On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

• Published in: Revista matemática complutense Vol. 31; no. 1; pp. 1 - 62
• Main Authors: Nazarov, Sergei A., Pérez, M. Eugenia
• Format: Journal Article
• Language: English
• Published: Milan Springer Milan 2018; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Asymptotic properties; Boundary value problems; Eigenvalues; Eigenvectors; Geometry; Mathematics; Mathematics and Statistics; Periodic variations; Singularity (mathematics); Topology; 35B27; 74H45; 35B25; 35P05; Concentrated masses; Corner singularities; Homogenization problems; Asymptotic splitting of eigenvalues; 47A75; 35B40; Spectral analysis; Steklov problem
• ISSN: 1139-1138; 1988-2807
• DOI: 10.1007/s13163-017-0243-4

We construct two-term asymptotics
λ
k
ε
=
ε
m
-
2
(
M
+
ε
μ
k
+
O
(
ε
3
/
2
)
)
of eigenvalues of a mixed boundary-value problem in
Ω
⊂
R
2
with many heavy (
m
>
2
) concentrated masses near a straight part
Γ
of the boundary
∂
Ω
.
ε
is a small positive parameter related to size and periodicity of the masses;
k
∈
N
. The main term
M
>
0
is common for all eigenvalues but the correction terms
μ
k
, which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on
Γ
, exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a “strongly” singular weight.