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Characterization of the essential spectrum of the Neumann-Poincaré operator in 2D domains with corner via Weyl sequences. (English) Zbl 1423.35268

Let \(\Omega\) be a planar domain with a smooth boundary, \(D\) a subdomain strictly contained in \(\Omega\). This work studies the spectrum of the Neumann-Poincaré operator \(\mathcal{K}_D^*:L^2(\partial D)\rightarrow L^2(\partial D)\), defined by \[ \mathcal{K}_D^*(\varphi)(x)=\int_{\partial D}\frac{\partial P}{\partial \nu_y}\phi(y)ds(y), \] where \(P(x,y)\) is the Poisson Kernel associated with \(\Omega\). In the case in which the boundary \(\partial D\) is sufficiently smooth the operator \(\mathcal{K}_D^*\) is compact and hence has only a discrete spectrum, but when the boundary is not smooth the Neumann-Poincaré operator may have and essential spectrum. Here the case in which \(\partial D\) has a corner is studied in detail, and the essential spectrum is explicitly identified as a closed interval whose endpoints depend on the size of the angle at the corner. The arguments involve the use of the theory of elliptic corner singularities and the construction of Weyl sequences.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J75 Singular elliptic equations
47A10 Spectrum, resolvent
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