Exponential decay estimates of the eigenvalues for the Neumann-Poincaré operator on analytic boundaries in two dimensions. (English) Zbl 1407.35147

The properties of the spectrum of the Neumann-Poincaré operator \[ K\varphi(x)=\int_{\Gamma}\frac{(x-y,\nu_{y})}{|x-y|^{2}}\varphi(y)\,dy \] are studied in the case of a bounded planar domain \(G\subset {\mathbb R}^{2}\) with analytic boundary \(\Gamma\). Here \(\nu_{y}\) is the outward unit normal vector at \(y\in \Gamma\). It is stated that if \(\lambda_{n}\) are eigenvalues of \(K\) such that \(|\lambda_{n}|\leq |\lambda_{n-1}|\) for all \(n\) then, for every \(\varepsilon<\varepsilon_{G}\), with \(\varepsilon_{G}\) the maximal Grauert radius of the boundary, there is a constant \(C\) such that \[ |\lambda_{2n-1}=|\lambda_{2n}|\leq C e^{-\varepsilon n}\;\;\forall n. \]


35P15 Estimates of eigenvalues in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35R30 Inverse problems for PDEs
Full Text: DOI arXiv Euclid


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