Homogenization in a thin domain with an oscillatory boundary

We consider a family {Ω ε } ε>0 of periodic domains in R 2 with waveguide geometry and analyse spectral properties of the Neumann Laplacian −∆ Ω ε on Ω ε . The waveguide Ω ε is a union of a thin straight strip of the width ε and a family of small protuberances with the so-called "room-and-passage" geometry. The protuberances are attached periodically, with a period ε, along the strip upper boundary. For ε → 0 we prove a (kind of) resolvent convergence of −∆ Ω ε to a certain ordinary differential operator. Also we demonstrate Hausdorff convergence of the spectrum. In particular, we conclude that if the sizes of "passages" are appropriately scaled the first spectral gap of −∆ Ω ε is determined exclusively by geometric properties of the protuberances. The proofs are carried out using methods of homogenization theory.

show abstract

“…where the function u 1 belongs to H 2 (R) and is a solution of the problem 2 be an arbitrary function. We introduce the function f ε by…”

Section: Setting Of the Problem And Main Resultsmentioning

confidence: 99%

Spectrum of a singularly perturbed periodic thin waveguide

Cardone

Khrabustovskyi

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…Indeed, in [9,10,8,11,47] thin domains with highly oscillatory boundaries have been studied in a theoretical framework and convergence results have been obtained, mostly for elliptic or stationary problems. In comparison with our work we do not assume a thin domain.…”

Section: Literature Reviewmentioning

confidence: 99%

Effective rough boundary parametrization for reaction-diffusion systems

Mocenni

Sparacino

Zubelli

2014

APPL ANAL DISCRETE M

View full text Add to dashboard Cite

We address the problem of parametrizing the boundary data for reactiondiffusion partial differential equations associated to distributed systems that possess rough boundaries. The boundaries are modeled as fast oscillating periodic structures and are endowed with Neumann or Dirichlet boundary conditions. Using techniques from homogenization theory and multiple-scale analysis we derive the effective equation and boundary conditions that are satisfied by the homogenized solution. We present numerical simulations that validate our theoretical results and compare it with the alternative approach based on solving the same equation with a smoothed version of the boundary. The numerical tests show the accuracy of the homogenized solution to the effective system vis a vis the numerical solution of the original differential equation. The homogenized solution is shown undergoing dynamical regime shifts, such as anticipation of pattern formation, obtained by varying the diffusion coefficient.

show abstract

“…Boundary value and spectral problems in thin domains are usually treated using the analysis of resolvents ( [FS09]), the method of asymptotic expansions (see for example [CD79], [Pan05], [BF10], [MP10], [Naz01], [PS13]), two-scale convergence ( [EP96], [MMP00], [PP11], [PP15]), Γ-convergence ( [MS95], [AB01], [BFF00], [Gau+02], [BMT07], [BMT12]), compensated compactness agrument ( [GM03]), and the unfolding method ( [BG08], [AP11], [AVP17]). The presented list of works devoted to the homogenization in thin structures is far from being complete, but our primary focus is the case of thin domains with locally periodic rapidly varying thickness, and to our best knowledge the works closely related to our study are [MP10], [AP11], [FS09], [BF10], and [NPT16]. We describe them briefly below.…”

Section: Introductionmentioning

confidence: 99%

“…The case of periodic rapidly oscillating boundary was considered in [MP10], where the authors studied the asymptotic behaviour of second-order self-adjoint elliptic operators with periodic coefficients, for different boundary conditions. In [AP11] the case of a locally periodic rapidly oscillating boundary was addressed, and the authors studied the Neumann boundary value problem for the Laplace operator in a two-dimensional thin domain by means of the unfolding method. Spectral asymptotics of the Laplace operator in thin domains with slowly varying thickness were considered in [FS09], [BF10], [NPT16], where under the Dirichlet boundary conditions the localization of eigenfunctions occur.…”

Section: Introductionmentioning

confidence: 99%

Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

Pettersson¹

2017

Differential Equations &Amp; Applications

View full text Add to dashboard Cite

Abstract. The aim of this paper is to adapt the notion of two-scale convergence in L p to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally periodic rapidly oscillating boundary shrinks to a segment, and the corresponding measure charging the cylinder converges to a one-dimensional Lebegues measure of an interval. The method is then applied to the asymptotic analysis of linear elliptic operators with locally periodic coefficients and a p-Laplacian stated in thin cylinders with locally periodic rapidly varying thickness.Mathematics subject classification (2010): Primary 35B27, secondary 74K10.

show abstract

Homogenization in a thin domain with an oscillatory boundary

Cited by 56 publications

References 6 publications

Spectrum of a singularly perturbed periodic thin waveguide

Spectrum of a singularly perturbed periodic thin waveguide

Effective rough boundary parametrization for reaction-diffusion systems

Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

Contact Info

Product

Resources

About