{"paper":{"title":"A decay estimate for the eigenvalues of the Neumann-Poincar\\'{e} operator in two dimensions using the Grunsky coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Mikyoung Lim, Younghoon Jung","submitted_at":"2018-11-13T02:09:12Z","abstract_excerpt":"We investigate the decay property of the eigenvalues of the Neumann-Poincar\\'{e} operator in two dimensions. As is well-known, this operator admits only a sequence of eigenvalues that accumulates to zero as its spectrum for a bounded domain having $C^{1,\\alpha}$ boundary with $\\alpha\\in (0,1)$. In this paper, we show that the eigenvalue $\\lambda_k$'s of the Neumann-Poincar\\'{e} operator ordered by size satisfy that $|\\lambda_k| = O(k^{-p-\\alpha+1/2})$ for an arbitrary simply connected domain having $C^{1+p,\\alpha}$ boundary with $p\\geq 0,~ \\alpha\\in(0,1)$ and $p+\\alpha>\\frac{1}{2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05070","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"}