{"id":"https://openalex.org/W7214710648","doi":"https://doi.org/10.48550/arxiv.2609.32561","title":"Formalising Linear Elliptic PDE Theory in Lean 4","display_name":"Formalising Linear Elliptic PDE Theory in Lean 4","publication_year":2026,"publication_date":"2026-09-26","ids":{"openalex":"https://openalex.org/W7214710648","doi":"https://doi.org/10.48550/arxiv.2609.32561"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.32561","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32561","pdf_url":"https://arxiv.org/pdf/2609.32561","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/2609.32561","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5147992546","display_name":"Alejandro Jose Soto Franco","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Franco, Alejandro Jos\u00e9 Soto","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5092513686","display_name":"Kobe Marshall-Stevens","orcid":"https://orcid.org/0000-0002-8625-1382","observed_orcids":["https://orcid.org/0000-0002-8625-1382"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marshall-Stevens, Kobe","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/T10126","display_name":"Logic, programming, and type systems","score":0.4584999978542328,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/T10126","display_name":"Logic, programming, and type systems","score":0.4584999978542328,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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":0.10329999774694443,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.08340000361204147,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/formal-verification","display_name":"formal verification","score":0.6919999718666077},{"id":"https://openalex.org/keywords/lean-4","display_name":"Lean 4","score":0.9869999885559082},{"id":"https://openalex.org/keywords/mathlib","display_name":"Mathlib","score":0.7540000081062317},{"id":"https://openalex.org/keywords/dirichlet-problem","display_name":"Dirichlet problem","score":0.871999979019165},{"id":"https://openalex.org/keywords/linear-elliptic-pdes","display_name":"linear elliptic PDEs","score":0.8339999914169312},{"id":"https://openalex.org/keywords/divergence-form-operators","display_name":"divergence form operators","score":0.9089999794960022},{"id":"https://openalex.org/keywords/poincare-inequality","display_name":"Poincar\u00e9 inequality","score":0.9079999923706055},{"id":"https://openalex.org/keywords/lax-milgram-theorem","display_name":"Lax-Milgram theorem","score":0.9629999995231628},{"id":"https://openalex.org/keywords/fredholm-alternative","display_name":"Fredholm alternative","score":0.9990000128746033},{"id":"https://openalex.org/keywords/spectral-theorem","display_name":"spectral theorem","score":0.9829999804496765},{"id":"https://openalex.org/keywords/sobolev-embedding-theorem","display_name":"Sobolev embedding theorem","score":0.9729999899864197},{"id":"https://openalex.org/keywords/sobolev-spaces","display_name":"Sobolev spaces","score":0.9649999737739563},{"id":"https://openalex.org/keywords/second-order-elliptic-operators","display_name":"second-order elliptic operators","score":0.7160000205039978}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8140000104904175},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.715399980545044},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5651999711990356},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.5356000065803528},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.49070000648498535},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.46619999408721924},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.36250001192092896},{"id":"https://openalex.org/C3020036199","wikidata":"https://www.wikidata.org/wiki/Q1644451","display_name":"Pseudodifferential operators","level":2,"score":0.3517000079154968},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.3411000072956085},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3393000066280365},{"id":"https://openalex.org/C179603306","wikidata":"https://www.wikidata.org/wiki/Q268493","display_name":"Elliptic curve","level":2,"score":0.3370000123977661},{"id":"https://openalex.org/C35435516","wikidata":"https://www.wikidata.org/wiki/Q5370016","display_name":"Embedding problem","level":3,"score":0.3368000090122223},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3034000098705292},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.2913999855518341},{"id":"https://openalex.org/C98306654","wikidata":"https://www.wikidata.org/wiki/Q2409122","display_name":"Spectral theory","level":3,"score":0.28209999203681946},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.26980000734329224},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2669999897480011},{"id":"https://openalex.org/C151756577","wikidata":"https://www.wikidata.org/wiki/Q36810","display_name":"Quadrilateral","level":3,"score":0.2612000107765198}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.32561","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32561","pdf_url":"https://arxiv.org/pdf/2609.32561","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.2609.32561","doi":"https://doi.org/10.48550/arxiv.2609.32561","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.32561","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.32561","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.32561","pdf_url":"https://arxiv.org/pdf/2609.32561","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214710648.pdf","grobid_xml":"https://content.openalex.org/works/W7214710648.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W1545761024","https://openalex.org/W1855169650","https://openalex.org/W1866311589","https://openalex.org/W2546463436","https://openalex.org/W2981188613","https://openalex.org/W3178506813","https://openalex.org/W4315630308","https://openalex.org/W6948201204","https://openalex.org/W7102412881","https://openalex.org/W7151856918","https://openalex.org/W7204185295"],"related_works":["https://openalex.org/W7151856918","https://openalex.org/W2026881464","https://openalex.org/W2017864885","https://openalex.org/W2002172566","https://openalex.org/W2012566450","https://openalex.org/W7221550196","https://openalex.org/W3036885384","https://openalex.org/W2890861933","https://openalex.org/W1997948069","https://openalex.org/W2021774451"],"abstract_inverted_index":{"We":[0],"formalise":[1],"in":[2,20,29],"Lean":[3,78,104],"4,":[4],"on":[5],"top":[6],"of":[7,11,38,68,84,89],"Mathlib,":[8],"the":[9,12,30,33,36,42,47,50,57,93,101],"solvability":[10,70],"Dirichlet":[13],"problem":[14],"for":[15,71],"second-order":[16],"linear":[17],"elliptic":[18],"operators":[19],"divergence":[21],"form.":[22],"The":[23],"machine-verified":[24],"results,":[25],"with":[26,100],"no":[27],"sorry":[28],"development,":[31],"include":[32],"Poincar\u00e9":[34],"inequality,":[35],"existence":[37],"weak":[39],"solutions":[40],"by":[41],"Lax-Milgram":[43],"theorem,":[44,52],"Rellich-Kondrachov":[45],"compactness,":[46],"Fredholm":[48],"alternative,":[49],"spectral":[51],"interior":[53],"regularity":[54],"estimates,":[55],"and":[56,75],"Sobolev":[58,85],"embedding":[59],"theorem.":[60],"From":[61],"these":[62],"results":[63],"we":[64,95],"obtain":[65],"a":[66,81],"formalisation":[67],"classical":[69],"sufficiently":[72],"regular":[73],"coefficients":[74],"data.":[76],"Our":[77],"library":[79],"includes":[80],"self-contained":[82],"theory":[83],"spaces":[86],"developed":[87],"independently":[88],"existing":[90],"formalisations.":[91],"Throughout":[92],"paper":[94],"associate":[96],"each":[97],"prose":[98],"statement":[99],"named":[102],"machine-checked":[103],"declaration":[105],"that":[106],"discharges":[107],"it.":[108]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}