{"id":"https://openalex.org/W1527700202","doi":"https://doi.org/10.1137/15m1010658","title":"A Compactness Lemma and Its Application to the Existence of Minimizers for the Liquid Drop Model","display_name":"A Compactness Lemma and Its Application to the Existence of Minimizers for the Liquid Drop Model","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W1527700202","doi":"https://doi.org/10.1137/15m1010658","mag":"1527700202"},"language":"en","primary_location":{"id":"doi:10.1137/15m1010658","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1010658","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical 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/A5016122295","display_name":"Rupert L. Frank","orcid":"https://orcid.org/0000-0001-7973-4688"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Rupert L. Frank","raw_affiliation_strings":["California  Institute of Technology"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"California  Institute of Technology","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072851506","display_name":"\u00c9lliott H. Lieb","orcid":"https://orcid.org/0000-0001-5843-3587"},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Elliott H. Lieb","raw_affiliation_strings":["Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Departments of Mathematics and Physics, Princeton University, Princeton, NJ 08544","institution_ids":["https://openalex.org/I20089843"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.2684,"has_fulltext":false,"cited_by_count":63,"citation_normalized_percentile":{"value":0.97507064,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"47","issue":"6","first_page":"4436","last_page":"4450"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9990000128746033,"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":0.9990000128746033,"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":0.9988999962806702,"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":0.9933000206947327,"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/mathematics","display_name":"Mathematics","score":0.8513606786727905},{"id":"https://openalex.org/keywords/compact-space","display_name":"Compact space","score":0.778752326965332},{"id":"https://openalex.org/keywords/ball","display_name":"Ball (mathematics)","score":0.7143447399139404},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6228460669517517},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.4789879322052002},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.42930951714515686},{"id":"https://openalex.org/keywords/semi-empirical-mass-formula","display_name":"Semi-empirical mass formula","score":0.42835646867752075},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4257553815841675},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.4227170944213867},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.4174523651599884},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3281664252281189},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.1004367470741272},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.09472423791885376},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08910781145095825}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8513606786727905},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.778752326965332},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.7143447399139404},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6228460669517517},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.4789879322052002},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.42930951714515686},{"id":"https://openalex.org/C83727373","wikidata":"https://www.wikidata.org/wiki/Q206451","display_name":"Semi-empirical mass formula","level":4,"score":0.42835646867752075},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4257553815841675},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.4227170944213867},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.4174523651599884},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3281664252281189},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.1004367470741272},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.09472423791885376},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08910781145095825},{"id":"https://openalex.org/C152568617","wikidata":"https://www.wikidata.org/wiki/Q2348","display_name":"Neutron","level":2,"score":0.0},{"id":"https://openalex.org/C12294094","wikidata":"https://www.wikidata.org/wiki/Q929080","display_name":"Fission","level":3,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/15m1010658","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1010658","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:authors.library.caltech.edu:63715","is_oa":false,"landing_page_url":"https://authors.library.caltech.edu/63715/","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":null,"sustainable_development_goals":[{"score":0.6700000166893005,"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W1480714962","https://openalex.org/W1492498476","https://openalex.org/W1578622189","https://openalex.org/W1659004305","https://openalex.org/W1968100044","https://openalex.org/W1977225278","https://openalex.org/W1987668553","https://openalex.org/W2002111682","https://openalex.org/W2009701727","https://openalex.org/W2013854242","https://openalex.org/W2027736906","https://openalex.org/W2028631122","https://openalex.org/W2033339654","https://openalex.org/W2037358889","https://openalex.org/W2038928461","https://openalex.org/W2063209308","https://openalex.org/W2066046051","https://openalex.org/W2076571148","https://openalex.org/W2079835250","https://openalex.org/W2156703756","https://openalex.org/W2164781628","https://openalex.org/W2808010733","https://openalex.org/W2963079241","https://openalex.org/W2963145277","https://openalex.org/W2963161858","https://openalex.org/W2964003705","https://openalex.org/W3101797525","https://openalex.org/W3103530514","https://openalex.org/W3106408446","https://openalex.org/W4233357497","https://openalex.org/W4235609472","https://openalex.org/W4246959100"],"related_works":["https://openalex.org/W2121543990","https://openalex.org/W2402800040","https://openalex.org/W2000020320","https://openalex.org/W4252969132","https://openalex.org/W1981078319","https://openalex.org/W2008917324","https://openalex.org/W2581541046","https://openalex.org/W238609530","https://openalex.org/W4387315838","https://openalex.org/W2805799113"],"abstract_inverted_index":{"The":[0],"ancient":[1],"Gamow":[2],"liquid":[3],"drop":[4],"model":[5],"of":[6,21,37,73,90,97,111,122,134,144],"nuclear":[7],"energies":[8],"has":[9,80],"had":[10],"a":[11,24,60,64,68,115],"renewed":[12],"life":[13],"as":[14],"an":[15],"interesting":[16,77],"problem":[17,79],"in":[18,84,125],"the":[19,35,49,53,56,71,88,91,98,120,132,135,142,145],"calculus":[20],"variations:":[22],"Find":[23],"set":[25],"$\\Omega":[26],"\\subset":[27],"\\mathbb{R}^3$":[28],"with":[29],"given":[30],"volume":[31],"$A$":[32,102],"that":[33,59],"minimizes":[34,48],"sum":[36],"its":[38,42],"surface":[39],"area":[40],"and":[41,51],"Coulomb":[43],"self":[44],"energy.":[45],"A":[46,108,138],"ball":[47,61],"former":[50],"maximizes":[52],"latter,":[54],"but":[55],"conjecture":[57],"is":[58,62,67,114,141],"always":[63],"minimizer---when":[65],"there":[66],"minimizer.":[69],"Even":[70],"existence":[72,89,121],"minimizers":[74],"for":[75,118],"this":[76],"geometric":[78,126],"not":[81],"been":[82],"shown":[83],"general.":[85],"We":[86],"prove":[87],"absolute":[92],"minimizer":[93],"(over":[94],"all":[95],"$A$)":[96],"energy":[99,105],"divided":[100],"by":[101],"(the":[103],"binding":[104],"per":[106],"particle).":[107],"second":[109],"result":[110],"our":[112],"work":[113],"general":[116],"method":[117],"showing":[119],"optimal":[123],"sets":[124],"minimization":[127],"problems,":[128],"which":[129],"we":[130],"call":[131],"\u201cmethod":[133],"missing":[136],"mass.\u201d":[137],"third":[139],"point":[140],"extension":[143],"pulling":[146],"back":[147],"compactness":[148],"lemma":[149],"[E.":[150],"H.":[151],"Lieb,":[152],"Invent.":[153],"Math.,":[154],"74":[155],"(1983),":[156],"pp.":[157],"441--448]":[158],"from":[159],"$W^{1,p}$":[160],"to":[161],"$BV$.":[162]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":10},{"year":2020,"cited_by_count":8},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
