Elliptic Problems With a Parameter and Parabolic Problems of General Type

Agranovich, M. S.; Vishik, M. I.

doi:10.1070/rm1964v019n03abeh001149

Cited by 445 publications

(462 citation statements)

References 17 publications

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“…If x 0 ∈ Γ we choose x 1 ∈ B δ (x 0 ) ∩ Ω and δ > 0 sufficiently small such that B δ (x 1 ) ⊂ B δ (x 0 ) ∩ Ω. In both cases, the statement of the lemma follows easily by arguments similar to those used in the proof of [AV,Lemma 4.2]. Proposition 2.3.…”

Section: Proof Of the Necessitymentioning

confidence: 83%

Necessity of Parameter-ellipticity for Multi-order Systems of Differential Equations

Denk

Faierman

2012

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

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Section: Proof Of the Necessitymentioning

confidence: 83%

Necessity of Parameter-ellipticity for Multi-order Systems of Differential Equations

Denk

Faierman

2012

Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations

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“…Therefore, by the results of [35], the eigenvalues of the pencil A lie inside the union of a strip and a double angle:…”

Section: Preliminary Description Of the Resultsmentioning

confidence: 99%

“…The Parseval identity for Fourier images reshapes (2.9) to a family of nonhomogeneous problems (2.9): 3 . Since the line R ⊂ C is free from the spectrum of the pencil A, the estimates [35] for solutions of elliptic problems with a parameter (see [12] or [16,Chapter 3] …”

Section: Spectrum Of the Pencil On The Periodicity Cellmentioning

confidence: 99%

Gap opening in the essential spectrum of the elasticity theory problem in a periodic half-layer

Назаров

2010

St. Petersburg Math. J.

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Abstract. Rayleigh waves are studied in an elastic half-layer with a periodic end and rigidly clamped faces. It is established that the essential spectrum of the corresponding problem of elasticity theory has a band structure, and an example of a waveguide is presented in which a gap opens in the essential spectrum; i.e., an interval arises that contains points of an at most discrete spectrum. §1. Introduction 1. Preamble. In the case of a homogeneous isotropic elastic half-space, surface waves were discovered by Lord Rayleigh [1], and since then many investigations devoted to similar effects have appeared (see a survey of modern literature in [2], and also the paper [3], which is absent in [2]). A Rayleigh wave is a plane wave of the forma vector-valued factor U (z) decaying exponentially as z = x 3 → −∞. The arising of such waves explains specific wave processes in elastic bodies.The wave number k ∈ R + = [0, +∞) determines a frequency cutoff ω † (k) above which, i.e., for ω ≥ ω † (k), the wave (1.1) exists necessarily. In the present paper we deal with a problem related to a similar phenomenon. Namely, we study an elastic, but not necessarily homogeneous and isotropic cushion Ω 0 having the form of a half-layer with a periodic end and rigidly clamped side faces (see Figure 1, where the clamped surface is shadowed). Some Rayleigh waves decaying exponentially as z → −∞ can propagate along the end of the cushion, and if Ω 0 is a cylinder ∆ × R, then we have a single cutoff ω † > 0; i.e., the corresponding operator of the elasticity theory system acquires a continuous spectrum [ω † , +∞). Our main goal in this paper is to show that, in the periodic case, a gap can open in the essential spectrum; i.e., an interval can exist the ends of which belong to the continuous spectrum, but inside which only points of the discrete spectrum may occur. Some of the results were announced earlier in [4].

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“…In the case µ = 0 this is the usual definition of ellipticity with parameter which was introduced by Agmon [1] and Agranovich-Vishik [3]. Therefore we may assume in the following that µ > 0.…”

Section: Introductionmentioning

confidence: 99%

“…In this case even for λ = 0 the principal symbol A (0) (x, ξ, λ) vanishes for ξ = 0 which causes the main difficulties in proving existence results and estimates. Note that the symbol A (0) (x, ξ, λ) is homogeneous in ξ and λ of degree 2m, as it is the case for the problems treated in [3].…”

Section: Introductionmentioning

confidence: 99%