Mixed Boundary Value Problems in Singularly Perturbed Two-Dimensional Domains with the Steklov Spectral Condition

We study the asymptotic behavior of the spectrum of the Laplace equation with the Steklov, Dirichlet, Neumann boundary conditions or their combination in a twodimensional domain with small holes of diameter O(ε) as ε → +0. We derive and justify asymptotic expansions of eigenvalues and eigenfunctions of two types: series in ʓ= | ln ε|−1 and power series with rational and holomorphic terms in ʓ respectively. For the overall Steklov problem we obtain asymptotic expansions in the low and middle frequency ranges of the spectrum. Bibliography: 18 titles

Steklov spectral problems in a set with a thin toroidal hole

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity $Gamma_e$ having the shape of a thin toroidal set, with a constant cross-section of diameter $ell 1$. We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable $|lne|^{-1}$, and we prove that the relative asymptotic error is of much smaller order $O(e|ln e|)$ as $e o 0^+$. The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve $Gamma$, the axis of the cavity $Gamma_e$

The stiff Neumann problem: Asymptotic specialty and "kissing" domains

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Omega subset of R-d which is divided into two subdomains: an annulus Omega(1) and a core Omega(0). The density and the stiffness constants are of order epsilon(-2m) and epsilon(-1) in Omega(0), while they are of order 1 in( )Omega(1). Here m is an element of R is fixed and epsilon > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as epsilon -> 0 for any m. In dimension 2 the case when Omega(0) touches the exterior boundary partial derivative Omega S and Omega(1) gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the "smooth" case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x -> O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given

Assessing Aggregate Human Exposure to Toluene In Europe

20th Annual Conference of the International-Society-for-Environmental-Epidemiology, Oct 12-16, 2008, Pasadena, C