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For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-\\Delta)^s_p$ with $0 < s \\le 1$. If $(-\\Delta)^s_p u_s = f_s $ in $\\Omega \\subset \\mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\\Omega$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \\to 1^-$"},"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":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.13389","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2026-05-13T11:46:54Z","cross_cats_sorted":[],"title_canon_sha256":"b45f2c94d902325197dc6b15369b20d2571ce0788bf527d47aaaa5aaf3acfb4b","abstract_canon_sha256":"24ff752d0fd8359c9476896a5b240810c072bff8310fbeababbf25591a3503c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:47.728630Z","signature_b64":"+lMX+Ozbh7wtu9HFxJ/dugDItNHaMUKpvHrxdsSLvwpS5rka4+s4ZN38FABfarxE695CZPr6IYXcPcibLE/yCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5e4e34f62c020aa58d12f5383b78d01c7f7e288a60730e426fcd6753b0648c2","last_reissued_at":"2026-05-18T02:44:47.728161Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:47.728161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal stability of complement value problems for p-L\\'evy operators","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Solutions to p-Lévy integro-differential equations converge strongly to local p-Laplacian limits in the optimal Sobolev norm as the nonlocality parameter s approaches 1 from below.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guy Foghem","submitted_at":"2026-05-13T11:46:54Z","abstract_excerpt":"We establish the optimal convergence of solutions to integro-differential equations (IDEs) governed by symmetric integrodifferential $p$-L\\'evy operators, $1 < p < \\infty$, in the presence of nonlocal Dirichlet or Neumann boundary conditions. For illustrative purposes, consider the particular case of the (fractional) $p$-Laplacian $(-\\Delta)^s_p$ with $0 < s \\le 1$. If $(-\\Delta)^s_p u_s = f_s $ in $\\Omega \\subset \\mathbb{R}^d,$ augmented with a Dirichlet or Neumann data $g_s$ then under suitable assumptions on $\\Omega$, $f_s$ and $g_s$, we show that $(u_s)_s$ strongly converges as $s \\to 1^-$"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"under suitable assumptions on Ω, f_s and g_s, we show that (u_s)_s strongly converges as s → 1^- in the optimal, that is, ||u_s - u_1||_{W^{s,p}(Ω)} → 0","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Suitable assumptions on the domain Ω, the right-hand sides f_s, and the boundary data g_s that allow the nonlocal-to-local passage; the precise regularity or compatibility conditions on these data are not detailed in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Solutions to fractional p-Laplacian equations converge optimally in W^{s,p} to local p-Laplacian solutions as s approaches 1.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Solutions to p-Lévy integro-differential equations converge strongly to local p-Laplacian limits in the optimal Sobolev norm as the nonlocality parameter s approaches 1 from below.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"2b60c5ff4dfd6351d6d03422529d5966a2fdb92535d2b8ce057d191c27feae9c"},"source":{"id":"2605.13389","kind":"arxiv","version":1},"verdict":{"id":"8e2c1867-1fa9-4e4d-b729-923446deb80d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:46:09.046246Z","strongest_claim":"under suitable assumptions on Ω, f_s and g_s, we show that (u_s)_s strongly converges as s → 1^- in the optimal, that is, ||u_s - u_1||_{W^{s,p}(Ω)} → 0","one_line_summary":"Solutions to fractional p-Laplacian equations converge optimally in W^{s,p} to local p-Laplacian solutions as s approaches 1.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Suitable assumptions on the domain Ω, the right-hand sides f_s, and the boundary data g_s that allow the nonlocal-to-local passage; the precise regularity or compatibility conditions on these data are not detailed in the abstract.","pith_extraction_headline":"Solutions to p-Lévy integro-differential equations converge strongly to local p-Laplacian limits in the optimal Sobolev norm as the nonlocality parameter s approaches 1 from below."},"references":{"count":57,"sample":[{"doi":"","year":1967,"title":"R. 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