{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:MWCTGBYUPKKDSKT2OJWGWDZJIK","short_pith_number":"pith:MWCTGBYU","schema_version":"1.0","canonical_sha256":"65853307147a94392a7a726c6b0f2942a8f8852716777fd73d9f309501be6f2b","source":{"kind":"arxiv","id":"2411.18852","version":1},"attestation_state":"computed","paper":{"title":"$W^{1,p}$ estimates for Schr\\\"odinger equation in the region above a convex graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ziyi Xu","submitted_at":"2024-11-28T01:43:44Z","abstract_excerpt":"We investigate the $W^{1,p}$ estimates of the Neumann problem for the Schr\\\"odinger equation $-\\Delta u+ V u={\\rm div}(f)$ in the region above a convex graph. For any $p>2$, we obtain a sufficient condition for the $W^{1,p}$ solvability. As a result, we obtain sharp $W^{1,p}$ estimate $$\\|\\nabla u\\|_{L^p(\\Omega)}+\\|V^\\frac{1}{2}u\\|_{L^p(\\Omega)}\\leq C\\|f\\|_{L^p(\\Omega)}$$\n for $1 2$, we obtain a sufficient condition for the $W^{1,p}$ solvability. As a result, we obtain sharp $W^{1,p}$ estimate $$\\|\\nabla u\\|_{L^p(\\Omega)}+\\|V^\\frac{1}{2}u\\|_{L^p(\\Omega)}\\leq C\\|f\\|_{L^p(\\Omega)}$$\n for $1