Spectral analysis of Neumann-Poincaré operator

Abstract

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincaré operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey.

arXiv commentsurvey

References in corpus (8)

• Schatten classes on compact manifolds: Kernel conditions
• The essential spectrum of the Neumann--Poincare operator on a domain with corners
• Asymptotic analysis for close evaluation of layer potentials
• Weyl's law for the eigenvalues of the Neumann--Poincaré operators in three dimensions: Willmore energy and surface geometry
• Spectral structure of the Neumann--Poincaré operator on tori
• Surface Riesz transforms and spectral property of elastic Neumann--Poincaé operators on less smooth domains in three dimensions
• A concavity condition for existence of a negative Neumann-Poincaré eigenvalue in three dimensions
• Convergence rate for eigenvalues of the elastic Neumann--Poincaré operator on smooth and real analytic boundaries in two dimensions

Cited by in corpus (2)

• Dyson gas on a curved contour
• Eigenvalue asymptotics for polynomially compact pseudodifferenial operators and applications