{"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"}