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Our results in this paper are two folds. First we prove $L^p$ convergence results for solutions of the above system, for non-oscillating operator, $A_\\e(x) =A(x)$, with the following convergence rate for all $1\\leq p <\\infty$ $$ \\|u_\\e - u_0\\|_{L^p(D)} \\leq C_p \\begin{cases} \\e^{1/2p}"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1209.0483","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-09-03T20:45:49Z","cross_cats_sorted":[],"title_canon_sha256":"ee78f7b52bf277d388d545e9da25f95a37bdef22d4141bb7cf8e372d03d3314e","abstract_canon_sha256":"58514b445099faf79c3ced7b71105400ae3f747a2fccbcdc52e3f52520c2211e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:09:59.892340Z","signature_b64":"j4q21k7xgackgjYjQ3IOKHAUWjMxE8GfnM5ww2vQdgOPtHdBn0btuLehauOsUCUKghsbOLos72Xo20RGpM5vDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"085849cbaf77cddfc53688045d575870ff52b8ef6a8ff588837179b7f4bcc2ff","last_reissued_at":"2026-05-18T03:09:59.891796Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:09:59.891796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Applications of Fourier analysis in homogenization of Dirichlet problem II. $L^p$ estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hayk Aleksanyan, Henrik Shahgholian, Per Sj\\\"olin","submitted_at":"2012-09-03T20:45:49Z","abstract_excerpt":"Let $u_\\e$ be a solution to the system $$ \\mathrm{div}(A_\\e(x) \\nabla u_{\\e}(x))=0 \\text{\\ in} D, \\qquad u_{\\e}(x)=g(x,x/\\e) \\text{\\ on}\\partial D, $$ where $D \\subset \\R^d $ ($d \\geq 2$), is a smooth uniformly convex domain, and $g$ is 1-periodic in its second variable, and both $A_\\e$ and $g$ reasonably smooth. Our results in this paper are two folds. First we prove $L^p$ convergence results for solutions of the above system, for non-oscillating operator, $A_\\e(x) =A(x)$, with the following convergence rate for all $1\\leq p <\\infty$ $$ \\|u_\\e - u_0\\|_{L^p(D)} \\leq C_p \\begin{cases} \\e^{1/2p}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0483","kind":"arxiv","version":2},"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":"1209.0483","created_at":"2026-05-18T03:09:59.891873+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.0483v2","created_at":"2026-05-18T03:09:59.891873+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0483","created_at":"2026-05-18T03:09:59.891873+00:00"},{"alias_kind":"pith_short_12","alias_value":"BBMETS5PO7G5","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"BBMETS5PO7G57RJW","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"BBMETS5P","created_at":"2026-05-18T12:26:58.693483+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/BBMETS5PO7G57RJWRACF2V2YOD","json":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD.json","graph_json":"https://pith.science/api/pith-number/BBMETS5PO7G57RJWRACF2V2YOD/graph.json","events_json":"https://pith.science/api/pith-number/BBMETS5PO7G57RJWRACF2V2YOD/events.json","paper":"https://pith.science/paper/BBMETS5P"},"agent_actions":{"view_html":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD","download_json":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD.json","view_paper":"https://pith.science/paper/BBMETS5P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.0483&json=true","fetch_graph":"https://pith.science/api/pith-number/BBMETS5PO7G57RJWRACF2V2YOD/graph.json","fetch_events":"https://pith.science/api/pith-number/BBMETS5PO7G57RJWRACF2V2YOD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD/action/storage_attestation","attest_author":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD/action/author_attestation","sign_citation":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD/action/citation_signature","submit_replication":"https://pith.science/pith/BBMETS5PO7G57RJWRACF2V2YOD/action/replication_record"}},"created_at":"2026-05-18T03:09:59.891873+00:00","updated_at":"2026-05-18T03:09:59.891873+00:00"}