{"paper":{"title":"Weyl's law for the eigenvalues of the Neumann--Poincar\\'e operators in three dimensions: Willmore energy and surface geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Yoshihisa Miyanishi","submitted_at":"2018-06-10T13:30:32Z","abstract_excerpt":"We deduce eigenvalue asymptotics of the Neumann--Poincar\\'e operators in three dimensions. The region $\\Omega$ is $C^{2, \\alpha}$ ($\\alpha>0$) bounded in ${\\mathbf R}^3$ and the Neumann--Poincar\\'e operator ${\\mathcal K}_{\\partial\\Omega} : L^2(\\partial \\Omega) \\rightarrow L^2(\\partial \\Omega) $ is defined by $$ {\\mathcal K}_{\\partial\\Omega}[\\psi]({\\bf x}) := \\frac{1}{4\\pi} \\int_{\\partial \\Omega} \\frac{\\langle {\\bf y}-{\\bf x}, {\\bf n}({\\bf y}) \\rangle}{|{\\bf x}-{\\bf y}|^3} \\psi({\\bf y})\\; dS_{\\bf y} $$ where $dS_{\\bf y}$ is the surface element and ${\\bf n}({\\bf y})$ is the outer normal vector o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03657","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":""},"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"}