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That is, $u\\in W^{1,p}(B_1(0))$, $2<p<\\infty$, $u\\ge 0$ and $$ \\d\\left( |\\nabla u|^{p-2}\\nabla u\\right)=\\chi_{\\{u>0\\}}\\textrm{ in }B_1(0) $$ where $u(x)\\ge 0$ and $\\chi_A$ is the characteristic function of the set $A$. Our main result is that for almost every free boundary point, with respect to the $(n-1)-$Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,\\beta}-$graph. That is, for $\\H^{n-1}-$a.e. point $x^0\\in \\partial \\{u>0\\}\\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\\cap \\par"},"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":"1611.04397","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-14T14:47:10Z","cross_cats_sorted":[],"title_canon_sha256":"bef55e358dfd3d1d62e203f64d7c3764a9d94eb2b36805f911e4de11796f6daf","abstract_canon_sha256":"2728807ac4787feb6ca688498751901bd55cdfa77c4da1831115ae77f6c1f0d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:16.761043Z","signature_b64":"uU6Hc2DhtYTAsiF91vS7VpqvW41cBLMnFf7AqzdcRgLXR69GsUBo+v2u197GDsLzOkOeM7XwUBerutkJNkYTBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"525f7bda5565a1fc66f8a2e4b388367421ef066ac507aa73ec811a5764eb49d9","last_reissued_at":"2026-05-18T00:59:16.760400Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:16.760400Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Almost Everywhere Regularity for the Free Boundary of the Normalized p-harmonic Obstacle problem $p>2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"John Andersson","submitted_at":"2016-11-14T14:47:10Z","abstract_excerpt":"Let $u$ be a solution to the normalized p-harmonic obstacle problem with $p>2$. That is, $u\\in W^{1,p}(B_1(0))$, $2<p<\\infty$, $u\\ge 0$ and $$ \\d\\left( |\\nabla u|^{p-2}\\nabla u\\right)=\\chi_{\\{u>0\\}}\\textrm{ in }B_1(0) $$ where $u(x)\\ge 0$ and $\\chi_A$ is the characteristic function of the set $A$. Our main result is that for almost every free boundary point, with respect to the $(n-1)-$Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,\\beta}-$graph. That is, for $\\H^{n-1}-$a.e. point $x^0\\in \\partial \\{u>0\\}\\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\\cap \\par"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04397","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":"1611.04397","created_at":"2026-05-18T00:59:16.760501+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.04397v1","created_at":"2026-05-18T00:59:16.760501+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04397","created_at":"2026-05-18T00:59:16.760501+00:00"},{"alias_kind":"pith_short_12","alias_value":"KJPXXWSVMWQ7","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KJPXXWSVMWQ7YZXY","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KJPXXWSV","created_at":"2026-05-18T12:30:25.849896+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/KJPXXWSVMWQ7YZXYULSLHCBWOQ","json":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ.json","graph_json":"https://pith.science/api/pith-number/KJPXXWSVMWQ7YZXYULSLHCBWOQ/graph.json","events_json":"https://pith.science/api/pith-number/KJPXXWSVMWQ7YZXYULSLHCBWOQ/events.json","paper":"https://pith.science/paper/KJPXXWSV"},"agent_actions":{"view_html":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ","download_json":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ.json","view_paper":"https://pith.science/paper/KJPXXWSV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.04397&json=true","fetch_graph":"https://pith.science/api/pith-number/KJPXXWSVMWQ7YZXYULSLHCBWOQ/graph.json","fetch_events":"https://pith.science/api/pith-number/KJPXXWSVMWQ7YZXYULSLHCBWOQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ/action/storage_attestation","attest_author":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ/action/author_attestation","sign_citation":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ/action/citation_signature","submit_replication":"https://pith.science/pith/KJPXXWSVMWQ7YZXYULSLHCBWOQ/action/replication_record"}},"created_at":"2026-05-18T00:59:16.760501+00:00","updated_at":"2026-05-18T00:59:16.760501+00:00"}