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We prove $C^{1,\\alpha}_{loc}$ regularity with nearly optimal $\\alpha$ for viscosity solutions of this problem. In the case $f\\in L^{\\infty}\\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\\in L^q\\cap C$, $q>\\max(n,\\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory."},"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":"1603.06391","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-21T11:32:28Z","cross_cats_sorted":[],"title_canon_sha256":"27d9c7d9ee8785feaa20d6bd84cb48ec357ef8867988b8c29427d1f2af18e9e8","abstract_canon_sha256":"9f63aaa2d3e97c2806d9b8da832b4f4d4d0bb7a81269866e3cb0cd2aaa2ce9c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:55.321361Z","signature_b64":"4pdkZZMu5N2L5WWPjvG0QNuCfSb/AMfCsX64ynlEIolvW/YqMBThbOj8caiKgoAgravBi05Kv48t11IaPplpCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df51585e9d5ed8457cfbac1dd4d11ed27fe4c5a84dc720bf81aab07d53645717","last_reissued_at":"2026-05-18T00:58:55.320873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:55.320873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$C^{1,\\alpha}$ regularity for the normalized $p$-Poisson problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Amal Attouchi, Eero Ruosteenoja, Mikko Parviainen","submitted_at":"2016-03-21T11:32:28Z","abstract_excerpt":"We consider the normalized $p$-Poisson problem $$-\\Delta^N_p u=f \\qquad \\text{in}\\quad \\Omega.$$ The normalized $p$-Laplacian $\\Delta_p^{N}u:=|D u|^{2-p}\\Delta_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,\\alpha}_{loc}$ regularity with nearly optimal $\\alpha$ for viscosity solutions of this problem. In the case $f\\in L^{\\infty}\\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\\in L^q\\cap C$, $q>\\max(n,\\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06391","kind":"arxiv","version":3},"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":"1603.06391","created_at":"2026-05-18T00:58:55.320946+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.06391v3","created_at":"2026-05-18T00:58:55.320946+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06391","created_at":"2026-05-18T00:58:55.320946+00:00"},{"alias_kind":"pith_short_12","alias_value":"35IVQXU5L3ME","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"35IVQXU5L3MEK7H3","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"35IVQXU5","created_at":"2026-05-18T12:29:55.572404+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/35IVQXU5L3MEK7H3VQO5JUI62J","json":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J.json","graph_json":"https://pith.science/api/pith-number/35IVQXU5L3MEK7H3VQO5JUI62J/graph.json","events_json":"https://pith.science/api/pith-number/35IVQXU5L3MEK7H3VQO5JUI62J/events.json","paper":"https://pith.science/paper/35IVQXU5"},"agent_actions":{"view_html":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J","download_json":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J.json","view_paper":"https://pith.science/paper/35IVQXU5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.06391&json=true","fetch_graph":"https://pith.science/api/pith-number/35IVQXU5L3MEK7H3VQO5JUI62J/graph.json","fetch_events":"https://pith.science/api/pith-number/35IVQXU5L3MEK7H3VQO5JUI62J/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J/action/timestamp_anchor","attest_storage":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J/action/storage_attestation","attest_author":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J/action/author_attestation","sign_citation":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J/action/citation_signature","submit_replication":"https://pith.science/pith/35IVQXU5L3MEK7H3VQO5JUI62J/action/replication_record"}},"created_at":"2026-05-18T00:58:55.320946+00:00","updated_at":"2026-05-18T00:58:55.320946+00:00"}