{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T17:59:22Z","timestamp":1772906362193,"version":"3.50.1"},"reference-count":58,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T00:00:00Z","timestamp":1733702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100016378","name":"Technische Universit\u00e4t Dortmund","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100016378","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider a variational problem modeling transition between flat and wrinkled region in a thin elastic sheet and identify the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-limit as the sheet thickness goes to\u00a00, thus extending the previous work of the first author (Bella et al. in Arch Ration Mech Anal 218(1):163\u2013217, 2015). The limiting problem is scalar and convex, but constrained and posed for measures. For the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma -\\liminf $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0393<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mo>lim inf<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> inequality, we first pass to quadratic variables so that the constraint becomes linear, and then obtain the lower bound using Reshetnyak\u2019s theorem. The construction of the recovery sequence for the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Gamma -\\limsup $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0393<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mo>lim sup<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> inequality relies on mollification of quadratic variables and careful choice of multiple construction parameters. Eventually for the limiting problem, we show existence of a minimizer and equipartition of the energy for each frequency.\n<\/jats:p>","DOI":"10.1007\/s00332-024-10114-7","type":"journal-article","created":{"date-parts":[[2024,12,9]],"date-time":"2024-12-09T03:35:21Z","timestamp":1733715321000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["$$\\Gamma $$-Convergence for Plane to Wrinkles Transition Problem"],"prefix":"10.1007","volume":"35","author":[{"given":"Peter","family":"Bella","sequence":"first","affiliation":[]},{"given":"Roberta","family":"Marziani","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,12,9]]},"reference":[{"key":"10114_CR1","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":"10114_CR2","doi-asserted-by":"crossref","unstructured":"Bach, A., Esposito, T., Marziani, R., Zeppieri, C.I.: Interaction between oscillations and singular perturbations in a one-dimensional phase-field model. AWM Volume - Research in the Mathematics of Materials Science (2022)","DOI":"10.1007\/978-3-031-04496-0_1"},{"key":"10114_CR3","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1137\/22M1499145","volume":"55","author":"A Bach","year":"2023","unstructured":"Bach, A., Esposito, T., Marziani, R., Zeppieri, C.I.: Gradient damage models for heterogeneous materials. SIAM J. Math. Anal. 55, 4 (2023)","journal-title":"SIAM J. Math. Anal."},{"key":"10114_CR4","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/s00526-023-02540-4","volume":"62","author":"A Bach","year":"2023","unstructured":"Bach, A., Marziani, R., Zeppieri, C.I.: $$\\Gamma $$-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals. Calc. Var. 62, 199 (2023)","journal-title":"Calc. Var."},{"issue":"3","key":"10114_CR5","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1002\/cpa.21540","volume":"68","author":"J Bedrossian","year":"2015","unstructured":"Bedrossian, J., Kohn, R.V.: Blister patterns and energy minimization in compressed thin films on compliant substrates. Commun. Pure Appl. Math. 68(3), 472\u2013510 (2015)","journal-title":"Commun. Pure Appl. Math."},{"issue":"4","key":"10114_CR6","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1017\/S0308210515000086","volume":"145","author":"P Bella","year":"2015","unstructured":"Bella, P., Goldman, M.: Nucleation barriers at corners for a cubic-to-tetragonal phase transformation. Proc. R. Soc. Edinb. Sect. A 145(4), 715\u2013724 (2015)","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"issue":"1","key":"10114_CR7","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s00205-015-0858-x","volume":"218","author":"P Bella","year":"2015","unstructured":"Bella, P., Goldman, M., Zwicknagl, B.: Study of island formation in epitaxially strained films on unbounded domains. Arch. Ration. Mech. Anal. 218(1), 163\u2013217 (2015)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"5","key":"10114_CR8","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1002\/cpa.21471","volume":"67","author":"P Bella","year":"2014","unstructured":"Bella, P., Kohn, R.V.: Wrinkles as the result of compressive stresses in an annular thin film. Commun. Pure Appl. Math. 67(5), 693\u2013747 (2014)","journal-title":"Commun. Pure Appl. Math."},{"key":"10114_CR9","unstructured":"Bella, P., Kohn, R.V.: The annulus problem, using a von K\u00e1rm\u00e1n membrane model. Preliminary version https:\/\/math.nyu.edu\/~kohn\/pcmi\/annulus-problem-notes-for-pcmi.pdf"},{"issue":"5","key":"10114_CR10","doi-asserted-by":"crossref","first-page":"978","DOI":"10.1002\/cpa.21643","volume":"70","author":"P Bella","year":"2017","unstructured":"Bella, P., Kohn, R.V.: Coarsening of folds in hanging drapes. Commun. Pure Appl. Math. 70(5), 978\u20131021 (2017)","journal-title":"Commun. Pure Appl. Math."},{"key":"10114_CR11","doi-asserted-by":"crossref","first-page":"2093","DOI":"10.1098\/rsta.2016.0157","volume":"375","author":"P Bella","year":"2017","unstructured":"Bella, P., Kohn, R.V.: Wrinkling of a thin circular sheet bonded to a spherical substrate. Philos. Trans. R. Soc. A 375, 2093 (2017)","journal-title":"Philos. Trans. R. Soc. A"},{"key":"10114_CR12","unstructured":"Bella, P.: Wrinkles as a relaxation of compressive stresses. ProQuest LLC, Ann Arbor, MI, 2012, Thesis (Ph.D.)\u2013New York University"},{"issue":"2","key":"10114_CR13","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1007\/s00205-014-0816-z","volume":"216","author":"P Bella","year":"2015","unstructured":"Bella, P.: The transition between planar and wrinkled regions in a uniaxially stretched thin elastic film. Arch. Ration. Mech. Anal. 216(2), 623\u2013672 (2015)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"6","key":"10114_CR14","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1007\/s00332-014-9214-9","volume":"24","author":"P Bella","year":"2014","unstructured":"Bella, P., Kohn, R.V.: Metric-induced wrinkling of a thin elastic sheet. J. Nonlinear Sci. 24(6), 1147\u20131176 (2014)","journal-title":"J. Nonlinear Sci."},{"key":"10114_CR15","unstructured":"Bella, P., Rom\u00e1n, C.: Wrinkling of an elastic sheet floating on a liquid sphere (2023)"},{"issue":"6","key":"10114_CR16","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/s003320010007","volume":"10","author":"H Ben Belgacem","year":"2000","unstructured":"Ben Belgacem, H., Conti, S., DeSimone, A., M\u00fcller, S.: Rigorous bounds for the F\u00f6ppl-von K\u00e1rm\u00e1n theory of isotropically compressed plates. J. Nonlinear Sci. 10(6), 661\u2013683 (2000)","journal-title":"J. Nonlinear Sci."},{"issue":"1","key":"10114_CR17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s002050200206","volume":"164","author":"H Ben Belgacem","year":"2002","unstructured":"Ben Belgacem, H., Conti, S., DeSimone, A., M\u00fcller, S.: Energy scaling of compressed elastic films\u2014three-dimensional elasticity and reduced theories. Arch. Ration. Mech. Anal. 164(1), 1\u201337 (2002)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"10114_CR18","doi-asserted-by":"crossref","unstructured":"Bourne, D., Conti, S., M\u00fcller, S.: Folding patterns in partially delaminated thin films. In: Innovative Numerical Approaches for Multi-Field and Multi-Scale Problems. Lecture Notes in Applied and Computational Mechanics, vol 81. Springer, Cham, pp.\u00a025\u201339 (2016)","DOI":"10.1007\/978-3-319-39022-2_2"},{"issue":"2","key":"10114_CR19","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s10659-012-9420-3","volume":"113","author":"J Brandman","year":"2013","unstructured":"Brandman, J., Kohn, R.V., Nguyen, H.-M.: Energy scaling laws for conically constrained thin elastic sheets. J. Elast. 113(2), 251\u2013264 (2013)","journal-title":"J. Elast."},{"issue":"4","key":"10114_CR20","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s00526-023-02459-w","volume":"62","author":"B Brietzke","year":"2023","unstructured":"Brietzke, B., Kn\u00fcpfer, H.: Onset of pattern formation in thin ferromagnetic films with perpendicular anisotropy. Calc. Var. Partial Differ. Equ. 62(4), 133 (2023)","journal-title":"Calc. Var. Partial Differ. Equ."},{"issue":"12","key":"10114_CR21","doi-asserted-by":"crossref","first-page":"1632","DOI":"10.1002\/cpa.20297","volume":"62","author":"A Capella","year":"2009","unstructured":"Capella, A., Otto, F.: A rigidity result for a perturbation of the geometrically linear three-well problem. Commun. Pure Appl. Math. 62(12), 1632\u20131669 (2009)","journal-title":"Commun. Pure Appl. Math."},{"issue":"2","key":"10114_CR22","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1017\/S0308210510000478","volume":"142","author":"A Capella","year":"2012","unstructured":"Capella, A., Otto, F.: A quantitative rigidity result for the cubic-to-tetragonal phase transition in the geometrically linear theory with interfacial energy. Proc. R. Soc. Edinb. Sect. A 142(2), 273\u2013327 (2012)","journal-title":"Proc. R. Soc. Edinb. Sect. A"},{"issue":"3","key":"10114_CR23","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s00332-001-0456-y","volume":"11","author":"R Choksi","year":"2001","unstructured":"Choksi, R.: Scaling laws in microphase separation of diblock copolymers. J. Nonlinear Sci. 11(3), 223\u2013236 (2001)","journal-title":"J. Nonlinear Sci."},{"issue":"11","key":"10114_CR24","doi-asserted-by":"crossref","first-page":"1448","DOI":"10.1002\/1097-0312(200011)53:11<1448::AID-CPA6>3.0.CO;2-C","volume":"53","author":"S Conti","year":"2000","unstructured":"Conti, S.: Branched microstructures: scaling and asymptotic self-similarity. Commun. Pure Appl. Math. 53(11), 1448\u20131474 (2000)","journal-title":"Commun. Pure Appl. Math."},{"key":"10114_CR25","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1007\/s00526-023-02552-0","volume":"62","author":"S Conti","year":"2023","unstructured":"Conti, S., Garroni, A., Marziani, R.: Line-tension limits for line singularities and application to the mixed-growth case. Calc. Var. PDE 62, 228 (2023)","journal-title":"Calc. Var. PDE"},{"issue":"1","key":"10114_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00205-007-0076-2","volume":"187","author":"S Conti","year":"2008","unstructured":"Conti, S., Maggi, F.: Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187(1), 1\u201348 (2008)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"10114_CR27","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.85.066115","volume":"85","author":"B Davidovitch","year":"2012","unstructured":"Davidovitch, B., Schroll, R.D., Cerda, E.: Nonperturbative model for wrinkling in highly bendable sheets. Phys. Rev. E 85, 066115 (2012)","journal-title":"Phys. Rev. E"},{"issue":"45","key":"10114_CR28","doi-asserted-by":"crossref","first-page":"18227","DOI":"10.1073\/pnas.1108553108","volume":"108","author":"B Davidovitch","year":"2011","unstructured":"Davidovitch, B., Schroll, R.D., Vella, D., Adda-Bedia, M., Cerda, E.: Prototypical model for tensional wrinkling in thin sheets. Proc. Natl. Acad. Sci. 108(45), 18227\u201318232 (2011)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"10114_CR29","doi-asserted-by":"crossref","first-page":"4252","DOI":"10.1137\/20M1330117","volume":"53","author":"A Garroni","year":"2021","unstructured":"Garroni, A., Marziani, R., Scala, R.: Derivation of a line-tension model for dislocations from a nonlinear three-dimensional energy: the case of quadratic growth. SIAM J. Math. Anal. 53, 4252\u20134302 (2021)","journal-title":"SIAM J. Math. Anal."},{"issue":"2","key":"10114_CR30","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1140\/epje\/i2004-10041-1","volume":"15","author":"J-C G\u00e9minard","year":"2004","unstructured":"G\u00e9minard, J.-C., Bernal, R., Melo, F.: Wrinkle formations in axi-symmetrically stretched membranes. Eur. Phys. J. E 15(2), 117\u2013126 (2004)","journal-title":"Eur. Phys. J. E"},{"key":"10114_CR31","volume":"105","author":"J Huang","year":"2010","unstructured":"Huang, J., Davidovitch, B., Santangelo, C.D., Russell, T.P., Menon, N.: Smooth cascade of wrinkles at the edge of a floating elastic film. Phys. Rev. Lett. 105, 038302 (2010)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"10114_CR32","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1063\/1.1316058","volume":"42","author":"W Jin","year":"2001","unstructured":"Jin, W., Sternberg, P.: Energy estimates for the von K\u00e1rm\u00e1n model of thin-film blistering. J. Math. Phys. 42(1), 192\u2013199 (2001)","journal-title":"J. Math. Phys."},{"issue":"6","key":"10114_CR33","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1002\/cpa.21448","volume":"66","author":"H Kn\u00fcpfer","year":"2013","unstructured":"Kn\u00fcpfer, H., Kohn, R.V., Otto, F.: Nucleation barriers for the cubic-to-tetragonal phase transformation. Commun. Pure Appl. Math. 66(6), 867\u2013904 (2013)","journal-title":"Commun. Pure Appl. Math."},{"issue":"4","key":"10114_CR34","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1002\/cpa.3160470402","volume":"47","author":"RV Kohn","year":"1994","unstructured":"Kohn, R.V., M\u00fcller, S.: Surface energy and microstructure in coherent phase transitions. Commun. Pure Appl. Math. 47(4), 405\u2013435 (1994)","journal-title":"Commun. Pure Appl. Math."},{"key":"10114_CR35","doi-asserted-by":"crossref","unstructured":"Kohn, R.V.: Energy-driven pattern formation, International Congress of Mathematicians, vol. I, pp. 359\u2013383. Eur. Math. Soc., Z\u00fcrich (2007)","DOI":"10.4171\/022-1\/15"},{"issue":"3","key":"10114_CR36","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s00332-012-9154-1","volume":"23","author":"RV Kohn","year":"2013","unstructured":"Kohn, R.V., Nguyen, H.-M.: Analysis of a compressed thin film bonded to a compliant substrate: the energy scaling law. J. Nonlinear Sci. 23(3), 343\u2013362 (2013)","journal-title":"J. Nonlinear Sci."},{"issue":"4","key":"10114_CR37","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.1007\/s00332-018-9447-0","volume":"28","author":"RV Kohn","year":"2018","unstructured":"Kohn, R.V., O\u2019Brien, E.: The wrinkling of a twisted ribbon. J. Nonlinear Sci. 28(4), 1221\u20131249 (2018)","journal-title":"J. Nonlinear Sci."},{"issue":"2170","key":"10114_CR38","first-page":"20140432","volume":"470","author":"RV Kohn","year":"2014","unstructured":"Kohn, R.V., Wirth, B.: Optimal fine-scale structures in compliance minimization for a uniaxial load. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470(2170), 20140432 (2014)","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"issue":"8","key":"10114_CR39","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1002\/cpa.21589","volume":"69","author":"RV Kohn","year":"2016","unstructured":"Kohn, R.V., Wirth, B.: Optimal fine-scale structures in compliance minimization for a shear load. Commun. Pure Appl. Math. 69(8), 1572\u20131610 (2016)","journal-title":"Commun. Pure Appl. Math."},{"issue":"2","key":"10114_CR40","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/s10659-018-9686-1","volume":"134","author":"C Maor","year":"2019","unstructured":"Maor, C., Shachar, A.: On the role of curvature in the elastic energy of non-Euclidean thin bodies. J. Elast. 134(2), 149\u2013173 (2019)","journal-title":"J. Elast."},{"key":"10114_CR41","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1051\/cocv\/2023030","volume":"29","author":"R Marziani","year":"2023","unstructured":"Marziani, R.: $$\\Gamma $$-convergence and stochastic homogenisation of phase-transition functionals. ESAIM Control Optim. Calc. Var. 29, 44 (2023)","journal-title":"ESAIM Control Optim. Calc. Var."},{"issue":"4","key":"10114_CR42","doi-asserted-by":"publisher","first-page":"1060","DOI":"10.1017\/prm.2023.51","volume":"154","author":"R Marziani","year":"2024","unstructured":"Marziani, R., Solombrino, F.: Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenization. Proc. R. Soc. Edinb. Sect. A Math. 154(4), 1060\u20131094 (2024). https:\/\/doi.org\/10.1017\/prm.2023.51","journal-title":"Proc. R. Soc. Edinb. Sect. A Math."},{"issue":"5","key":"10114_CR43","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.2140\/apde.2018.11.1285","volume":"11","author":"H Olbermann","year":"2018","unstructured":"Olbermann, H.: The shape of low energy configurations of a thin elastic sheet with a single disclination. Anal. PDE 11(5), 1285\u20131302 (2018)","journal-title":"Anal. PDE"},{"issue":"1","key":"10114_CR44","doi-asserted-by":"crossref","first-page":"245","DOI":"10.2140\/apde.2019.12.245","volume":"12","author":"H Olbermann","year":"2019","unstructured":"Olbermann, H.: On a boundary value problem for conically deformed thin elastic sheets. Anal. PDE 12(1), 245\u2013258 (2019)","journal-title":"Anal. PDE"},{"key":"10114_CR45","unstructured":"Otto, F.: Cross-over in scaling laws: a simple example from micromagnetics. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, pp.\u00a0829\u2013838 (2002)"},{"issue":"8","key":"10114_CR46","doi-asserted-by":"crossref","first-page":"4956","DOI":"10.1002\/mma.6161","volume":"43","author":"D Padilla-Garza","year":"2020","unstructured":"Padilla-Garza, D.: An asymptotic variational problem modeling a thin elastic sheet on a liquid, lifted at one end. Math. Methods Appl. Sci. 43(8), 4956\u20134973 (2020)","journal-title":"Math. Methods Appl. Sci."},{"issue":"3","key":"10114_CR47","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1017\/S1474748015000201","volume":"16","author":"M Petrache","year":"2017","unstructured":"Petrache, M., Serfaty, S.: Next order asymptotics and renormalized energy for Riesz interactions. J. Inst. Math. Jussieu 16(3), 501\u2013569 (2017)","journal-title":"J. Inst. Math. Jussieu"},{"issue":"3","key":"10114_CR48","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1093\/imamat\/52.3.297","volume":"52","author":"AC Pipkin","year":"1994","unstructured":"Pipkin, A.C.: Relaxed energy densities for large deformations of membranes. IMA J. Appl. Math. 52(3), 297\u2013308 (1994)","journal-title":"IMA J. Appl. Math."},{"key":"10114_CR49","doi-asserted-by":"publisher","unstructured":"Ried, T., Rom\u00e1n, C.: Domain branching in micromagnetism: scaling law for the global and local energies. Rev. Mat. Iberoam. (2024). https:\/\/doi.org\/10.4171\/RMI\/1508","DOI":"10.4171\/RMI\/1508"},{"issue":"3","key":"10114_CR50","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1112\/plms.12505","volume":"126","author":"C Rom\u00e1n","year":"2023","unstructured":"Rom\u00e1n, C., Sandier, E., Serfaty, S.: Bounded vorticity for the 3D Ginzburg-Landau model and an isoflux problem. Proc. Lond. Math. Soc. (3) 126(3), 1015\u20131062 (2023)","journal-title":"Proc. Lond. Math. Soc. (3)"},{"issue":"2","key":"10114_CR51","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s00332-022-09879-6","volume":"33","author":"A R\u00fcland","year":"2023","unstructured":"R\u00fcland, A., Tribuzio, A.: On scaling laws for multi-well nucleation problems without gauge invariances. J. Nonlinear Sci. 33(2), 25 (2023a)","journal-title":"J. Nonlinear Sci."},{"key":"10114_CR52","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1051\/cocv\/2023047","volume":"29","author":"A R\u00fcland","year":"2023","unstructured":"R\u00fcland, A., Tribuzio, A.: On the energy scaling behaviour of singular perturbation models with prescribed Dirichlet data involving higher order laminates. ESAIM Control Optim. Calc. Var. 29, 68 (2023b)","journal-title":"ESAIM Control Optim. Calc. Var."},{"key":"10114_CR53","doi-asserted-by":"crossref","unstructured":"Santambrogio, F.: Optimal transport for applied mathematicians. Progress in Nonlinear Differential Equations and their Applications, vol.\u00a087, Birkh\u00e4user\/Springer, Cham (2015), Calculus of variations, PDEs, and modeling","DOI":"10.1007\/978-3-319-20828-2"},{"key":"10114_CR54","doi-asserted-by":"crossref","unstructured":"Serfaty, S.: Coulomb gases and Ginzburg-Landau vortices, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z\u00fcrich (2015)","DOI":"10.4171\/152"},{"issue":"2","key":"10114_CR55","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1002\/cpa.21704","volume":"71","author":"I Tobasco","year":"2018","unstructured":"Tobasco, I.: Axial compression of a thin elastic cylinder: bounds on the minimum energy scaling law. Commun. Pure Appl. Math. 71(2), 304\u2013355 (2018)","journal-title":"Commun. Pure Appl. Math."},{"issue":"3","key":"10114_CR56","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.1007\/s00205-020-01566-8","volume":"239","author":"I Tobasco","year":"2021","unstructured":"Tobasco, I.: Curvature-driven wrinkling of thin elastic shells. Arch. Ration. Mech. Anal. 239(3), 1211\u20131325 (2021)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"10114_CR57","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.106.224301","volume":"106","author":"H Vandeparre","year":"2011","unstructured":"Vandeparre, H., Pi\u00f1eirua, M., Brau, F., Roman, B., Bico, J., Gay, C., Bao, W., Lau, C.N., Reis, P.M., Damman, P.: Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains. Phys. Rev. Lett. 106, 224301 (2011)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"10114_CR58","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1088\/0951-7715\/17\/1\/017","volume":"17","author":"S Venkataramani","year":"2004","unstructured":"Venkataramani, S.: Lower bounds for the energy in a crumpled elastic sheet\u2014a minimal ridge. Nonlinearity 17(1), 301\u2013312 (2004)","journal-title":"Nonlinearity"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-024-10114-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-024-10114-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-024-10114-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,10]],"date-time":"2025-02-10T13:10:53Z","timestamp":1739193053000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-024-10114-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,9]]},"references-count":58,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,2]]}},"alternative-id":["10114"],"URL":"https:\/\/doi.org\/10.1007\/s00332-024-10114-7","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,9]]},"assertion":[{"value":"23 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 December 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"23"}}