Layer Potential Techniques

The asymptotic theory for elasticity imaging described in this book relies on layer potential techniques. In this chapter we prepare the way by reviewing a number of basic facts and preliminary results regarding the layer potentials associated with both the static and time-harmonic elasticity systems. The most important results in this chapter are on one hand the decomposition formulas for the solutions to transmission problems in elasticity and characterization of eigenvalues of the elasticity system as characteristic values of layer potentials and on the other hand, the Helmholtz-Kirchhoff identities. As will be shown later, the Helmholtz-Kirchhoff identities play a key role in the analysis of resolution in elastic wave imaging. We also note that when dealing with exterior problems for harmonic elasticity, one should introduce a radiation condition, known as the Sommerfeld radiation condition, in order to select the physical solution to the problem.

This chapter is organized as follows. In Section 1.1 we first review commonly used function spaces. Then we introduce in Section 1.2 equations of linear elasticity and use the Helmholtz decomposition theorem to decompose the displacement field into the sum of an irrotational (curl-free) and a solenoidal (divergence-free) field. Section 1.3 is devoted to the radiation condition for the time-harmonic elastic waves, which is used to select the physical solution to exterior problems. In Section 1.4 we introduce the layer potentials associated with the operators of static and time-harmonic elasticity, study their mapping properties, and prove decomposition formulas for the displacement fields. In Section 1.5 we derive the Helmholtz-Kirchhoff identities, which play a key role in the resolution analysis in Chapters 7 and 8. In Section 1.6 we characterize the eigenvalues of the elasticity operator on a bounded domain with Neumann or Dirichlet boundary conditions as the characteristic values of certain layer potentials which are meromorphic operator-valued functions. We also introduce Neumann and Dirichlet functions and write their spectral decompositions. These results will be used in Chapter 11. Finally, in Section 1.7 we state a generalization of Meyer's theorem concerning the regularity of solutions to the equations of linear elasticity, which will be needed in Chapter 11 in order to establish an asymptotic theory of eigenvalue elastic problems.

1.1 SOBOLEV SPACES

Throughout the book, symbols of scalar quantities are printed in italic type, symbols of vectors are printed in bold italic type, symbols of matrices or 2-tensors are printed in bold type, and symbols of 4-tensors are printed in blackboard bold type.

The following Sobolev spaces are needed for the study of mapping properties of layer potentials for elasticity equations.

Let [partial derivative]i denote [partial derivative]/[partial derivative]xi. We use [nabla] = ([partial derivative]i)di=1 and [partial derivative]2 = ([partial derivative]2ij))di,j=1 to denote the gradient and the Hessian, respectively.

Let Ω be a smooth domain in Rd, with d = 2 or 3. We define the Hilbert space H1(Ω) by

H1(Ω) = {u [member of] L2(Ω) : [nabla]u [member of] L2(Ω)},

where [nabla]u is interpreted as a distribution and L2(Ω) is defined in the usual way, with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The space H1(Ω) is equipped with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If Ω is bounded, another Banach space H10(Ω) arises by taking the closure of C∞0(Ω), the set of infinitely differentiable functions with compact support in Ω, in H1(Ω). We will also need the space H1loc(Rd\[bar.Ω]) of functions u [member of] L2loc(Rd\[bar.Ω]), the set of locally square summable functions in Rd\[bar.Ω], such that

h u [member of] H1 (Rd\[bar.Ω]) [for all] h [member of] C∞0(Rd\[bar.Ω].

Furthermore, we define H2(Ω) as the space of functions u [member of] H1(Ω) such that [partial derivative]2ij u [member of] L2(Ω), for i, j = 1, ..., d, and the space H3/2(Ω) as the interpolation space [H1(Ω), H2(Ω)]1/2 (see, for example, the book by Bergh and Löfström [49]).

It is known that the trace operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a bounded linear surjective operator from H1(Ω) into H1/2([partial derivative]Ω), where H1/2([partial derivative]Ω) is the collection of functions f [member of] L2([partial derivative]Ω) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We set H-1/2([partial derivative]Ω) = (H1/2([partial derivative]Ω))* and let < , >1/2, -1/2 denote the duality pair between these dual spaces.

We introduce a weighted norm, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in two dimensions. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

This weighted norm is introduced because, as will be shown later, the solutions of the static elasticity equation behave like O(|x|-1) in two dimensions as |x| -> ∞. For convenience, we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

In three dimensions, W(Rd\[bar.Ω]) is the usual Sobolev space.

We also define the Banach space W1,∞(Ω) by

W1,∞(Ω) = {u [member of] L∞(Ω) : [nabla]u [member of] L∞(Ω) (1.3)

where [nabla]u is interpreted as a distribution and L∞(Ω) is defined in the usual way, with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will need the following Hilbert spaces for deriving the Helmholtz decomposition theorem

Hcurl(Ω):= {u [member of] L2(Ω)d, [nabla] × u [member of] L2 (Ω)d},

equipped with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and

Hdiv(Ω) := {u [member of] L2(Ω)d, [nabla] · u [member of] L2(Ω)},

equipped with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, let T1, ..., Td-1 be an orthonormal basis for the tangent plane to [partial derivative]Ω at x and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

denote the tangential derivative on [partial derivative]Ω. We say that f [member of] H1([partial derivative]Ω) if f [member of] L2([partial derivative]Ω) and [partial derivative]f/[partial derivatve]T [member of] L2([partial derivative]Ω)d-1. Furthermore, we define H-1([partial derivative]Ω) as the dual of H1([partial derivative]Ω) and the space Hs([partial derivative]Ω), for 0 ≥ s ≥ 1, as the interpolation space [L2([partial derivative]Ω), H1([partial derivative]Ω)]s or, equivalently, as the set of functions f [member of] L2([partial derivative]Ω) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

see again [49].

1.2 ELASTICITY EQUATIONS

Let Ω be a domain in Rd, d = 2,3. Let λ and μ be the Lamé constants for Ω satisfying the strong convexity condition

μ > 0 and dλ + 2μ > 0. (1.5)

The constants [lamda] and μ are respectively referred to as the compression modulus and the shear modulus. The compression modulus measures the resistance of the material to compression and the shear modulus measures the resistance to shearing. We also introduce the bulk modulus β := λ + 2μ/d. We refer the reader to [122, p.11] for an explanation of the physical significance of (1.5).

In a homogeneous isotropic elastic medium, the elastostatic operator corresponding to the Lamé constants λ, μ, is given by

Lλ,μ u := μΔu + (λ + μ)[nabla][nabla] · u, u : Ω -> Rd. (1.6)

If Ω is bounded with a connected Lipschitz boundary, then we define the conormal derivative [partial derivative]u/[partial derivative]v by

[partial derivative]u/[partial derivative]v = λ([nabla] · u)n + μ([nabla]u + [nabla]ut)n, (1.7)

where [nabla]u is the matrix ([partial derivative]jui)di,j=1 with ui being the i-th component of u, the superscript t denotes the transpose, and n is the outward unit normal to the boundary [partial derivative]Ω.

Note that the conormal derivative has a direct physical meaning:

[partial derivative]u/[partial derivative]v = traction on [partial derivative]Ω.

The vector u is the displacement field of the elastic medium having the Lamé coefficients λ and μ, and the symmetric gradient

[nabla]su := ([nabla]u + [nabla]ut)/2

is the strain tensor.

In Rd, d = 2, 3, let

I := δij ei [cross product] ej,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with (e1, ..., ed) being the canonical basis of Rd and [cross product] denoting the tensor product between vectors in Rd. Here, I is the d × d identity matrix or 2-tensor while I is the identity 4-tensor.

Define the elasticity tensor C = (Cijkl)di,j,k,l=1 for Rd by

Cijkl = λδij δkl + μ([delt]ikδjl + δilδjk), (1.8)

which can be written as

C := λI [cross product] I + 2μI.

With this notation, we have

Lλ,μu = [nabla] · C[nabla]su,

and

[partial derivative]u/[partial derivative]v = (C[nabla]su)n = σ(u)n,

where σ(u) is the stress tensor given by

σ(u) = C[nabla]s u.

Now, we consider the elastic wave equation

ρ[partial derivative]2tU - Lλ,μU = 0,

where the positive constant ρ is the density of the medium. Then, we obtain a time-harmonic solution U(x, t) = Re]e-iωtu(x)] if the space-dependent part u satisfies the time-harmonic elasticity equation for the displacement field,

(Lλ,μ + ω2ρ)u = 0, (1.9)