Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance

Helsing, Johan; Kang, Hyeonbae; Lim, Mikyoung

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 991-1011.

Nous étudions les propriétés spectrales de l'opérateur Neumann–Poincaré sur les domaines planaires avec coins. Un accent particulier est mis sur l'existence d'un spectre continu et d'un point isolé du spectre. Nous montrons que le taux de résonance du spectre continu est différent de celui des valeurs propres. Nous dérivons ensuite une méthode pour distinguer spectre continu et valeurs propres. Nous effectuons des expériences numériques afin de voir si le spectre continu et les valeurs propres apparaissent pour des domaines avec coins. Pour les calculs, nous utilisons une modification de la méthode de Nyström. Elle permet la discrétisation convergente de l'opérateur Neumann–Poincaré d'ordre élevé sur des domaines avec coins. Les résultats des expériences montrent que tous les trois spectres possibles, spectre absolument continu, spectre singulier et point isolé du spectre, peuvent apparaître en fonction des domaines. Nous montrons aussi rigoureusement deux propriétés spectrales qui sont suggérées par des expériences numériques : la symétrie du spectre (y compris spectre continu), et de l'existence des valeurs propres sur des rectangles ayant des rapports d'aspect élevés.

We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.

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