EIGENVALUES AND EIGENFUNCTIONS OF DOUBLE LAYER POTENTIALS.

Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let Ω be a C² bounded region in Rⁿ (n = 2, 3). The double layer potential K : L²(∂Ω) → L²(∂Ω) is defined by (Kψ)(x) ≡ ∫ _{∂ Ω} ψ(y)·v_yE(x, y) ds_y, where E(x, y) = ∫_{1/2π log1/∣x-y∣} , if n = 2, ^{1/π log1/∣x-y∣} , if n = 3, ds_y is the line or surface element and v_y is the outer normal derivative on ∂Ω. It is known that K is a compact operator on L²(∂Ω) and consists of at most a countable number of eigenvalues, with 0 as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of ∂Ω. [ABSTRACT FROM AUTHOR]