{"id":"https://openalex.org/W2170799968","doi":"https://doi.org/10.3233/asy-2008-0897","title":"Gradient theory of phase transitions with a rapidly oscillating forcing term","display_name":"Gradient theory of phase transitions with a rapidly oscillating forcing term","publication_year":2008,"publication_date":"2008-08-01","ids":{"openalex":"https://openalex.org/W2170799968","doi":"https://doi.org/10.3233/asy-2008-0897","mag":"2170799968"},"language":"en","primary_location":{"id":"doi:10.3233/asy-2008-0897","is_oa":false,"landing_page_url":"https://doi.org/10.3233/asy-2008-0897","pdf_url":null,"source":{"id":"https://openalex.org/S141283892","display_name":"Asymptotic Analysis","issn_l":"0921-7134","issn":["0921-7134","1875-8576"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Asymptotic Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005257686","display_name":"Nicolas Dirr","orcid":"https://orcid.org/0000-0003-3634-7367"},"institutions":[{"id":"https://openalex.org/I4210091327","display_name":"Max Planck Institute for Mathematics in the Sciences","ror":"https://ror.org/00ez2he07","country_code":"DE","type":"facility","lineage":["https://openalex.org/I149899117","https://openalex.org/I4210091327"]},{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["DE","GB"],"is_corresponding":true,"raw_author_name":"Nicolas Dirr","raw_affiliation_strings":["Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, D-04103 Leipzig, Germany and Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. E-mail: N.Dirr@maths.bath.ac.uk"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, D-04103 Leipzig, Germany and Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. E-mail: N.Dirr@maths.bath.ac.uk","institution_ids":["https://openalex.org/I4210091327","https://openalex.org/I51601045"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5032152748","display_name":"Marcello Lucia","orcid":"https://orcid.org/0000-0002-3446-3906"},"institutions":[{"id":"https://openalex.org/I142393192","display_name":"College of Staten Island","ror":"https://ror.org/02p179j44","country_code":"US","type":"education","lineage":["https://openalex.org/I142393192"]},{"id":"https://openalex.org/I174216632","display_name":"City University of New York","ror":"https://ror.org/00453a208","country_code":"US","type":"education","lineage":["https://openalex.org/I174216632"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Marcello Lucia","raw_affiliation_strings":["Mathematics Department, The City University of New York, CSI, Staten Island, NY 10314, USA. E-mail: mlucia@math.csi.cuny.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Department, The City University of New York, CSI, Staten Island, NY 10314, USA. E-mail: mlucia@math.csi.cuny.edu","institution_ids":["https://openalex.org/I142393192","https://openalex.org/I174216632"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036258696","display_name":"Matteo Novaga","orcid":"https://orcid.org/0000-0002-0812-3780"},"institutions":[{"id":"https://openalex.org/I108290504","display_name":"University of Pisa","ror":"https://ror.org/03ad39j10","country_code":"IT","type":"education","lineage":["https://openalex.org/I108290504"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Matteo Novaga","raw_affiliation_strings":["Dipartimento di Matematica, Universit\u00e0 di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy. E-mail: novaga@dm.unipi.it"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica, Universit\u00e0 di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy. E-mail: novaga@dm.unipi.it","institution_ids":["https://openalex.org/I108290504"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5005257686"],"corresponding_institution_ids":["https://openalex.org/I4210091327","https://openalex.org/I51601045"],"apc_list":null,"apc_paid":null,"fwci":1.6555,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.85498062,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"60","issue":"1-2","first_page":"29","last_page":"59"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9965999722480774,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9851999878883362,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.7686138153076172},{"id":"https://openalex.org/keywords/forcing","display_name":"Forcing (mathematics)","score":0.6903177499771118},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.6803489327430725},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.6380559802055359},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.597492516040802},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.5571872591972351},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5545729398727417},{"id":"https://openalex.org/keywords/phase-transition","display_name":"Phase transition","score":0.5163360834121704},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.5146611332893372},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4329882264137268},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42366084456443787},{"id":"https://openalex.org/keywords/vector-potential","display_name":"Vector potential","score":0.4197339117527008},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.3308638036251068},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.20104897022247314},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.16343411803245544},{"id":"https://openalex.org/keywords/condensed-matter-physics","display_name":"Condensed matter physics","score":0.15761145949363708},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1551479697227478},{"id":"https://openalex.org/keywords/magnetic-field","display_name":"Magnetic field","score":0.10665252804756165},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10261386632919312}],"concepts":[{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.7686138153076172},{"id":"https://openalex.org/C197115733","wikidata":"https://www.wikidata.org/wiki/Q1003136","display_name":"Forcing (mathematics)","level":2,"score":0.6903177499771118},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.6803489327430725},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.6380559802055359},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.597492516040802},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5571872591972351},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5545729398727417},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.5163360834121704},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.5146611332893372},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4329882264137268},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42366084456443787},{"id":"https://openalex.org/C2027888","wikidata":"https://www.wikidata.org/wiki/Q521126","display_name":"Vector potential","level":3,"score":0.4197339117527008},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3308638036251068},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.20104897022247314},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.16343411803245544},{"id":"https://openalex.org/C26873012","wikidata":"https://www.wikidata.org/wiki/Q214781","display_name":"Condensed matter physics","level":1,"score":0.15761145949363708},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1551479697227478},{"id":"https://openalex.org/C115260700","wikidata":"https://www.wikidata.org/wiki/Q11408","display_name":"Magnetic field","level":2,"score":0.10665252804756165},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10261386632919312}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3233/asy-2008-0897","is_oa":false,"landing_page_url":"https://doi.org/10.3233/asy-2008-0897","pdf_url":null,"source":{"id":"https://openalex.org/S141283892","display_name":"Asymptotic Analysis","issn_l":"0921-7134","issn":["0921-7134","1875-8576"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Asymptotic Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arpi.unipi.it:11568/119245","is_oa":false,"landing_page_url":"http://hdl.handle.net/11568/119245","pdf_url":null,"source":{"id":"https://openalex.org/S4377196265","display_name":"CINECA IRIS Institutial research information system (University of Pisa)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I108290504","host_organization_name":"University of Pisa","host_organization_lineage":["https://openalex.org/I108290504"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:https://orca.cardiff.ac.uk:13075","is_oa":false,"landing_page_url":"https://orca.cardiff.ac.uk/id/eprint/13075/","pdf_url":null,"source":{"id":"https://openalex.org/S7407055383","display_name":"ORCA Online Research @Cardiff","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},{"id":"pmh:oai:www.research.unipd.it:11577/2268348","is_oa":false,"landing_page_url":"http://hdl.handle.net/11577/2268348","pdf_url":null,"source":{"id":"https://openalex.org/S4377196283","display_name":"Research Padua  Archive (University of Padua)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I138689650","host_organization_name":"University of Padua","host_organization_lineage":["https://openalex.org/I138689650"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W656727543","https://openalex.org/W1539347723","https://openalex.org/W1581406479","https://openalex.org/W1593038947","https://openalex.org/W1866311589","https://openalex.org/W1990541396","https://openalex.org/W1990620164","https://openalex.org/W2003166268","https://openalex.org/W2057992369","https://openalex.org/W2088186886","https://openalex.org/W2107729689","https://openalex.org/W2141130687","https://openalex.org/W2143936988","https://openalex.org/W2147717759"],"related_works":["https://openalex.org/W2990084823","https://openalex.org/W2025438663","https://openalex.org/W2478780871","https://openalex.org/W2985393691","https://openalex.org/W2001646560","https://openalex.org/W4386771804","https://openalex.org/W4387787906","https://openalex.org/W2057523058","https://openalex.org/W4366307899","https://openalex.org/W4210331724"],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,11,51,55,58,64,70,73,78,86],"\u0393-limit":[3],"of":[4,7,13,30,57,77],"a":[5,14,21,31,34,38],"family":[6],"functionals":[8,28,79],"which":[9],"model":[10],"interaction":[12],"material":[15],"that":[16,45,72],"undergoes":[17],"phase":[18],"transition":[19],"with":[20],"rapidly":[22],"oscillating":[23],"conservative":[24],"vector":[25,39,59],"field.":[26,40],"These":[27],"consist":[29],"gradient":[32],"term,":[33],"double-well":[35],"potential":[36],"and":[37,54,82],"The":[41],"scaling":[42],"is":[43,61],"such":[44],"all":[46],"three":[47],"terms":[48],"scale":[49],"in":[50,85],"same":[52],"way":[53],"frequency":[56],"field":[60],"equal":[62],"to":[63],"interface":[65],"thickness.":[66],"Difficulties":[67],"arise":[68],"from":[69],"fact":[71],"two":[74],"global":[75],"minimizers":[76],"are":[80],"nonconstant":[81],"converge":[83],"only":[84],"weak":[87],"L":[88],"2":[89],"-topology.":[90]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2018,"cited_by_count":1}],"updated_date":"2026-05-26T13:28:51.108037","created_date":"2016-06-24T00:00:00"}