Steiner, J.: The concertina pattern. Calc. Var. Partial Differ. Equ. 39, 139\u2013181 (2010)","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"9717_CR38","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4171\/JEMS\/70","volume":"9","author":"A Poliakovsky","year":"2007","unstructured":"Poliakovsky, A.: Upper bounds for singular perturbation problems involving gradient fields. J. Eur. Math. Soc. (JEMS) 9, 1\u201343 (2007)","journal-title":"J. Eur. Math. Soc. (JEMS)"},{"key":"9717_CR39","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1007\/s11854-008-0024-6","volume":"104","author":"A Poliakovsky","year":"2008","unstructured":"Poliakovsky, A.: A general technique to prove upper bounds for singular perturbation problems. J. Anal. Math. 104, 247\u2013290 (2008)","journal-title":"J. Anal. Math."},{"key":"9717_CR40","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/0550-3213(76)90424-7","volume":"110","author":"M Prasad","year":"1976","unstructured":"Prasad, M., Sommerfield, C\u00a0.M.: Solutions of classical gauge field theories with spin and internal symmetry. Nucl. Phys. B 110, 153\u2013172 (1976)","journal-title":"Nucl. Phys. B"},{"key":"9717_CR41","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1080\/14645180601168117","volume":"15","author":"CD Santangelo","year":"2006","unstructured":"Santangelo, C.D.: Geometry and the nonlinear elasticity of defects in smectic liquid crystals. Liq. Cryst. Today 15, 11\u201318 (2006)","journal-title":"Liq. Cryst. Today"},{"key":"9717_CR42","doi-asserted-by":"publisher","first-page":"045506","DOI":"10.1103\/PhysRevLett.91.045506","volume":"91","author":"CD Santangelo","year":"2003","unstructured":"Santangelo, C.D., Kamien, R.D.: Bogomol\u2019nyi, Prasad, and Sommerfield configurations in smectics. Phys. Rev. Lett. 91, 045506 (2003)","journal-title":"Phys. Rev. Lett."},{"key":"9717_CR43","first-page":"2911","volume":"461","author":"CD Santangelo","year":"2005","unstructured":"Santangelo, C.D., Kamien, R.D.: Curvature and topology in smectic-A liquid crystals. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, 2911\u20132921 (2005)","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"issue":"12","key":"9717_CR44","doi-asserted-by":"publisher","first-page":"011702","DOI":"10.1103\/PhysRevE.75.011702","volume":"75","author":"C\u00a0D Santangelo","year":"2007","unstructured":"Santangelo, C\u00a0.D., Kamien, R\u00a0.D.: Triply periodic smectic liquid crystals. Phys. Rev. E (3) 75(12), 011702 (2007)","journal-title":"Phys. Rev. E (3)"},{"key":"9717_CR45","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/BF00253122","volume":"101","author":"P Sternberg","year":"1988","unstructured":"Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209\u2013260 (1988)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9717_CR46","unstructured":"Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics: Heriot\u2013Watt Symposium, vol. IV, Volume\u00a039 of Research Notes in Mathematics, pp.\u00a0136\u2013212. Pitman, Boston, London (1979)"},{"key":"9717_CR47","doi-asserted-by":"crossref","unstructured":"Tartar, L.: The compensated compactness method applied to systems of conservation laws. In: Systems of Nonlinear Partial Differential Equations (Oxford, 1982), Volume\u00a0111 of NATO Advanced Science Institute Series C: Mathematical and Physical Sciences, pp.\u00a0263\u2013285. Reidel, Dordrecht (1983)","DOI":"10.1007\/978-94-009-7189-9_13"},{"key":"9717_CR48","unstructured":"Tartar, L.: The compensated compactness method for a scalar hyperbolic equation. Carnegie Mellon University Lecture Notes 87\u201320 (1987)"},{"issue":"29","key":"9717_CR49","first-page":"395","volume":"5","author":"L Tartar","year":"2005","unstructured":"Tartar, L.: Compensation effects in partial differential equations. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 5(29), 395\u2013453 (2005)","journal-title":"Rend. Accad. Naz. Sci. XL Mem. Mat. Appl."},{"key":"9717_CR50","unstructured":"Temam, R.: Navier\u2013Stokes Equations, Volume 2 of Studies in Mathematics and Its Applications, revised edn. North-Holland Publishing Co., Amsterdam-New York. Theory and numerical analysis. With an appendix by F. Thomasset (1979)"},{"issue":"115","key":"9717_CR51","first-page":"255","volume":"73","author":"AI Vol\u2019pert","year":"1967","unstructured":"Vol\u2019pert, A.I.: Spaces $${\\rm BV}$$ and quasilinear equations. Mat. Sb. (N.S.) 73(115), 255\u2013302 (1967)","journal-title":"Mat. Sb. (N.S.)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09717-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-021-09717-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-021-09717-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,30]],"date-time":"2024-08-30T21:59:33Z","timestamp":1725055173000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-021-09717-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,14]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,6]]}},"alternative-id":["9717"],"URL":"https:\/\/doi.org\/10.1007\/s00332-021-09717-1","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,14]]},"assertion":[{"value":"15 July 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"60"}}