The scattering theory of C. Wilcox in elasticity

Mabrouk, Mongi; Helali, Zouheir

doi:10.1002/mma.325

Cited by 4 publications

(7 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, it can be shown [15] moreover that u 0 (t; x) converges to u ∞ 0 (t; x) also in the energy norm as t → ∞. In particular (x 0 = t):…”

Section: Asymptotics In Pulse Modementioning

confidence: 94%

“…In fact, this extension, especially if we want to have the great level of generality as here, is far from being straightforward and presents many tedious hard technicalities. For these reasons, we have devoted to this an entire paper [15], the results of which are summarized in the appendix and used throughout the present paper. The appendix contains a brief exposition of the time-dependent theory of scattering for the pair of operators A 0 and A corresponding, respectively, to the elasticity problem in the entire space (the free problem) and in the exterior domain , under the assumption that veriÿes a local energy compactness criterion (the condition ELC) and is of Korn's type.…”

Section: Theoremmentioning

confidence: 99%

See 1 more Smart Citation

The elastic echo problem

Mabrouk¹,

Helali

2002

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

SUMMARYWe consider a homogeneous isotropic unbounded linear elastic medium ⊂ R 3 , having a free boundary . A forcing f(t; x) creates an incident displacement ÿeld u 0 (t; x). This primary ÿeld is scattered by giving rise to a secondary ÿeld or echo, for which we determine the asymptotic behaviour in time. NOTATIONSLet us ÿrst explain the notations which are used throughout the text. Some of them are deÿned more precisely in the appendix. We recommend to the reader which is not familiarized with the scattering theory of Wilcox to readÿrst this appendix.The space R 3 is endowed with the canonical basis {e 1 = (1; 0; 0); e 2 = (0; 1; 0); e 3 = (0; 0; 1), the origin O and the co-ordinate system (x 1 ; x 2 ; x 3 ). The canonical scalar product of R N is denoted by a dot. I is the identity map. For a matrix or a map A, For any vector u ∈ C 1 ( ), we deÿne the gradient tensor ∇u(x) and the linearized deformation tensor (u) = 1 2 (∇U + t ∇u) with components ij (u) = 1 2 (@u i =@x j + @u j =@x i ) and (u) will be the linearized stress tensor of u. By (: ; :), we denote the scalar product in L 2 . We deÿne also the following Hilbert spaces:2 ( * ; ) {u; u satisfait (NG)} where (NG) is the generalized Neumann condition (NG):We use also the localized counterparts (FrÃ echet spaces) L ; L N; loc ; L loc 2 ( * ; ) for functions which are locally in the mentioned spaces.Let X a Hilbert space and I ⊆ R. Then C(I; X ) is the Banach space of continuous X -valued functions and H 1 (I; X ) is the Hilbert space of X -valued functions, which are square integrable together with their ÿrst derivative on I .∇ is the usual di erential operator. If u is a scalar function, ∇u is a vector ( the gradient), ∇ 2 u = ∇∇u is the Hessian matrix of u, with elements @ 2 u=@x i x j , and ∇u is a matrix if u is a vectorial function.A 0 (resp., A) is the abstract operator associated with the free space (resp. exterior ) problem (see the appendix). For an operator T , D(T ) denotes the domain, T * the adjoint and T −1 the inverse. is the Fourier transform, * its adjoint (and inverse). The dual variable of x is p. Thus if u(x) ∈ L 2 (R 3 );û(p) = u(p) is its Fourier transform. If (·) is some expression, we shall also denote sometimes by (·) ∧ its Fourier transform.grad div is the LamÃ e operator, ; are the LamÃ e constants. T(n; u) = (u)n is the traction vector at the point x of the boundary where the normal is n. [L 2 (R 3 )] 2 respectively)P 1 = Á ⊗ Á,P 2 = I −Á ⊗ Á; Á = p=|p|. We deÿne then j =P j ;is the Green's function of the free space, unique solution of the problem ( * x + z)G 0 (x; x ; z) = − (x − x )I; ∀x; x ∈ R 3 . For real positive , we deÿne the two matrices G ± 0 (x; x ; ) = G 0 (x; x ; ± i0) by the limiting absorption technique:

show abstract

“…, it can be shown [15] moreover that u 0 (t; x) converges to u ∞ 0 (t; x) also in the energy norm as t → ∞. In particular (x 0 = t):…”

Section: Asymptotics In Pulse Modementioning

confidence: 94%

Section: Theoremmentioning

confidence: 99%

The elastic echo problem

Mabrouk¹,

Helali

2002

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…The class of domains considered by Mabrouk and Helali is wider than ours, and corresponds to domains which satisfy a condition which they call the elastic local compactness property (we refer to [12] for the definition). The methods used in [12] are independent to those in the present work. In particular, the radiation conditions considered in the construction of the perturbed (distorted) plane waves are different.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the radiation conditions considered in the construction of the perturbed (distorted) plane waves are different. Also, the generalized eigenfunctions in [12] are some 3 × 3 matrices (called the distorted plane waves in that reference) that satisfy column-wise equations analogous to (3) for elasticity; as a result of this, the formula in [12] corresponding to (2) involves matrix multiplications. Our choice of the vector fields V j as the eigenfunctions gives a formula for the spectral resolution which uses the inner product in C 3 instead; our approach gives an analogue to (2) which, we feel, turns out to be quite natural.…”

Section: Introductionmentioning

confidence: 99%

“…The spectral resolution of the operator (1) acting on an exterior domain with Neumann boundary conditions, is written in terms of an expansion of generalized eigenfunctions in a recent paper by M. Mabrouk and Z. Helali (see [12,Theorem 13]). The class of domains considered by Mabrouk and Helali is wider than ours, and corresponds to domains which satisfy a condition which they call the elastic local compactness property (we refer to [12] for the definition).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation