Spectral stiff problems in domains surrounded by thin
stiff and heavy bands
D. Gómez
Departamento de Matemáticas, Estadı́stica y Computación
Universidad de Cantabria. Av. de los Castros s/n. 39005 Santander. Spain
gomezdel@unican.es
Sesión especial: “Homogeneización y Perturbaciones Espectrales”
This a joint work with S. A. Nazarov (Russian Academy of Sciences, St. Peterburg) and
E. Pérez (Universidad de Cantabria, Santander)
Let Ω be a bounded domain of R2 with a smooth boundary Γ and let (ν, τ ) be the
natural orthogonal curvilinear coordinates in a neighborhood of Γ: τ is the arc length and
ν the distance along the outer normal to Γ. Let also ` denote the length of Γ and κ(τ )
the curvature of the curve Γ at the point τ . We assume that the domain Ω is surrounded
by a curvilinear strip defined by ωε = {x : 0 < ν < εh(τ )} where ε > 0 is a small
parameter and h a strictly positive function of the τ variable, `-periodic, h ∈ C ∞ (S` )
where S` stands for the circumference of length `. Let Ωε be the domain Ωε = Ω ∪ ωε ∪ Γ
and Γε the boundary of Ωε .
We consider the spectral Neumann problem in Ωε for a second order differential operator with piecewise constants coefficients:

−A∆x U ε = λε U ε
in Ω,



−t
ε
ε
−t−m
ε


−aε ∆x u = λ ε
u in ωε ,


ε
ε
U =u
on Γ,
(1)



A∂ν U ε = aε−t ∂ν uε
on Γ,



 −t
aε ∂n uε = 0
on Γε .
Here, A and a are two positive constants while ∂ν and ∂n denote the derivatives along
the outward normal vectors ν and n to the curves Γ and Γε , respectively. We consider
t ≥ 0 and m ≥ 0, and assume additionally that either t > 0 or m > 0. We study
the asymptotic behavior, as ε → 0, of the eigenvalues λε of problem (1) and of their
corresponding eigenfunctions {U ε , uε }. Depending on the parameters t and m, and the
function h, we highlight local effects for the eigenfunctions associated with certain range of
frequencies. We also consider the case where ωε is defined by ωε = {x : 0 < ν < εhε (τ )}
with hε (τ ) = h(τ /ε) and h a periodic function. See [1] and [2] for further descriptions of
the problem.
Acknowledgements: partially supported by MICINN, MTM2009-12628
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References
[1] D. Gómez, S. A. Nazarov and E. Pérez. Spectral stiff problems in domains
with strongly oscillating boundary. Integral Methods in Science and Engineering,
Birkhäuser, 2011.
[2] D. Gómez, S. A. Nazarov and E. Pérez. Spectral stiff problems in domains surrounded
by thin stiff and heavy bands: local effects for the eigenfunctions. Networks and
heterogeneous media, 6, 1–35, 2011.
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