TY - JOUR

T1 - Eigenvalues of the Neumann–Poincare operator in dimension 3: Weyl's law and geometry.

AU - Miyanishi, Y.

AU - Rozenblum, G.

N1 - Y. Miyanishi, G. Rozenblum, “Eigenvalues of the Neumann–Poincare operator in dimension 3: Weyl's law and geometry”, Алгебра и анализ, 31:2 (2019), 248–268

PY - 2019

Y1 - 2019

N2 - We consider the asymptotic properties of the eigenvalues of the Neumann- Poincare (NP) operator in three dimensions. The region Ω C R3 is bounded by a compact surface Γ = ƏΩ, with certain smoothness conditions imposed. The NP operat or K г, called often ‘the direct value of the double layer potential’, acting in L2(Γ), is defined by Kг[ϕ]:=1/4π ∫((y-x,n (y)))/(|x-y|3)ϕ(y)dSy where dSy is the surface element and n(y) is the outer unit normal on Γ. The first-named author proved in [27] that the singular numbers sj (Kг) of Kr and the ordered moduli of its eigenvalues λj (Kr) satisfy the Weyl law Si (K(Г))~|λj (К(Г))~{(3W (Г)-2πX(Г))/128π}1/2j-1/2 under the condition that Γ belongs to the class C2,a with α > 0, where W(Γ) and χ(Γ) denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface Γ. 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 Γ. 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.

AB - We consider the asymptotic properties of the eigenvalues of the Neumann- Poincare (NP) operator in three dimensions. The region Ω C R3 is bounded by a compact surface Γ = ƏΩ, with certain smoothness conditions imposed. The NP operat or K г, called often ‘the direct value of the double layer potential’, acting in L2(Γ), is defined by Kг[ϕ]:=1/4π ∫((y-x,n (y)))/(|x-y|3)ϕ(y)dSy where dSy is the surface element and n(y) is the outer unit normal on Γ. The first-named author proved in [27] that the singular numbers sj (Kг) of Kr and the ordered moduli of its eigenvalues λj (Kr) satisfy the Weyl law Si (K(Г))~|λj (К(Г))~{(3W (Г)-2πX(Г))/128π}1/2j-1/2 under the condition that Γ belongs to the class C2,a with α > 0, where W(Γ) and χ(Γ) denote, respectively, the Willmore energy and the Euler characteristic of the boundary surface Γ. 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 Γ. 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.

KW - Potential theory

KW - spectral theory

KW - Spectral Geometry

KW - Neumann–Poincaré operator

KW - eigenvalues

KW - Weyl's law

KW - pseudodifferential operators

KW - Willmore energy

UR - http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=1648&option_lang=rus

UR - https://elibrary.ru/item.asp?id=37078099

M3 - Article

VL - 31

SP - 248

EP - 268

JO - АЛГЕБРА И АНАЛИЗ

JF - АЛГЕБРА И АНАЛИЗ

SN - 0234-0852

IS - 2

ER -