EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR IN DIMENSION 3: WEYL'S LAW AND GEOMETRY

Journal article, 2020

Asymptotic properties of the eigenvalues of the Neumann-Poincare (NP) operator in three dimensions are treated. The region Omega subset of R-3 is bounded by a compact surface Gamma = partial derivative Omega, with certain smoothness conditions imposed. The NP operator K-Gamma, called often 'the direct value of the double layer potential', acting in L-2(Gamma), is defined by K-Gamma[psi](X) := 1/4 pi integral(Gamma) < y - x, n(y)>/vertical bar x - y vertical bar(3)psi(y) dS(y), where dS(y) is the surface element and n(y) is the outer unit normal on F. The firstnamed author proved in [27] that the singular numbers s(j) (K-Gamma) of K-Gamma and the ordered moduli of its eigenvalues lambda(j) (K-Gamma) satisfy the Weyl law s(j)(K(Gamma)) similar to vertical bar lambda(j)(K-Gamma)vertical bar similar to {3W(Gamma) - 2 pi chi(Gamma)/128 pi}(1/2) j(-1/2), under the condition that Gamma belongs to the class C-2,C-alpha with alpha > 0, where W(Gamma) and chi(Gamma) denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface Gamma. Although the NP operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular numbers exists), the ordered moduli of the eigenvalues of K-Gamma satisfy the same asymptotic relation. The main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary Gamma. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of K-Gamma. A more sophisticated estimate allows us to give a natural extension of the Weyl law for the case of a smooth boundary.