{"id":"https://openalex.org/W7208730633","doi":"https://doi.org/10.48550/arxiv.2609.03339","title":"Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation","display_name":"Exterior Serrin Rigidity for the Homogeneous{k}-Hessian Equation","publication_year":2026,"publication_date":"2026-09-03","ids":{"openalex":"https://openalex.org/W7208730633","doi":"https://doi.org/10.48550/arxiv.2609.03339"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.03339","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03339","pdf_url":"https://arxiv.org/pdf/2609.03339","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","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.03339","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5120547470","display_name":"\u5fd7\u8f89 \u5f20","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhang, Zhihui","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5120547470"],"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9115999937057495,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9115999937057495,"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":0.06599999964237213,"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/T13080","display_name":"Advanced Differential Geometry Research","score":0.0013000000035390258,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/overdetermined-problems","display_name":"overdetermined problems","score":0.9559999704360962},{"id":"https://openalex.org/keywords/exterior-dirichlet-problem","display_name":"exterior Dirichlet problem","score":0.5720000267028809},{"id":"https://openalex.org/keywords/k-convexity","display_name":"K-convexity","score":0.6819999814033508},{"id":"https://openalex.org/keywords/star-shaped-domains","display_name":"star-shaped domains","score":0.8299999833106995}],"concepts":[{"id":"https://openalex.org/C81901731","wikidata":"https://www.wikidata.org/wiki/Q331348","display_name":"Overdetermined system","level":2,"score":0.7889999747276306},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7688999772071838},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.6662999987602234},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.5360000133514404},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.46880000829696655},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.45559999346733093},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.43799999356269836},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4147000014781952},{"id":"https://openalex.org/C45137528","wikidata":"https://www.wikidata.org/wiki/Q7977966","display_name":"Weak solution","level":2,"score":0.3610000014305115},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3544999957084656},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.35109999775886536},{"id":"https://openalex.org/C79464548","wikidata":"https://www.wikidata.org/wiki/Q464794","display_name":"Minkowski space","level":2,"score":0.35089999437332153},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.3391999900341034},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C110342517","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Constant coefficients","level":2,"score":0.3253999948501587},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.31299999356269836},{"id":"https://openalex.org/C57489055","wikidata":"https://www.wikidata.org/wiki/Q190046","display_name":"Ellipsoid","level":2,"score":0.2994999885559082},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.29440000653266907},{"id":"https://openalex.org/C27016315","wikidata":"https://www.wikidata.org/wiki/Q580101","display_name":"Integral equation","level":2,"score":0.2937000095844269},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.2888000011444092},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.2874999940395355},{"id":"https://openalex.org/C153919410","wikidata":"https://www.wikidata.org/wiki/Q9993851","display_name":"Weak formulation","level":3,"score":0.27390000224113464},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.2727000117301941},{"id":"https://openalex.org/C153658351","wikidata":"https://www.wikidata.org/wiki/Q746264","display_name":"Constant function","level":3,"score":0.2709999978542328},{"id":"https://openalex.org/C127172972","wikidata":"https://www.wikidata.org/wiki/Q128709","display_name":"Viscosity","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C7980502","wikidata":"https://www.wikidata.org/wiki/Q322339","display_name":"Heaviside step function","level":2,"score":0.2565000057220459}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.03339","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03339","pdf_url":"https://arxiv.org/pdf/2609.03339","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.2609.03339","doi":"https://doi.org/10.48550/arxiv.2609.03339","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.03339","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:2609.03339","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.03339","pdf_url":"https://arxiv.org/pdf/2609.03339","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":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7208730633.pdf","grobid_xml":"https://content.openalex.org/works/W7208730633.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4285428163","https://openalex.org/W4315647935","https://openalex.org/W4414876839"],"related_works":["https://openalex.org/W4414876839","https://openalex.org/W4417372352","https://openalex.org/W4288724958","https://openalex.org/W7147546411","https://openalex.org/W7213606690","https://openalex.org/W6922485564","https://openalex.org/W4393404943","https://openalex.org/W4315647935","https://openalex.org/W2876014120","https://openalex.org/W7171362967"],"abstract_inverted_index":{"In":[0,39],"this":[1,152],"paper,":[2],"we":[3,140],"study":[4],"the":[5,10,22,59,68,73,94,111,115,156],"exterior":[6,157],"overdetermined":[7],"problems":[8],"for":[9,42,122,128,135],"homogeneous":[11],"k-Hessian":[12],"equations":[13],"$$\u03c3_k(D^2u)=0\\quad\\text{in}~\\mathbb{R}^n\\setminus\\overline\u03a9$$":[14],"in":[15],"three":[16,63],"dimensional":[17],"regimes.":[18],"For":[19],"$2\\le":[20],"k\\frac{n}{2}$,":[21],"solution":[23,60],"with":[24],"boundary":[25,99],"$1$":[26],"and":[27,57,72,84,130],"fundamental":[28],"power-growth":[29],"at":[30,114],"infinity":[31],"is":[32],"considered":[33],"under":[34],"strict":[35,145],"$(k-1)$-convexity":[36],"of":[37],"$\\partial\u03a9$.":[38],"each":[40],"case,":[41],"a":[43,54,124,131,143],"positive":[44],"constant":[45],"$c$,":[46],"$|Du|=c>0$":[47],"on":[48],"$\\partial\u03a9$":[49],"forces":[50],"$\u03a9$":[51],"to":[52,162],"be":[53],"Euclidean":[55],"ball":[56],"determines":[58],"explicitly.":[61],"The":[62,106],"arguments":[64],"are":[65,120],"organized":[66],"by":[67,149],"single":[69],"parameter":[70],"$q=(n-k)/k$":[71],"scale":[74],"function":[75,148],"\\[":[76],"F_q(t)=\\frac{t^{1-q}-1}{1-q},\\qquad":[77],"F_1(t)=\\log":[78],"t.":[79],"\\]":[80],"After":[81],"normalization,":[82],"$U=F_q(v)$":[83],"$M_q[v]=vD^2v-qDv\\otimes":[85],"Dv$":[86],"satisfies":[87],"$\u03c3_k(M_q[v])=0$.":[88],"A":[89],"unified":[90],"contact-point":[91],"calculation":[92],"gives":[93],"sharp":[95],"bound":[96],"$|Dv|\\le":[97],"b$;a":[98],"viscosity":[100],"contact":[101],"then":[102],"yields":[103],"$H_k\\ge":[104],"qbH_{k-1}$.":[105],"integral":[107],"mechanism":[108],"that":[109],"closes":[110],"argument":[112],"changes":[113],"critical":[116],"exponent:":[117],"Rellich--Pohozaev":[118],"identities":[119],"used":[121],"$q>1$,":[123],"scale-invariant":[125],"Wronskian--Newton":[126],"current":[127],"$q=1$,":[129],"renormalized":[132],"Newton--Jacobi":[133],"mass":[134],"$q<1$.":[136],"When":[137],"$q\\leq":[138],"1,$":[139],"also":[141],"obtain":[142],"global":[144],"k-convex":[146],"defining":[147],"Minkowski":[150],"gauge,":[151],"help":[153],"us":[154],"extend":[155],"construction":[158],"from":[159],"convex":[160],"domains":[161],"strictly":[163],"star-shaped":[164],"domains.":[165]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-05T00:00:00"}