SPECTRAL STRUCTURE OF THE NEUMANN--POINCARE\' OPERATOR ON THIN ELLIPSOIDS AND FLAT DOMAINS

초록

We investigate the spectral structure of the Neumann--Poincare'\ operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate spheroids and flat oblate ellipsoids. We show that eigenvalues of the Neumann--Poincare'\ operators on a sequence of the prolate spheroids are densely distributed in the interval [0, 1/2] as their eccentricities tend to 1, namely as they become longer. We then prove that eigenvalues of the Neumann--Poincare'\ operators on the oblate ellipsoids are densely distributed in the interval [ - 1/2, 1/2] as the ellipsoids become flatter. This shows, in particular, that even if there are at most finitely many negative eigenvalues on the oblate ellipsoids, more and more negative eigenvalues appear as the ellipsoids become flatter. We also show a similar spectral property for flat three-dimensional domains.

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