Статья «EIGENVALUES OF THE NEUMANN-POINCARE OPERATOR IN DIMENSION 3: WEYL’S LAW AND GEOMETRY, "Алгебра и анализ"»

We consider the asymptotic properties of the eigenvalues of the Neumann- Poincare (NP) operator in three dimensions. The region ? C R is bounded by a compact surface ? = ??, with certain smoothness conditions imposed. The NP operat or K, called often ‘the direct value of the double layer potential’, acting in L(?), is defined by K[?]:=1/4? ?((y-x,n (y)))/(|x-y|3)?(y)dS where dS is the surface element and n(y) is the outer unit normal on ?. The first-named author proved in [27] that the singular numbers sj (K) of Kr and the ordered moduli of its eigenvalues ?j (Kr) satisfy the Weyl law S (K(Г))?|?j (К(Г))?{(3W (Г)-2?X(Г))/128?}1/2j- under the condition that ? belongs to the class C with ? > 0, where W(?) and ?(?) denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface ?. Although the NP operator is not selfadjoint (and therefore no general relationships between eigenvalues and singular number exists), the ordered moduli of the eigenvalues of Kr satisfy the same asymptotic relation. Our main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary ?. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of Kr. A more sophisticated estimation allows us to give a natural extension of the Weyl law for the case of a smooth boundary.