Some perspectives on (non)local phase transitions and minimal surfaces
We present here some classical and modern results about phase transitions and minimal surfaces, which are quite intertwined topics. We start from scratch, revisiting the theory of phase transitions as put forth by Lev Landau. Then, we relate the short-range phase transitions to the classical minimal...
Saved in:
| Published in | Bulletin of mathematical sciences Vol. 13; no. 1 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
World Scientific Publishing Company
01.04.2023
World Scientific Publishing |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1664-3607 1664-3615 |
| DOI | 10.1142/S1664360723300013 |
Cover
Loading…
| Abstract | We present here some classical and modern results about phase transitions and minimal surfaces, which are quite intertwined topics. We start from scratch, revisiting the theory of phase transitions as put forth by Lev Landau. Then, we relate the short-range phase transitions to the classical minimal surfaces, whose basic regularity theory is presented, also in connection with a celebrated conjecture by Ennio De Giorgi. With this, we explore the recently developed subject of long-range phase transitions and relate its genuinely nonlocal regime to the analysis of fractional minimal surfaces. |
|---|---|
| AbstractList | We present here some classical and modern results about phase transitions and minimal surfaces, which are quite intertwined topics. We start from scratch, revisiting the theory of phase transitions as put forth by Lev Landau. Then, we relate the short-range phase transitions to the classical minimal surfaces, whose basic regularity theory is presented, also in connection with a celebrated conjecture by Ennio De Giorgi. With this, we explore the recently developed subject of long-range phase transitions and relate its genuinely nonlocal regime to the analysis of fractional minimal surfaces. |
| Author | Dipierro, Serena Valdinoci, Enrico |
| Author_xml | – sequence: 1 givenname: Serena surname: Dipierro fullname: Dipierro, Serena – sequence: 2 givenname: Enrico surname: Valdinoci fullname: Valdinoci, Enrico |
| BookMark | eNp9kE1PwzAMhiMEEmPsB3DrEQ6DfLbNEU0MJk3iMDhHTuJCpq6ZkgLi39Mx4AASJ1u2n9d-fUIOu9ghIWeMXjIm-dWKlaUUJa24EJRSJg7IaFeaipKpw5-cVsdkkvN6GKGq0rrSIzJfxQ0WW0x5i64Pr5iL2BXnw4KLNjpoi-0zZCz6BF0OfYhdLqDzxSZ0YTN080tqwGE-JUcNtBknX3FMHuc3D7O76fL-djG7Xk6d0EpMGbMOsaaK-lp65I3SzoOupBJOKdE4Ti0AeF5rFNaC91Cj1BKttVrqUozJYq_rI6zNNg1HpHcTIZjPQkxPBlIfXIsGpaeqpLXnlkvLSwDpK0FLjo2HqoFBq9pruRRzTtgYF3rYeRzchtYwanbfNX--O5DsF_l9yX8M3TNvMbU-u4BdH5rgftC_yAc91o3T |
| CitedBy_id | crossref_primary_10_1007_s12215_024_01035_1 |
| Cites_doi | 10.2307/1970520 10.1007/s002080050159 10.1090/S0002-9947-2014-05906-0 10.1023/A:1010602715526 10.1007/BF01475472 10.2307/1970556 10.3934/dcds.2010.28.1179 10.1090/S0002-9947-2010-05021-4 10.1007/978-3-030-18921-1_1 10.4310/jdg/1357141507 10.1007/s00033-008-7093-3 10.1007/s00222-019-00904-2 10.1007/s10231-011-0243-9 10.1007/s00526-009-0225-6 10.1002/cpa.20093 10.1142/9789812834744_0004 10.1093/acprof:oso/9780198507840.001.0001 10.1090/S0894-0347-00-00345-3 10.1007/978-3-030-33116-0_5 10.1007/BF02849427 10.1007/s00526-012-0539-7 10.1215/S0012-7094-00-10331-6 10.1515/crelle-2019-0045 10.1016/j.jfa.2009.01.020 10.1088/1361-6544/aab89d 10.1016/0040-9383(82)90023-4 10.1016/j.anihpc.2012.01.006 10.1016/j.jfa.2016.11.016 10.1007/s00526-012-0580-6 10.1016/j.na.2007.04.031 10.1007/s00220-020-03771-8 10.1016/0362-546X(91)90177-3 10.1016/j.aim.2007.03.013 10.1090/S0002-9947-1972-0308905-4 10.1007/978-1-4684-9486-0 10.1142/5573 10.4310/jdg/1563242471 10.1080/14786446808639951 10.1215/00127094-2018-0052 10.1515/crelle-2015-0006 10.1002/cpa.20331 10.1137/17M114933X 10.1007/s00526-017-1295-5 10.1016/0001-6160(72)90037-5 10.1063/1.1744102 10.1007/BF00251230 10.1016/j.aim.2013.08.007 10.1016/0167-2789(83)90233-6 10.1016/j.jfa.2013.02.005 10.4007/annals.2011.174.3.3 10.1515/crelle-2019-0005 10.1007/BF00250724 10.4007/annals.2009.169.41 10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.0.CO;2-U 10.1016/j.anihpc.2013.02.001 10.1016/s0294-1449(16)30328-6 10.1016/j.anihpc.2018.08.003 10.1016/s0294-1449(16)30387-0 10.1353/ajm.2020.0032 10.2140/apde.2018.11.1881 10.1007/s005260000036 10.1002/cpa.3160480101 10.1090/S0002-9904-1974-13461-0 10.1007/BF01404309 10.1007/978-3-319-28739-3 10.4208/ata.OA-0008 10.1007/PL00013453 10.1080/03605302.2015.1135165 10.1090/S0002-9947-1924-1501263-9 10.1007/s002080050196 10.4310/jdg/1525399218 10.1016/j.matpur.2013.05.001 |
| ContentType | Journal Article |
| Copyright | 2023, The Author(s) |
| Copyright_xml | – notice: 2023, The Author(s) |
| DBID | ADCHV AAYXX CITATION DOA |
| DOI | 10.1142/S1664360723300013 |
| DatabaseName | World Scientific Open CrossRef DOAJ Directory of Open Access Journals |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| Database_xml | – sequence: 1 dbid: DOA name: DOAJ Directory of Open Access Journals url: https://www.doaj.org/ sourceTypes: Open Website |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1664-3615 |
| ExternalDocumentID | oai_doaj_org_article_e4d05608d2b24b26aa4d73062efda7fa 10_1142_S1664360723300013 S1664360723300013 |
| GroupedDBID | -A0 2VQ 4.4 40G 5VS 8FE 8FG AAKKN AAYZH AAYZJ ABEEZ ABJCF ACACY ACGFS ACIPV ACIWK ACULB ADBBV ADCHV ADINQ AENEX AFGXO AFKRA AHBXF AHBYD AHSBF AHYZX ALMA_UNASSIGNED_HOLDINGS AMKLP BCNDV BENPR BGLVJ C24 C6C CCPQU EBS EJD GROUPED_DOAJ H4N HCIFZ HZ~ IAO ISR ITC J9A KQ8 L6V M7S M~E O9- OK1 PIMPY PROAC PTHSS RSV RWJ SMT SOJ U2A AAYXX CITATION PHGZM PHGZT |
| ID | FETCH-LOGICAL-c3953-11bcee8050d84de2f59cda97453c553fc20baaad289e3bbadda8e494ebbb94963 |
| IEDL.DBID | DOA |
| ISSN | 1664-3607 |
| IngestDate | Wed Aug 27 01:32:57 EDT 2025 Tue Jul 01 01:39:10 EDT 2025 Thu Apr 24 23:04:17 EDT 2025 Mon Nov 25 02:43:21 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Keywords | regularity theory minimal surfaces Phase transition |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c3953-11bcee8050d84de2f59cda97453c553fc20baaad289e3bbadda8e494ebbb94963 |
| OpenAccessLink | https://doaj.org/article/e4d05608d2b24b26aa4d73062efda7fa |
| ParticipantIDs | doaj_primary_oai_doaj_org_article_e4d05608d2b24b26aa4d73062efda7fa worldscientific_primary_S1664360723300013 crossref_primary_10_1142_S1664360723300013 crossref_citationtrail_10_1142_S1664360723300013 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | 20230400 |
| PublicationDateYYYYMMDD | 2023-04-01 |
| PublicationDate_xml | – month: 04 year: 2023 text: 20230400 |
| PublicationDecade | 2020 |
| PublicationTitle | Bulletin of mathematical sciences |
| PublicationYear | 2023 |
| Publisher | World Scientific Publishing Company World Scientific Publishing |
| Publisher_xml | – name: World Scientific Publishing Company – name: World Scientific Publishing |
| References | S1664360723300013BIB003 S1664360723300013BIB047 S1664360723300013BIB004 S1664360723300013BIB048 S1664360723300013BIB005 S1664360723300013BIB049 S1664360723300013BIB006 Farina A. (S1664360723300013BIB054) 2008; 7 S1664360723300013BIB007 S1664360723300013BIB008 S1664360723300013BIB009 Dipierro S. (S1664360723300013BIB050) 2018 S1664360723300013BIB085 Berestycki H. (S1664360723300013BIB014) 1997; 25 S1664360723300013BIB043 Haas A. E. (S1664360723300013BIB068) 2015 S1664360723300013BIB087 S1664360723300013BIB044 Mahon B. (S1664360723300013BIB074) 2004 S1664360723300013BIB088 S1664360723300013BIB001 S1664360723300013BIB089 S1664360723300013BIB002 S1664360723300013BIB046 Barrios B. (S1664360723300013BIB012) 2014; 13 S1664360723300013BIB080 S1664360723300013BIB081 S1664360723300013BIB082 del Pino M. (S1664360723300013BIB084) 2013; 93 S1664360723300013BIB015 S1664360723300013BIB059 S1664360723300013BIB016 S1664360723300013BIB017 S1664360723300013BIB019 Caffarelli L. (S1664360723300013BIB031) 2010; 63 Fujimoto M. (S1664360723300013BIB062) 2005 S1664360723300013BIB051 S1664360723300013BIB095 S1664360723300013BIB096 S1664360723300013BIB097 S1664360723300013BIB010 De Giorgi E. (S1664360723300013BIB040) 1975; 8 S1664360723300013BIB055 S1664360723300013BIB056 S1664360723300013BIB057 Benarros D. (S1664360723300013BIB013) 1995 S1664360723300013BIB090 S1664360723300013BIB091 S1664360723300013BIB092 S1664360723300013BIB093 S1664360723300013BIB025 Dipierro S. (S1664360723300013BIB042) 2020; 67 S1664360723300013BIB069 S1664360723300013BIB026 S1664360723300013BIB027 S1664360723300013BIB028 S1664360723300013BIB029 Modica L. (S1664360723300013BIB078) 1977; 14 Dávila J. (S1664360723300013BIB037) 2018; 109 S1664360723300013BIB061 S1664360723300013BIB063 S1664360723300013BIB020 S1664360723300013BIB064 S1664360723300013BIB021 S1664360723300013BIB065 S1664360723300013BIB022 S1664360723300013BIB066 S1664360723300013BIB023 S1664360723300013BIB067 S1664360723300013BIB024 Dipierro S. (S1664360723300013BIB045) 2022; 150 De Giorgi E. (S1664360723300013BIB039) 1965; 19 S1664360723300013BIB060 De Giorgi E. (S1664360723300013BIB041) 1979 Baronowitz B. (S1664360723300013BIB011) 2022 S1664360723300013BIB036 Farina A. (S1664360723300013BIB058) 2019; 19 Valdinoci E. (S1664360723300013BIB098) 2004; 574 S1664360723300013BIB072 S1664360723300013BIB073 S1664360723300013BIB030 Landau L. D. (S1664360723300013BIB071) 1937; 7 S1664360723300013BIB075 S1664360723300013BIB032 S1664360723300013BIB076 S1664360723300013BIB033 S1664360723300013BIB077 S1664360723300013BIB079 De Giorgi E. (S1664360723300013BIB038) 1965; 19 S1664360723300013BIB070 Bistafa S. R. (S1664360723300013BIB018) 2018; 36 Cinti E. (S1664360723300013BIB035) 2019; 112 |
| References_xml | – ident: S1664360723300013BIB006 doi: 10.2307/1970520 – ident: S1664360723300013BIB003 doi: 10.1007/s002080050159 – ident: S1664360723300013BIB028 doi: 10.1090/S0002-9947-2014-05906-0 – ident: S1664360723300013BIB002 doi: 10.1023/A:1010602715526 – ident: S1664360723300013BIB016 doi: 10.1007/BF01475472 – ident: S1664360723300013BIB096 doi: 10.2307/1970556 – ident: S1664360723300013BIB023 doi: 10.3934/dcds.2010.28.1179 – ident: S1664360723300013BIB056 doi: 10.1090/S0002-9947-2010-05021-4 – ident: S1664360723300013BIB001 doi: 10.1007/978-3-030-18921-1_1 – volume: 93 start-page: 67 issue: 1 year: 2013 ident: S1664360723300013BIB084 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1357141507 – volume: 7 start-page: 19 year: 1937 ident: S1664360723300013BIB071 publication-title: Zh. Eksp. Teor. Fiz. – start-page: 44 volume-title: Advances in Geometric Analysis and Continuum Mechanics (Stanford, CA, 1993). year: 1995 ident: S1664360723300013BIB013 – ident: S1664360723300013BIB070 doi: 10.1007/s00033-008-7093-3 – ident: S1664360723300013BIB059 doi: 10.1007/s00222-019-00904-2 – ident: S1664360723300013BIB082 doi: 10.1007/s10231-011-0243-9 – ident: S1664360723300013BIB066 doi: 10.1007/s00526-009-0225-6 – ident: S1664360723300013BIB029 doi: 10.1002/cpa.20093 – ident: S1664360723300013BIB055 doi: 10.1142/9789812834744_0004 – ident: S1664360723300013BIB020 doi: 10.1093/acprof:oso/9780198507840.001.0001 – ident: S1664360723300013BIB007 doi: 10.1090/S0894-0347-00-00345-3 – volume: 67 start-page: 1324 issue: 9 year: 2020 ident: S1664360723300013BIB042 publication-title: Not. Am. Math. Soc. – ident: S1664360723300013BIB051 doi: 10.1007/978-3-030-33116-0_5 – ident: S1664360723300013BIB061 doi: 10.1007/BF02849427 – ident: S1664360723300013BIB092 doi: 10.1007/s00526-012-0539-7 – ident: S1664360723300013BIB015 doi: 10.1215/S0012-7094-00-10331-6 – ident: S1664360723300013BIB048 doi: 10.1515/crelle-2019-0045 – ident: S1664360723300013BIB097 doi: 10.1016/j.jfa.2009.01.020 – ident: S1664360723300013BIB036 doi: 10.1088/1361-6544/aab89d – ident: S1664360723300013BIB075 doi: 10.1016/0040-9383(82)90023-4 – ident: S1664360723300013BIB090 doi: 10.1016/j.anihpc.2012.01.006 – ident: S1664360723300013BIB046 doi: 10.1016/j.jfa.2016.11.016 – ident: S1664360723300013BIB024 doi: 10.1007/s00526-012-0580-6 – ident: S1664360723300013BIB017 doi: 10.1016/j.na.2007.04.031 – ident: S1664360723300013BIB047 doi: 10.1007/s00220-020-03771-8 – ident: S1664360723300013BIB081 doi: 10.1016/0362-546X(91)90177-3 – ident: S1664360723300013BIB073 doi: 10.1016/j.aim.2007.03.013 – ident: S1664360723300013BIB072 doi: 10.1090/S0002-9947-1972-0308905-4 – volume: 36 start-page: 1 year: 2018 ident: S1664360723300013BIB018 publication-title: J. Brazil. Soc. Mech. Sci. Eng. – ident: S1664360723300013BIB065 doi: 10.1007/978-1-4684-9486-0 – year: 2022 ident: S1664360723300013BIB011 publication-title: Discrete Contin. Dyn. Syst. – volume-title: The Physics of Structural Phase Transitions year: 2005 ident: S1664360723300013BIB062 – volume: 19 start-page: 1281 issue: 4 year: 2019 ident: S1664360723300013BIB058 publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – ident: S1664360723300013BIB064 doi: 10.1142/5573 – volume: 112 start-page: 447 issue: 3 year: 2019 ident: S1664360723300013BIB035 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1563242471 – ident: S1664360723300013BIB076 doi: 10.1080/14786446808639951 – ident: S1664360723300013BIB026 doi: 10.1215/00127094-2018-0052 – ident: S1664360723300013BIB060 doi: 10.1515/crelle-2015-0006 – volume: 574 start-page: 147 year: 2004 ident: S1664360723300013BIB098 publication-title: J. Reine Angew. Math. – volume: 63 start-page: 1111 issue: 9 year: 2010 ident: S1664360723300013BIB031 publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20331 – volume: 19 start-page: 79 year: 1965 ident: S1664360723300013BIB038 publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – ident: S1664360723300013BIB044 doi: 10.1137/17M114933X – ident: S1664360723300013BIB043 doi: 10.1007/s00526-017-1295-5 – ident: S1664360723300013BIB005 doi: 10.1016/0001-6160(72)90037-5 – ident: S1664360723300013BIB033 doi: 10.1063/1.1744102 – ident: S1664360723300013BIB077 doi: 10.1007/BF00251230 – start-page: 165 volume-title: Recent Developments in Nonlocal Theory year: 2018 ident: S1664360723300013BIB050 – volume-title: The Man Who Changed Everything: The Life of James Clerk Maxwell year: 2004 ident: S1664360723300013BIB074 – ident: S1664360723300013BIB032 doi: 10.1016/j.aim.2013.08.007 – ident: S1664360723300013BIB008 doi: 10.1016/0167-2789(83)90233-6 – ident: S1664360723300013BIB091 doi: 10.1016/j.jfa.2013.02.005 – ident: S1664360723300013BIB085 doi: 10.4007/annals.2011.174.3.3 – ident: S1664360723300013BIB025 doi: 10.1515/crelle-2019-0005 – ident: S1664360723300013BIB067 doi: 10.1007/BF00250724 – ident: S1664360723300013BIB087 doi: 10.4007/annals.2009.169.41 – volume: 7 start-page: 741 issue: 4 year: 2008 ident: S1664360723300013BIB054 publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – ident: S1664360723300013BIB010 doi: 10.1002/1097-0312(200008)53:8<1007::AID-CPA3>3.0.CO;2-U – volume: 14 start-page: 526 issue: 3 year: 1977 ident: S1664360723300013BIB078 publication-title: Boll. Un. Mat. Ital. A – ident: S1664360723300013BIB027 doi: 10.1016/j.anihpc.2013.02.001 – volume: 150 start-page: 2223 issue: 5 year: 2022 ident: S1664360723300013BIB045 publication-title: Proc. Amer. Math. Soc. – ident: S1664360723300013BIB009 doi: 10.1016/s0294-1449(16)30328-6 – ident: S1664360723300013BIB021 doi: 10.1016/j.anihpc.2018.08.003 – start-page: 131 volume-title: Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) year: 1979 ident: S1664360723300013BIB041 – ident: S1664360723300013BIB080 doi: 10.1016/s0294-1449(16)30387-0 – ident: S1664360723300013BIB049 doi: 10.1353/ajm.2020.0032 – ident: S1664360723300013BIB088 doi: 10.2140/apde.2018.11.1881 – ident: S1664360723300013BIB004 doi: 10.1007/s005260000036 – ident: S1664360723300013BIB030 doi: 10.1002/cpa.3160480101 – volume: 19 start-page: 463 issue: 3 year: 1965 ident: S1664360723300013BIB039 publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – ident: S1664360723300013BIB095 doi: 10.1090/S0002-9904-1974-13461-0 – ident: S1664360723300013BIB019 doi: 10.1007/BF01404309 – ident: S1664360723300013BIB022 doi: 10.1007/978-3-319-28739-3 – ident: S1664360723300013BIB089 doi: 10.4208/ata.OA-0008 – ident: S1664360723300013BIB069 doi: 10.1007/PL00013453 – volume: 13 start-page: 609 issue: 3 year: 2014 ident: S1664360723300013BIB012 publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – volume: 8 start-page: 277 year: 1975 ident: S1664360723300013BIB040 publication-title: Rend. Mat. – ident: S1664360723300013BIB057 doi: 10.1080/03605302.2015.1135165 – ident: S1664360723300013BIB079 doi: 10.1090/S0002-9947-1924-1501263-9 – ident: S1664360723300013BIB063 doi: 10.1007/s002080050196 – volume: 109 start-page: 111 issue: 1 year: 2018 ident: S1664360723300013BIB037 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1525399218 – volume: 25 start-page: 69 issue: 1 year: 1997 ident: S1664360723300013BIB014 publication-title: Ann. Sc. Norm. Super. Pisa Cl. Sci. – volume-title: A Commentary on the Scientific Writings of J. Willard Gibbs year: 2015 ident: S1664360723300013BIB068 – ident: S1664360723300013BIB093 doi: 10.1016/j.matpur.2013.05.001 |
| SSID | ssj0000579979 ssib050729703 ssib041538293 ssib059270981 |
| Score | 2.2425942 |
| SecondaryResourceType | review_article |
| Snippet | We present here some classical and modern results about phase transitions and minimal surfaces, which are quite intertwined topics. We start from scratch,... |
| SourceID | doaj crossref worldscientific |
| SourceType | Open Website Enrichment Source Index Database Publisher |
| SubjectTerms | minimal surfaces Phase transition regularity theory |
| Title | Some perspectives on (non)local phase transitions and minimal surfaces |
| URI | http://www.worldscientific.com/doi/abs/10.1142/S1664360723300013 https://doaj.org/article/e4d05608d2b24b26aa4d73062efda7fa |
| Volume | 13 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV07T8MwELagEx0qnqI8Kg8MFClq4lfiEVBLhVSWUqlbZMeOGGha9bHy2znHoS1UgoUlg-PEzuXs706--w6hm4TLUCdGBIAmzkHJeKBFDsudRlxnhhtWVmsYvIj-iD2P-Xir1JeLCfP0wF5wHcsMYHSYGKIJ00QoxQxopSA2NyrOS9MIMG_LmQJNYm4dbwEZd_zY8ea8kUsSh7LKuPQs4LGUJTFfJAQLqAjj6gg0YqQzdI2ujYD374ymbyBWcv3XUaOkOPVpjC7KZwumeoeoUdmX-N5_1xHas8Uxqg_W5KyLE9QbTicWzzZZlgs8LfBtMS3aJbLh2RtAG146FPMBXVgVBjsSkgncXazmuYvjOkWjXvf1sR9U5RSCjEpOgyjSgIgJCMMkzFiSc5kZBf4EpxnnNM9IqJVSBlwwS7WGjU8llklmtdaSwUI9QzWYij1HWCiew45tuCKaRbGSCVVgZ1CVEQomC22i8Es-aVZxjbuSF--pz4Mm6Y5Im-hu_cjME2381vnBCX3d0XFklw2gOWmlOelfmtNE7R-_bP2-nQEv_mPAS3TgitX7uJ8rVFvOV_YaTJqlbqH9pPfUKnUYroOP7ieo8-ve |
| linkProvider | Directory of Open Access Journals |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Some+perspectives+on+%28non%29local+phase+transitions+and+minimal+surfaces&rft.jtitle=Bulletin+of+mathematical+sciences&rft.au=Dipierro%2C+Serena&rft.au=Valdinoci%2C+Enrico&rft.date=2023-04-01&rft.pub=World+Scientific+Publishing+Company&rft.issn=1664-3607&rft.eissn=1664-3615&rft.volume=13&rft.issue=1&rft_id=info:doi/10.1142%2FS1664360723300013&rft.externalDocID=S1664360723300013 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1664-3607&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1664-3607&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1664-3607&client=summon |