{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,29]],"date-time":"2025-03-29T16:42:11Z","timestamp":1743266531891,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:00:00Z","timestamp":1584576000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T00:00:00Z","timestamp":1584576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100009767","name":"Heriot-Watt University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100009767","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":[[2020,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A novel general framework for the study of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-limit of these kind of functionals by knowing the<jats:inline-formula><jats:alternatives><jats:tex-math>$$ \\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u0393<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-limit of the underlying energies. In particular, the interaction between the functionals and the underlying energies results, in the case these latter converge to a non-continuous energy, in an additional effect in the relaxation process. This study was motivated by a question in the context of epitaxial growth evolution with adatoms. Interesting cases of application of the general theory are also presented.<\/jats:p>","DOI":"10.1007\/s00332-020-09623-y","type":"journal-article","created":{"date-parts":[[2020,3,19]],"date-time":"2020-03-19T19:03:38Z","timestamp":1584644618000},"page":"1723-1769","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the Gamma Convergence of Functionals Defined Over Pairs of Measures and Energy-Measures"],"prefix":"10.1007","volume":"30","author":[{"given":"Marco","family":"Caroccia","sequence":"first","affiliation":[]},{"given":"Riccardo","family":"Cristoferi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,3,19]]},"reference":[{"key":"9623_CR1","doi-asserted-by":"publisher","first-page":"1111","DOI":"10.1016\/j.anihpc.2007.09.004","volume":"25","author":"E Acerbi","year":"2008","unstructured":"Acerbi, E., Bouchitt\u00e9, G.: A general class of phase transition models with weighted interface energy. Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire 25, 1111\u20131143 (2008)","journal-title":"Ann. Inst. H. Poincar\u00e9 Anal. Non Lin\u00e9aire"},{"key":"9623_CR2","doi-asserted-by":"publisher","first-page":"65","DOI":"10.4171\/IFB\/274","volume":"14","author":"R Alicandro","year":"2012","unstructured":"Alicandro, R., Cicalese, M., Sigalotti, L.: Phase transitions in presence of surfactants: from discrete to continuum. Interfaces Free Bound. 14, 65\u2013103 (2012)","journal-title":"Interfaces Free Bound."},{"key":"9623_CR3","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":"9623_CR4","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1093\/qmath\/hax009","volume":"68","author":"M Ba\u00eda","year":"2017","unstructured":"Ba\u00eda, M., Barroso, A.C., Matias, J.: A model for phase transitions with competing terms. Q. J. Math. 68, 957\u20131000 (2017)","journal-title":"Q. J. Math."},{"key":"9623_CR5","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198507840.001.0001","volume-title":"Gamma-Convergence for Beginners","author":"A Braides","year":"2002","unstructured":"Braides, A.: Gamma-Convergence for Beginners, vol. 22. Clarendon Press, Oxford (2002)"},{"key":"9623_CR6","doi-asserted-by":"crossref","unstructured":"Bresson, X., Laurent, T., Uminsky, D., von Brecht, J.H.: Multiclass total variation clustering. In: Proceedings of the 26th International Conference on Neural Information Processing Systems (NIPS), pp.\u00a01421\u20131429 (2013)","DOI":"10.21236\/ADA612811"},{"key":"9623_CR7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4310\/CMS.2006.v4.n1.a1","volume":"4","author":"M Burger","year":"2006","unstructured":"Burger, M.: Surface diffusion including adatoms. Commun. Math. Sci. 4, 1\u201351 (2006)","journal-title":"Commun. Math. Sci."},{"key":"9623_CR8","unstructured":"Buttazzo, G.: Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, vol.\u00a0207 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York (1989)"},{"issue":"159","key":"9623_CR9","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/BF01766298","volume":"4","author":"G Buttazzo","year":"1991","unstructured":"Buttazzo, G., Freddi, L.: Functionals defined on measures and applications to non-equi-uniformly elliptic problems. Ann. Mat. Pura Appl. 4(159), 133\u2013149 (1991)","journal-title":"Ann. Mat. Pura Appl."},{"key":"9623_CR10","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1007\/s10851-016-0678-0","volume":"57","author":"L Calatroni","year":"2017","unstructured":"Calatroni, L., van Gennip, Y., Sch\u00f6nlieb, C.-B., Rowland, H.M., Flenner, A.: Graph clustering, variational image segmentation methods and Hough transform scale detection for object measurement in images. J. Math. Imaging Vis. 57, 269\u2013291 (2017)","journal-title":"J. Math. Imaging Vis."},{"key":"9623_CR11","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1007\/s00205-018-1258-9","volume":"230","author":"M Caroccia","year":"2018","unstructured":"Caroccia, M., Cristoferi, R., Dietrich, L.: Equilibria configurations for epitaxial crystal growth with adatoms. Arch. Ration. Mech. Anal. 230, 785\u2013838 (2018)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9623_CR12","doi-asserted-by":"crossref","unstructured":"Caroccia, M., Chambolle, A., Mumford Slep\u010dev, D.: Mumford-Shah functionals on graphs and their asymptotics. Preprint (2020)","DOI":"10.1088\/1361-6544\/ab81ee"},{"key":"9623_CR13","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1051\/m2an\/2009044","volume":"44","author":"A Chambolle","year":"2010","unstructured":"Chambolle, A., Giacomini, A., Lussardi, L.: Continuous limits of discrete perimeters. M2AN Math. Model. Numer. Anal. 44, 207\u2013230 (2010)","journal-title":"M2AN Math. Model. Numer. Anal."},{"key":"9623_CR14","unstructured":"Cristoferi, R., Thorpe, M.: Large data limit for a phase transition model with the p-Laplacian on point clouds, to appear on European Journal of Applied Mathematics, ArXiv preprint ArXiv:1802.08703 (2018)"},{"key":"9623_CR15","unstructured":"Dal Maso, G.: An Introduction to $$\\Gamma $$-Convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkh\u00e4user, Boston (1993)"},{"key":"9623_CR16","doi-asserted-by":"crossref","unstructured":"De\u00a0Lellis, C.: Rectifiable sets, densities and tangent measures. In: Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z\u00fcrich (2008)","DOI":"10.4171\/044"},{"key":"9623_CR17","volume-title":"Modern Methods in the Calculus of Variations: $$L^p$$ Spaces, Springer Monographs in Mathematics","author":"I Fonseca","year":"2007","unstructured":"Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: $$L^p$$ Spaces, Springer Monographs in Mathematics. Springer, New York (2007)"},{"key":"9623_CR18","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1007\/s00205-007-0082-4","volume":"186","author":"I Fonseca","year":"2007","unstructured":"Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Equilibrium configurations of epitaxially strained crystalline films: existence and regularity results. Arch. Ration. Mech. Anal. 186, 477\u2013537 (2007a)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9623_CR19","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/s00205-006-0012-x","volume":"183","author":"I Fonseca","year":"2007","unstructured":"Fonseca, I., Morini, M., Slastikov, V.: Surfactants in foam stability: a phase-field model. Arch. Ration. Mech. Anal. 183, 411\u2013456 (2007b)","journal-title":"Arch. Ration. Mech. Anal."},{"issue":"96","key":"9623_CR20","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1016\/j.matpur.2011.07.003","volume":"9","author":"I Fonseca","year":"2011","unstructured":"Fonseca, I., Fusco, N., Leoni, G., Millot, V.: Material voids in elastic solids with anisotropic surface energies. J. Math. Pures Appl. 9(96), 591\u2013639 (2011)","journal-title":"J. Math. Pures Appl."},{"key":"9623_CR21","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1007\/s00205-012-0509-4","volume":"205","author":"I Fonseca","year":"2012","unstructured":"Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Motion of elastic thin films by anisotropic surface diffusion with curvature regularization. Arch. Ration. Mech. Anal. 205, 425\u2013466 (2012)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9623_CR22","doi-asserted-by":"publisher","first-page":"373","DOI":"10.2140\/apde.2015.8.373","volume":"8","author":"I Fonseca","year":"2015","unstructured":"Fonseca, I., Fusco, N., Leoni, G., Morini, M.: Motion of three-dimensional elastic films by anisotropic surface diffusion with curvature regularization. Anal. PDE 8, 373\u2013423 (2015)","journal-title":"Anal. PDE"},{"key":"9623_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0065-2156(04)40001-5","volume":"40","author":"E Fried","year":"2004","unstructured":"Fried, E., Gurtin, M.E.: A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundaries and epitaxy. Adv. Appl. Mech. 40, 1\u2013177 (2004)","journal-title":"Adv. Appl. Mech."},{"key":"9623_CR24","doi-asserted-by":"publisher","first-page":"1358","DOI":"10.4153\/CJM-2014-044-6","volume":"67","author":"N Garc\u00eda Trillos","year":"2015","unstructured":"Garc\u00eda Trillos, N., Slep\u010dev, D.: On the rate of convergence of empirical measures in $$\\infty $$-transportation distance. Can. J. Math. 67, 1358\u20131383 (2015)","journal-title":"Can. J. Math."},{"issue":"2","key":"9623_CR25","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.acha.2016.09.003","volume":"45","author":"N Garc\u00eda Trillos","year":"2016","unstructured":"Garc\u00eda Trillos, N., Slep\u010dev, D.: A variational approach to the consistency of spectral clustering. Appl. Comput. Harmonic Anal. 45(2), 239\u2013281 (2016)","journal-title":"Appl. Comput. Harmonic Anal."},{"key":"9623_CR26","doi-asserted-by":"crossref","unstructured":"Maggi, F.: Sets of Finite Perimeter and Geometric Variational Problems, vol.\u00a0135 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (2012). An introduction to geometric measure theory","DOI":"10.1017\/CBO9781139108133"},{"key":"9623_CR27","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1016\/S0294-1449(16)30360-2","volume":"4","author":"L Modica","year":"1987","unstructured":"Modica, L.: Gradient theory of phase transitions with boundary contact energy. Annales de l\u2019Institut Henri Poincare (C) Non Linear Anal. 4, 487\u2013512 (1987)","journal-title":"Annales de l\u2019Institut Henri Poincare (C) Non Linear Anal."},{"issue":"14","key":"9623_CR28","first-page":"526","volume":"5","author":"L Modica","year":"1977","unstructured":"Modica, L., Mortola, S.: Il limite nella $$\\Gamma $$-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A 5(14), 526\u2013529 (1977)","journal-title":"Boll. Un. Mat. Ital. A"},{"key":"9623_CR29","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1088\/0951-7715\/20\/1\/011","volume":"20","author":"A R\u00e4tz","year":"2006","unstructured":"R\u00e4tz, A., Voigt, A.: A diffuse-interface approximation for surface diffusion including adatoms. Nonlinearity 20, 177 (2006)","journal-title":"Nonlinearity"},{"key":"9623_CR30","doi-asserted-by":"publisher","first-page":"1627","DOI":"10.1002\/cpa.20046","volume":"57","author":"E Sandier","year":"2004","unstructured":"Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg\u2013Landau. Commun. Pure Appl. Math. 57, 1627\u20131672 (2004)","journal-title":"Commun. Pure Appl. Math."},{"key":"9623_CR31","doi-asserted-by":"publisher","first-page":"1427","DOI":"10.3934\/dcds.2011.31.1427","volume":"31","author":"S Serfaty","year":"2011","unstructured":"Serfaty, S.: Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Discrete Contin. Dyn. Syst. 31, 1427\u20131451 (2011)","journal-title":"Discrete Contin. Dyn. Syst."},{"key":"9623_CR32","doi-asserted-by":"publisher","first-page":"073114","DOI":"10.1063\/1.3318256","volume":"96","author":"B Spencer","year":"2010","unstructured":"Spencer, B., Tersoff, J.: Asymmetry and shape transitions of epitaxially strained islands on vicinal surfaces. Appl. Phys. Lett. 96, 073114 (2010)","journal-title":"Appl. Phys. Lett."},{"key":"9623_CR33","unstructured":"Szlam, A., Bresson, X.: Total variation and Cheeger cuts. In: Proceedings of the 27th International Conference on International Conference on Machine Learning (ICML), pp.\u00a01039\u20131046 (2010)"},{"key":"9623_CR34","unstructured":"Thorpe, M., Slep\u010dev, D.: Transportation $$L^p$$ distances: properties and extensions, In preparation (2017)"},{"key":"9623_CR35","doi-asserted-by":"crossref","unstructured":"Thorpe, M., Theil, F.: Asymptotic analysis of the Ginzburg\u2013Landau functional on point clouds, to appear in the Proceedings of the Royal Society of Edinburgh Section A: Mathematics, arXiv preprint arXiv:1604.04930 (2017)","DOI":"10.1017\/prm.2018.32"},{"key":"9623_CR36","doi-asserted-by":"crossref","unstructured":"Thorpe, M., Park, S., Kolouri, S., Rohde, G.K., Slep\u010dev, D.: A transportation $$L^p$$ distance for signal analysis, to appear in the Journal of Mathematical Imaging and Vision, arXiv preprint arXiv:1609.08669 (2017)","DOI":"10.1007\/s10851-017-0726-4"},{"key":"9623_CR37","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s00205-015-0929-z","volume":"220","author":"NG Trillos","year":"2016","unstructured":"Trillos, N.G., Slep\u010dev, D.: Continuum limit of total variation on point clouds. Arch. Ration. Mech. Anal. 220, 193\u2013241 (2016)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9623_CR38","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1017\/S0956792517000304","volume":"28","author":"Y van Gennip","year":"2017","unstructured":"van Gennip, Y., Sch\u00f6nlieb, C.-B.: Introduction: big data and partial differential equations. Eur. J. Appl. Math. 28, 877\u2013885 (2017)","journal-title":"Eur. J. Appl. Math."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09623-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-020-09623-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-020-09623-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,2]],"date-time":"2024-08-02T15:06:57Z","timestamp":1722611217000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-020-09623-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,19]]},"references-count":38,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["9623"],"URL":"https:\/\/doi.org\/10.1007\/s00332-020-09623-y","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2020,3,19]]},"assertion":[{"value":"6 September 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}