Layer Potential Techniques in Spectral Analysis

Ammari, Habib; Kang, Hyeonbae; Lee, Hyundae

doi:10.1090/surv/153

Cited by 134 publications

(239 citation statements)

References 5 publications

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“…Recall from Theorem 7.5 that the eigenvalues λ α j (k) = (β α j (1/k)) −1 , for j ∈ N. The high coupling limit expansion for λ α j (k) is written in terms of the expansion β α j (z) = β j (0)+zβ α j,1 +· · · as 5) where λ j (0) = (β j (0)) −1 is the j th Dirichlet eigenvalue for the Laplacian on D. This naturally agrees with the formula for the leading order terms presented in [1].…”

Section: Layer Potential Representation Of Operators In Power Seriessupporting

confidence: 73%

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Bloch waves in crystals and periodic high contrast media

Lipton

Viator

2017

ESAIM: M2AN

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Analytic representation formulas and power series are developed describing the band structure inside periodic photonic and acoustic crystals made from high contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasi-periodic source free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. Convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation.

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Section: Layer Potential Representation Of Operators In Power Seriessupporting

confidence: 73%

“…We introduce the d-dimensional α-quasi-periodic Green's function, d = 2, 3 given by, see, e.g., [1],…”

Section: Hilbert Space Setting Quasi-periodic Resonances and Represementioning

confidence: 99%

Bloch waves in crystals and periodic high contrast media

Lipton

Viator

2017

ESAIM: M2AN

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“…Thus for example, one could resort to Asymptotic Analysis and may succeed to write out an asymptotic expansion for u(ǫ, x) and u(ǫ, p + ǫt). In this sense, we mention the work of Ammari and Kang [2], Ammari, Kang, and Lee [3], Ammari, Kang, and Touibi [4], Ammari, Kang, and Lim [5], Maz'ya and Movchan [6], Maz'ya, Nazarov, and Plamenewskij [7,8], Maz'ya, Movchan, and Nieves [9]. We also mention the extensive literature of Calculus of Variations and of Homogenization Theory, and in particular the contributions of Bakhvalov and Panasenko [10], Cioranescu and Murat [11,12], Jikov, Kozlov, and Oleȋnik [13], Marčenko and Khruslov [14].…”

Section: Introductionmentioning

confidence: 99%

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

Riva

Musolino

2013

Math Methods in App Sciences

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Abstract:We consider a periodically perforated domain obtained by making in R n a periodic set of holes, each of them of size proportional to ǫ. Then we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector valued function u which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ǫ contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then our aim is to describe what happens to the displacement vector function u when ǫ tends to 0. Under suitable assumptions we prove the existence of a family of solutions {u(ǫ, ·)} ǫ∈]0,ǫ ′ [ with a prescribed limiting behaviour when ǫ approaches 0. Moreover, the family {u(ǫ, ·)} ǫ∈]0,ǫ ′ [ is in a sense locally unique and can be continued real analytically for negative values of ǫ.

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“…Evolution boundary value problems may be also considered as the eigenvalue problem. Using the single layer potential technique and the polarization matrix approach, in the monograph by Ammari et al (2009) asymptotic expansions to solutions of eigenvalue problems have been provided.…”

Section: Introductionmentioning

confidence: 99%

Topology optimization of quasistatic contact problems

Myźliński¹

2012

International Journal of Applied Mathematics and Computer Science

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This paper deals with the formulation of a necessary optimality condition for a topology optimization problem for an elastic contact problem with Tresca friction. In the paper a quasistatic contact model is considered, rather than a stationary one used in the literature. The functional approximating the normal contact stress is chosen as the shape functional. The aim of the topology optimization problem considered is to find the optimal material distribution inside a design domain occupied by the body in unilateral contact with the rigid foundation to obtain the optimally shaped domain for which the normal contact stress along the contact boundary is minimized. The volume of the body is assumed to be bounded. Using the material derivative and asymptotic expansion methods as well as the results concerning the differentiability of solutions to quasistatic variational inequalities, the topological derivative of the shape functional is calculated and a necessary optimality condition is formulated.

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Layer Potential Techniques in Spectral Analysis

Abstract: Layer potential techniques in spectral analysis / Habib Ammari, Hyeonbae Kang, Hyundae Lee. p. cm.-(Mathematical surveys and monographs ; v. 153) Includes bibliographical references and index.

Cited by 134 publications

References 5 publications

Bloch waves in crystals and periodic high contrast media

Bloch waves in crystals and periodic high contrast media

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

Topology optimization of quasistatic contact problems

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