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Carleman factorization of layer potentials on smooth domains

Kazunori Ando, Hyeonbae Kang, Yoshihisa Miyanishi, Mihai Putinar

Abstract

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincaré operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincaré operator to the amenable $L^2$-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric-microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.

Carleman factorization of layer potentials on smooth domains

Abstract

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincaré operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincaré operator to the amenable -space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric-microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.
Paper Structure (23 sections, 38 theorems, 226 equations)

This paper contains 23 sections, 38 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Potentials and jump formulae
  4. The Dirichlet to Neumann map
  5. Pseudo-differential calculus
  6. Integral operators and their kernels
  7. Symmetrizable operators
  8. Spectral analysis
  9. Factorization of a symmetrizable operator
  10. Functional calculus with smooth functions
  11. Improved spectral resolution
  12. Biorthogonal systems of vectors in Hilbert space
  13. Spectral synthesis
  14. Analysis of the Neumann-Poincaré operator
  15. Spectral resolution of the Neumann-Poincaré operator
  16. ...and 8 more sections

Key Result

Theorem 3.1

(Theorem 3 in Krein) Assume $K \neq 0$ is a compact symmetrizable operator. Then both operators $K \in {\mathcal{L}}(H_1)$ and $K^\ast \in {\mathcal{L}}(H_{-1})$ are compact. The spectrum $\sigma(K)$ of $K$ is real with non-zero eigenvalues. For every $\lambda \in \sigma(K) \setminus \{0\}$ the asso

Theorems & Definitions (62)

  • Theorem 3.1
  • Corollary 3.2
  • proof
  • Example 3.3
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • Proposition 3.6
  • proof
  • ...and 52 more