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Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\\varepsilon.$$\n  Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for\n  $$\\frac{3}{2}-\\varepsilon<p<3+\\varepsilon.$$\n  As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\\vare"},"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":"2411.18458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-27T15:47:56Z","cross_cats_sorted":[],"title_canon_sha256":"14ad986a0e8618d78b9577ae981b1ce7d0dce437bca4afbbb15b8b229f068476","abstract_canon_sha256":"fe0e48e996c42b7f888f269d76a73098ce68e3f65a96c879be9da43aa66209a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:41:16.696138Z","signature_b64":"mYuMMzpZl4219D9RoUsSch1+rE5rm2fzrz5KUUB9gRxXvl3RSsuGDxaQqbzxNqp83hy9L4cG3RxR6fY6qAttAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e16155f945e04cfafd5b6e5cb62183baf56cbe65620e54aaa4b1678ed168806","last_reissued_at":"2026-07-05T09:41:16.695660Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:41:16.695660Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Schr\\\"odinger equations with $B_\\infty$ potentials in the region above a Lipschitz graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jun Geng, Ziyi Xu","submitted_at":"2024-11-27T15:47:56Z","abstract_excerpt":"In this paper we investigate the $L^p$ regularity, $L^p$ Neumann and $W^{1,p}$ problems for generalized Schr\\\"odinger operator $-\\text{div}(A\\nabla )+ V $ in the region above a Lipschitz graph under the assumption that $A$ is elliptic, symmetric and $x_d-$independent. Specifically, we prove that the $L^p$ regularity problem is uniquely solvable for $$1<p<2+\\varepsilon.$$\n  Moreover, we also establish the $W^{1,p}$ estimate for Neumann problem for\n  $$\\frac{3}{2}-\\varepsilon<p<3+\\varepsilon.$$\n  As a by-product, we also obtain that the $L^p$ Neumann problem is uniquely solvable for $1<p<2+\\vare"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.18458","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.18458/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2411.18458","created_at":"2026-07-05T09:41:16.695714+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.18458v1","created_at":"2026-07-05T09:41:16.695714+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.18458","created_at":"2026-07-05T09:41:16.695714+00:00"},{"alias_kind":"pith_short_12","alias_value":"RYLBKX4ULYCM","created_at":"2026-07-05T09:41:16.695714+00:00"},{"alias_kind":"pith_short_16","alias_value":"RYLBKX4ULYCM7L6V","created_at":"2026-07-05T09:41:16.695714+00:00"},{"alias_kind":"pith_short_8","alias_value":"RYLBKX4U","created_at":"2026-07-05T09:41:16.695714+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/RYLBKX4ULYCM7L6VW3S4WYQYHO","json":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO.json","graph_json":"https://pith.science/api/pith-number/RYLBKX4ULYCM7L6VW3S4WYQYHO/graph.json","events_json":"https://pith.science/api/pith-number/RYLBKX4ULYCM7L6VW3S4WYQYHO/events.json","paper":"https://pith.science/paper/RYLBKX4U"},"agent_actions":{"view_html":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO","download_json":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO.json","view_paper":"https://pith.science/paper/RYLBKX4U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.18458&json=true","fetch_graph":"https://pith.science/api/pith-number/RYLBKX4ULYCM7L6VW3S4WYQYHO/graph.json","fetch_events":"https://pith.science/api/pith-number/RYLBKX4ULYCM7L6VW3S4WYQYHO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO/action/storage_attestation","attest_author":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO/action/author_attestation","sign_citation":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO/action/citation_signature","submit_replication":"https://pith.science/pith/RYLBKX4ULYCM7L6VW3S4WYQYHO/action/replication_record"}},"created_at":"2026-07-05T09:41:16.695714+00:00","updated_at":"2026-07-05T09:41:16.695714+00:00"}