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The $\\infty$-Laplacian is the PDE system \\[ \\tag{1} \\label{1} \\Delta_\\infty u \\, :=\\, \\Big(Du \\otimes Du + |Du|^2[Du]^\\bot\\! \\otimes I\\Big) :D^2u\\, =\\, 0, \\] where $[Du]^\\bot := \\text{Proj}_{R(Du)^\\bot}$. \\eqref{1} constitutes the fundamental equation of vectorial Calculus of Variations in $L^\\infty$, associated to the model functional \\[ \\tag{2} \\label{2} E_\\infty (u,\\Omega')\\, =\\, \\big\\| |Du|^2\\big\\|_{L^\\infty(\\Omega')} ,\\ \\ \\ \\Omega' \\Subset \\Omega. \\] We show that generalised solut"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.01811","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-06T13:30:17Z","cross_cats_sorted":[],"title_canon_sha256":"334ae751742af233ba695a9b1c6582d8e9f83cac6030734207259bbbf4016596","abstract_canon_sha256":"8337d4598f6878d397d1da3b6b6c2e595521f6d062e2432e3b305cfc07769292"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:01.885441Z","signature_b64":"admNr5Q0Y13892oIxISuO4tdO9uuwKjPrM/Dlo/kSPbAKJtJyu0hCVwRK+tQLDmx/CU2fMzy42Td60eitp4eDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"840170fa899cf63a225d839e22ae0a81a22bc543475d965e5b3394876a4d3aca","last_reissued_at":"2026-05-18T00:50:01.884821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:01.884821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A New Characterisation of $\\infty$-Harmonic and $p$-Harmonic Maps via Affine Variations in $L^\\infty$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nikos Katzourakis (Reading, UK)","submitted_at":"2015-09-06T13:30:17Z","abstract_excerpt":"Let $u: \\Omega \\subseteq \\mathbb{R}^n \\longrightarrow \\mathbb{R}^N$ be a smooth map and $n,N \\in \\mathbb{N}$. The $\\infty$-Laplacian is the PDE system \\[ \\tag{1} \\label{1} \\Delta_\\infty u \\, :=\\, \\Big(Du \\otimes Du + |Du|^2[Du]^\\bot\\! \\otimes I\\Big) :D^2u\\, =\\, 0, \\] where $[Du]^\\bot := \\text{Proj}_{R(Du)^\\bot}$. \\eqref{1} constitutes the fundamental equation of vectorial Calculus of Variations in $L^\\infty$, associated to the model functional \\[ \\tag{2} \\label{2} E_\\infty (u,\\Omega')\\, =\\, \\big\\| |Du|^2\\big\\|_{L^\\infty(\\Omega')} ,\\ \\ \\ \\Omega' \\Subset \\Omega. \\] We show that generalised solut"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01811","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.01811","created_at":"2026-05-18T00:50:01.884930+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.01811v4","created_at":"2026-05-18T00:50:01.884930+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01811","created_at":"2026-05-18T00:50:01.884930+00:00"},{"alias_kind":"pith_short_12","alias_value":"QQAXB6UJTT3D","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_16","alias_value":"QQAXB6UJTT3DUIS5","created_at":"2026-05-18T12:29:37.295048+00:00"},{"alias_kind":"pith_short_8","alias_value":"QQAXB6UJ","created_at":"2026-05-18T12:29:37.295048+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG","json":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG.json","graph_json":"https://pith.science/api/pith-number/QQAXB6UJTT3DUIS5QOPCFLQKQG/graph.json","events_json":"https://pith.science/api/pith-number/QQAXB6UJTT3DUIS5QOPCFLQKQG/events.json","paper":"https://pith.science/paper/QQAXB6UJ"},"agent_actions":{"view_html":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG","download_json":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG.json","view_paper":"https://pith.science/paper/QQAXB6UJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.01811&json=true","fetch_graph":"https://pith.science/api/pith-number/QQAXB6UJTT3DUIS5QOPCFLQKQG/graph.json","fetch_events":"https://pith.science/api/pith-number/QQAXB6UJTT3DUIS5QOPCFLQKQG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG/action/storage_attestation","attest_author":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG/action/author_attestation","sign_citation":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG/action/citation_signature","submit_replication":"https://pith.science/pith/QQAXB6UJTT3DUIS5QOPCFLQKQG/action/replication_record"}},"created_at":"2026-05-18T00:50:01.884930+00:00","updated_at":"2026-05-18T00:50:01.884930+00:00"}