{"id":"https://openalex.org/W7220999542","doi":"https://doi.org/10.48550/arxiv.2610.08499","title":"Sheeting property and rigidity for stable solutions to the Allen--Cahn equation","display_name":"Sheeting property and rigidity for stable solutions to the Allen--Cahn equation","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220999542","doi":"https://doi.org/10.48550/arxiv.2610.08499"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08499","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08499","pdf_url":"https://arxiv.org/pdf/2610.08499","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.08499","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157983334","display_name":"Yong Liu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Yong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113233283","display_name":"Tianci Luo","orcid":"https://orcid.org/0009-0000-8157-6471","observed_orcids":["https://orcid.org/0009-0000-8157-6471"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Luo, Tianci","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5017410460","display_name":"Kelei Wang","orcid":"https://orcid.org/0000-0002-2815-0495","observed_orcids":["https://orcid.org/0000-0002-2815-0495","https://orcid.org/0009-0006-7384-8725","https://orcid.org/0009-0004-0519-3005"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Kelei","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5157938503","display_name":"Juncheng Wei","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wei, Juncheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5036619608","display_name":"Yong Wei","orcid":"https://orcid.org/0000-0002-9460-9217","observed_orcids":["https://orcid.org/0000-0002-9460-9217"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wei, Yong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5157970196","display_name":"Ke Wu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wu, Ke","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9726999998092651,"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"}},"topics":[{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9726999998092651,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.024299999698996544,"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/T11087","display_name":"Solidification and crystal growth phenomena","score":0.0010000000474974513,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/allen-cahn-equation","display_name":"Allen-Cahn equation","score":0.9860000014305115},{"id":"https://openalex.org/keywords/stable-solutions","display_name":"stable solutions","score":0.9750000238418579},{"id":"https://openalex.org/keywords/rigidity","display_name":"rigidity","score":0.8399999737739563},{"id":"https://openalex.org/keywords/zero-sets","display_name":"zero sets","score":0.6850000023841858},{"id":"https://openalex.org/keywords/multiplicity-one","display_name":"multiplicity one","score":0.6010000109672546},{"id":"https://openalex.org/keywords/morse-index","display_name":"Morse index","score":0.9150000214576721},{"id":"https://openalex.org/keywords/energy-demand-growth","display_name":"energy demand growth","score":0.6620000004768372},{"id":"https://openalex.org/keywords/windthrow","display_name":"windthrow","score":0.39500001072883606},{"id":"https://openalex.org/keywords/even-potential","display_name":"even potential","score":0.6819999814033508},{"id":"https://openalex.org/keywords/closed-manifolds","display_name":"closed manifolds","score":0.9039999842643738},{"id":"https://openalex.org/keywords/positive-ricci-curvature","display_name":"positive Ricci curvature","score":0.859000027179718},{"id":"https://openalex.org/keywords/weak-convergence-of-measures","display_name":"weak convergence of measures","score":0.5740000009536743},{"id":"https://openalex.org/keywords/entire-solutions","display_name":"entire solutions","score":0.39899998903274536}],"concepts":[{"id":"https://openalex.org/C156004811","wikidata":"https://www.wikidata.org/wiki/Q2228257","display_name":"Multiplicity (mathematics)","level":2,"score":0.853600025177002},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.809499979019165},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6205000281333923},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5931000113487244},{"id":"https://openalex.org/C156730297","wikidata":"https://www.wikidata.org/wiki/Q2296951","display_name":"Morse theory","level":2,"score":0.5252000093460083},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4925999939441681},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.45649999380111694},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.388700008392334},{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.36160001158714294},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3379000127315521},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.3336000144481659},{"id":"https://openalex.org/C72319582","wikidata":"https://www.wikidata.org/wiki/Q584304","display_name":"Degenerate energy levels","level":2,"score":0.32010000944137573},{"id":"https://openalex.org/C137877099","wikidata":"https://www.wikidata.org/wiki/Q1332977","display_name":"Subsequence","level":3,"score":0.319599986076355},{"id":"https://openalex.org/C109663097","wikidata":"https://www.wikidata.org/wiki/Q182453","display_name":"Adiabatic process","level":2,"score":0.3172000050544739},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.2992999851703644},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.2791000008583069},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.2750999927520752},{"id":"https://openalex.org/C196298200","wikidata":"https://www.wikidata.org/wiki/Q577705","display_name":"Critical point (mathematics)","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.2596000134944916},{"id":"https://openalex.org/C60033838","wikidata":"https://www.wikidata.org/wiki/Q1751685","display_name":"Solution set","level":3,"score":0.25949999690055847}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08499","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08499","pdf_url":"https://arxiv.org/pdf/2610.08499","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},{"id":"doi:10.48550/arxiv.2610.08499","doi":"https://doi.org/10.48550/arxiv.2610.08499","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08499","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.08499","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08499","pdf_url":"https://arxiv.org/pdf/2610.08499","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"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":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220999542.pdf","grobid_xml":"https://content.openalex.org/works/W7220999542.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7223117198","https://openalex.org/W7213898499","https://openalex.org/W7140305572","https://openalex.org/W4417409169","https://openalex.org/W7214291465","https://openalex.org/W2033992582","https://openalex.org/W2960968585","https://openalex.org/W3031487561","https://openalex.org/W3002342385","https://openalex.org/W3211683681"],"abstract_inverted_index":{"We":[0,18],"prove":[1],"a":[2,28,43],"sheeting":[3],"theorem":[4],"for":[5,77,88],"stable":[6,45],"solutions":[7,47],"to":[8,27],"the":[9,24,35],"Allen--Cahn":[10],"equation":[11],"in":[12,48],"ambient":[13],"dimensions":[14,100],"$2$":[15],"through":[16,102],"$10$.":[17],"show":[19],"that":[20,34],"weak":[21],"convergence":[22],"of":[23,62,99],"measures":[25],"$\\varepsilon|Du_\\varepsilon|^2\\,dX$":[26],"hyperplane":[29],"with":[30,52,66,90],"integer":[31],"multiplicity":[32,75,86],"implies":[33],"zero":[36],"sets":[37],"are":[38,56],"disjoint":[39],"smooth":[40],"graphs.":[41],"As":[42],"consequence,":[44],"entire":[46,64],"$\\mathbb":[49],"R^N$,":[50],"$N\\le7$,":[51],"$O(R^{N-1})$":[53],"energy":[54,72,92],"growth":[55,73],"one-dimensional.":[57],"When":[58],"$W''(-1)=W''(1)$,":[59],"every":[60],"blowdown":[61],"an":[63],"solution":[65],"finite":[67],"Morse":[68,94],"index":[69,95],"and":[70,93],"this":[71],"has":[74],"one":[76,87],"$4\\le":[78],"N\\le10$.":[79],"For":[80],"even":[81],"potentials,":[82],"we":[83],"also":[84],"obtain":[85],"limits":[89],"bounded":[91],"on":[96],"closed":[97],"manifolds":[98],"$3$":[101],"$7$,":[103],"under":[104],"bumpiness":[105],"or":[106],"positive":[107],"Ricci":[108],"curvature.":[109]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-09T00:00:00"}
