{"paper":{"title":"On the Hardy-Schr\\\"odinger operator with a boundary singularity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fr\\'ed\\'eric Robert, Nassif Ghoussoub","submitted_at":"2014-10-07T21:08:58Z","abstract_excerpt":"We investigate the Hardy-Schr\\\"odinger operator $L_\\gamma=-\\Delta -\\frac{\\gamma}{|x|^2}$ on domains $\\Omega\\subset\\rn$, whose boundary contain the singularity $0$. The situation is quite different from the well-studied case when $0$ is in the interior of $\\Omega$. For one, if $0\\in\\Omega$, then $L_\\gamma$ is positive if and only if $\\gamma<\\frac{(n-2)^2}{4}$, while if $0\\in\\partial\\Omega$ the operator $L_{\\gamma}$ could be positive for larger value of $\\gamma$, potentially reaching the maximal constant $\\frac{n^2}{4}$ on convex domains.\n  We prove optimal regularity and a Hopf-type Lemma for v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1913","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"}