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Asymptotic models for scattering from unbounded media with high conductivity

Houssem Haddar, Armin Lechleiter (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We analyze the accuracy and well-posedness of generalized impedance boundary value problems in the framework of scattering problems from unbounded highly absorbing media. We restrict ourselves in this first work to the scalar problem (E-mode for electromagnetic scattering problems). Compared to earlier works, the unboundedness of the rough absorbing layer introduces severe difficulties in the analysis for the generalized impedance boundary conditions, since classical compactness arguments are no...

Asymptotics for multifractal conservation laws

Piotr Biler, Grzegorz Karch, Wojbor Woyczynski (1999)

Studia Mathematica

We study asymptotic behavior of solutions to multifractal Burgers-type equation u t + f ( u ) x = A u , where the operator A is a linear combination of fractional powers of the second derivative - 2 / x 2 and f is a polynomial nonlinearity. Such equations appear in continuum mechanics as models with fractal diffusion. The results include decay rates of the L p -norms, 1 ≤ p ≤ ∞, of solutions as time tends to infinity, as well as determination of two successive terms of the asymptotic expansion of solutions.

Behaviour of the first eigenvalue of the p-Laplacian in a domain with a hole

M. Sango (2001)

Colloquium Mathematicae

We investigate the behaviour of a sequence λ s , s = 1,2,..., of eigenvalues of the Dirichlet problem for the p-Laplacian in the domains Ω s , s = 1,2,..., obtained by removing from a given domain Ω a set E s whose diameter vanishes when s → ∞. We estimate the deviation of λ s from the eigenvalue of the limit problem. For the derivation of our results we construct an appropriate asymptotic expansion for the sequence of solutions of the original eigenvalue problem.

Borel summable solutions of the Burgers equation

Grzegorz Łysik (2009)

Annales Polonici Mathematici

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation t u - x ² u = x ( u ² ) to be convergent or Borel summable.

Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem

Clair Poignard (2012)

ESAIM: Proceedings

We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be εα–periodic, ε being the magnitude of the mean thickness of the layer, and α a positive parameter describing the degree of roughness. For ε tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent to...

Boundary layer for Chaffee-Infante type equation

Roger Temam, Xiaoming Wang (1998)

Archivum Mathematicum

This article is concerned with the nonlinear singular perturbation problem due to small diffusivity in nonlinear evolution equations of Chaffee-Infante type. The boundary layer appearing at the boundary of the domain is fully described by a corrector which is “explicitly" constructed. This corrector allows us to obtain convergence in Sobolev spaces up to the boundary.

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