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Here, $\\Delta_p u=\\textrm{div}\\bigl(|\\nabla u|^{p-2}\\nabla u\\bigr)$, and $p\\in(1,2)\\cup(2,\\infty)$.\n  Near those free boundary points where $\\nabla \\varphi\\neq0$, the operator $\\Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\\nabla \\varphi=0$ then $\\Delta_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points.\n  Here we stu"},"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":"1701.05262","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-01-19T00:26:41Z","cross_cats_sorted":[],"title_canon_sha256":"eaaf7308f584ed7923f2aa6196b21a3d79b208e96fdc25912d579895e0159d8f","abstract_canon_sha256":"459dba33bfa477dc5e78f3a723b7ea0d0c8d850304ba409c42ffad3fa3190def"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:30.405281Z","signature_b64":"4vNc+rS25XgeV50NI1ytjKagPs/GcJfTew2SpOKGOXMwEXOn1MU43GyM7qCUfPmMy0emFWOHVAKA3Fq5jq1SCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"764279500b29181781fd75b74c51f960d24dd3e4e5088d4d670dbe5d81e7f168","last_reissued_at":"2026-05-18T00:52:30.404864Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:30.404864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the regularity of the free boundary in the $p$-Laplacian obstacle problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Brian Krummel, Xavier Ros-Oton","submitted_at":"2017-01-19T00:26:41Z","abstract_excerpt":"We study the regularity of the free boundary in the obstacle for the $p$-Laplacian, $\\min\\bigl\\{-\\Delta_p u,\\,u-\\varphi\\bigr\\}=0$ in $\\Omega\\subset\\mathbb R^n$. Here, $\\Delta_p u=\\textrm{div}\\bigl(|\\nabla u|^{p-2}\\nabla u\\bigr)$, and $p\\in(1,2)\\cup(2,\\infty)$.\n  Near those free boundary points where $\\nabla \\varphi\\neq0$, the operator $\\Delta_p$ is uniformly elliptic and smooth, and hence the free boundary is well understood. However, when $\\nabla \\varphi=0$ then $\\Delta_p$ is singular or degenerate, and nothing was known about the regularity of the free boundary at those points.\n  Here we stu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05262","kind":"arxiv","version":1},"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":"1701.05262","created_at":"2026-05-18T00:52:30.404937+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.05262v1","created_at":"2026-05-18T00:52:30.404937+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.05262","created_at":"2026-05-18T00:52:30.404937+00:00"},{"alias_kind":"pith_short_12","alias_value":"OZBHSUALFEMB","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"OZBHSUALFEMBPAP5","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"OZBHSUAL","created_at":"2026-05-18T12:31:34.259226+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/OZBHSUALFEMBPAP5OW3UYUPZMD","json":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD.json","graph_json":"https://pith.science/api/pith-number/OZBHSUALFEMBPAP5OW3UYUPZMD/graph.json","events_json":"https://pith.science/api/pith-number/OZBHSUALFEMBPAP5OW3UYUPZMD/events.json","paper":"https://pith.science/paper/OZBHSUAL"},"agent_actions":{"view_html":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD","download_json":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD.json","view_paper":"https://pith.science/paper/OZBHSUAL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.05262&json=true","fetch_graph":"https://pith.science/api/pith-number/OZBHSUALFEMBPAP5OW3UYUPZMD/graph.json","fetch_events":"https://pith.science/api/pith-number/OZBHSUALFEMBPAP5OW3UYUPZMD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD/action/storage_attestation","attest_author":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD/action/author_attestation","sign_citation":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD/action/citation_signature","submit_replication":"https://pith.science/pith/OZBHSUALFEMBPAP5OW3UYUPZMD/action/replication_record"}},"created_at":"2026-05-18T00:52:30.404937+00:00","updated_at":"2026-05-18T00:52:30.404937+00:00"}