Volume 35, 2005, 91–127
D. Natroshvili, G. Sadunishvili, I. Sigua and Z. Tediashvili
FLUID-SOLID INTERACTION:
ACOUSTIC SCATTERING BY AN ELASTIC OBSTACLE WITH LIPSCHITZ BOUNDARY
interface problems of the theory of acoustic scattering by an elastic obstacle which are also known as fluid-solid (fluid-structure) interaction problems.
It is assumed that the obstacle has a Lipschitz boundary. The sought for field functions belong to spaces havingL2integrable nontangential maximal functions on the interface and the transmission conditions are understood in the sense of nontangential convergence almost everywhere. The unique- ness and existence questions are investigated. The solutions are represented by potential type integrals. The solvability of the direct problem is shown for arbitrary wave numbers and for arbitrary incident wave functions. It is established that the scalar acoustic (pressure) field in the exterior domain is defined uniquely, while the elastic (displacement) vector field in the in- terior domain is defined modulo Jones modes, in general. On the basis of the results obtained it is proved that the inverse fluid-structure interaction problem admits at most one solution.
2000 Mathematics Subject Classification. 35J05, 35J25, 35J55, 35P25, 47A40, 74F10, 74J20.
Key words and phrases. Fluid-solid interaction, Elasticity theory, Helmholtz equation, Potential theory, Interface problems, Steady state os- cillations.
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1. Introduction
Direct and inverse problems related to the interaction between vector fields of different dimension have received much attention in the mathemat- ical and engineering scientific literature and have been intensively investi- gated for the past years. They arise in many physical and mechanical models describing the interaction of two different media where the whole process is characterized by a vector-function of dimension kin one medium and by a vector-function of dimensionnin another one (for example, fluid-structure interaction where a streamlined body is an elastic obstacle, scattering of acoustic and electromagnetic waves by an elastic obstacle, interaction be- tween an elastic body and seismic waves, etc.).
Quite many authors have considered and studied in detail the direct problems of the interaction between an elastic isotropic body which occu- pies a bounded region Ω+ (where a three-dimensional elastic vector field is to be defined) and some isotropic medium (fluid say) which occupies the unbounded exterior region, the complement of Ω+ (where a scalar field is to be defined). The time-harmonic dependent unknown vector and scalar fields are coupled by some kinematic and dynamic conditions on the bound- ary ∂Ω+. Main attention has been given to the problems determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in acompressible inviscid fluid. An exhaustive information in this direction concerning theoretical and numerical results can be found in [3], [4], [5], [6], [7], [21], [12], [13], [14], [17], [19], [20], [29], [40].
The case ofanisotropicobstacle have been treated in [37], [22], [23], [36].
In [36] the corresponding inverse problem is also considered. This kind of problems arise in detecting and identifying submerged objects.
In all the above papers the boundary of the region occupied by an elastic obstacle is assumed to besufficiently smoothand the transmission conditions are considered either in the classical, or in the usual Sobolev or generalized functional trace sense (in the case of weak setting).
In the present paper our main goal is to generalize the results of the above cited works to Lipschitz domains when the transmission conditions are understood in the sense of nontangential convergence almost everywhere.
Following the approach for the case of smooth interface, we propose as solution an ansatz of combinations of single and double layer potentials.
By the special representation formulas of the sought for acoustic and elas- tic fields we reduce equivalently the original transmission problem to the system of integral equations. In the case of Lipschitz interface, however, the lack of smoothness introduce essential difficulties in the analysis of the integral equations obtained. These are overcome through the use of har- monic analysis technique together with a careful study of the properties of the boundary integral operators generated by the single and double layer acoustic and elastic potentials. We essentially employ the results obtained in papers [43], [11], [41], [31], [38], [32], [33], [1], [2].
In particular, by the potential method we have derived the necessary and sufficient conditions of solvability of the original transmission problem and shown that the direct scattering problems are solvable for arbitrary values of the frequency parameter and for arbitrary incident wave functions. It is established that the scalar radiating acoustic (pressure) field in the exterior domain is defined uniquely, while the elastic (displacement) vector field in the interior domain is defined modulo Jones modes, in general.
On the basis of the results obtained and applying the approach developed in [9], [27], and [36] we have proved the uniqueness of solution to the inverse fluid-structure interaction (scattering) problem.
2. Mathematical Formulation of the Interface Problem.
Properties of Potentials
2.1. Elastic field. Let Ω+⊂R3be a bounded domain (diam Ω+<+∞) with a connected boundaryS =∂Ω+ and Ω−=R3\Ω+, Ω+ = Ω+∪S.
Throughout the paper we assume that the boundary S is a Lipschitz surface (if not otherwise stated).
The region Ω+ is supposed to be filled up by a homogeneous isotropic medium with the elastic coefficients (Lam´e constants) λ and µ, and the density%1= const>0.
The homogeneous system of steady state oscillation equations of the lin- ear elasticity reads as follows (see, e.g., [28])
A(∂, ω)u(x) :=A(∂)u(x) +%1ω2u(x) =
=µ∆u+ (λ+µ) grad divu+%1ω2u(x) = 0, (2.1) where u= (u1, u2, u3)> is the complex-valued displacement vector (ampli- tude),ω >0 is the oscillation (frequency) parameter,
A(∂, ω) :=A(∂) +%1ω2I3, A(∂) := [Akj(∂) ]3×3,
Akj(∂) =µ δkj∆ + (λ+µ)∂k∂j, ∂= (∂1, ∂2, ∂3), ∂j= ∂
∂xj
; here and in what follows I3 stands for the unit 3×3 matrix, δkp is the Kronecker delta, ∆ =∂12+∂22+∂32 is the Laplace operator, the superscript
> denotes transposition.
The stress tensor{σkj}and the strain tensor{εkj}are related by Hook’s law
σkj =δkjλ(ε11+ε22+ε33) + 2µ εkj, εkj = 2−1(∂kuj+∂juk).
As usual, the quadratic form corresponding to the density of potential energy is assumed to be positive definite in the symmetric real variables εkj =εjk (see, e.g., [28], [15])
E(u, u) =σkjεkj ≥δ1εkjεkj, δ1= const>0, (2.2)
implying the inequalitiesµ >0, 3λ+ 2µ >0.Clearly, we also have
E(u, u) =σkjεkj ≥δ1[ε0kjε0kj+ε00kjε00kj], (2.3) where an over-bar denotes complex conjugation, and whereεkj =ε0kj+i ε00kj are the entries of the complex strain tensor corresponding to the vector u=u0+i u00, i=√
−1.Here and in what follows we employ the summation over repeated indices from 1 to 3, unless otherwise stated.
The inequality (2.2) implies the positive definiteness of the matrixA(ξ) forξ∈R3\ {0}
A(ξ)ζ·ζ =Akj(ξ)ζjζk ≥δ2|ξ|2|ζ|2, δ2= const>0,
whereζis an arbitrary three-dimensional complex vector,ζ ∈C3. Through- out the papera·b=Pm
k=1akbk denotes the scalar product of two vectors inCm.
Further we introduce the stress operator
T(∂, n) = [Tkj(∂, n) ]3×3, Tkj(∂, n) =λ nk∂j+µ nj∂k+µ δkj∂n, wheren= (n1, n2, n3) is a unit vector and∂n denotes the directional deriv- ative along the vectorn.
Thek-th component of the stress vector acting on a surface element with the unit normal vectornis calculated by the formula
[T(∂, n)u]k=σkjnj=h
2µ ∂nu+λ ndivu+µ[n×curlu]i
k, (2.4) where [· × ·] denotes the cross product of two vectors.
Note that throughout the paper we will employ the notationL2,W2s(s≥ 0), andH2r(r∈R) for the usual Lebesgue, Sobolev-Slobodetski, and Bessel potential function spaces respectively. Recall that L2 = W20 = H20 and W2s = H2s for s ≥ 0, and for a Lipschitz surface S the space H2r(S) is defined correctly only for−1≤r≤1. By||u||X we denote the norm of the elementuin the spaceX.
2.2. Scalar field. We assume that the exterior, simply connected domain Ω−is filled up by a homogeneous anisotropic medium (compressible viscid fluid say) with the constant density %2. Further, let some physical process (the propagation of acoustic waves say) in Ω− be described by a complex-valued scalar function (scalar pressure field)w(x) being a solution of the homogeneous ”wave equation” (generalized Helmholtz equation)
a(∂, ω)w:=a(∂)w+%2ω2w= 0, (2.5) wherea(∂) =akj∂k∂j, the real constantsakj =ajkdefine a positive definite matrixea= [akj]3×3, i.e.,
e
a ζ·ζ =akjζjζk≥δ3|ζ|2, δ3= const>0, (2.6) for arbitraryζ ∈C3.
Denote bySωthe characteristic surface (ellipsoid) given by the equation e
a ξ·ξ−%2ω2= 0, ξ∈R3.
For an arbitrary vector η ∈ R3 with |η| = 1 there exists only one point ξ(η)∈ Sω such that the outward unit normal vectorn(ξ(η)) toSω at the pointξ(η) has the same direction asη, i.e.,n(ξ(η)) =η. Note thatξ(−η) =
−ξ(η)∈Sωand n(−ξ(η)) =−η.
It can be easily verified that
ξ(η) =ω√%2 ea−1η·η−1/2
e
a−1η, (2.7)
whereea−1is the matrix inverse toea.
Now we are in the position to define the class Som(Ω−) of complex-valued functions satisfying the generalized Sommerfeld type radiation conditions (see, e.g., [42]).
A function w belongs to Som(Ω−) if w ∈ C1(Ω−) and for sufficiently large|x|
w(x) =O(|x|−1), ∂kw(x)−i ξk(η)w(x) =O(|x|−2), k= 1,2,3, (2.8) whereξ(η)∈Sω corresponds to the vectorη=x/|x| (i.e.,ξ(η) is given by (2.7) withη= ˆx:=x/|x|).
The conditions (2.8) are equivalent to the classical Sommerfeld radiation conditions for the Helmholtz equation if the a(∂) is the Laplace operator (see, e.g., [42], [8]). In the sequel, elements of the class Som(Ω−) will also be referred to asradiating functions.
We have the following analogue of the classical Rellich-Vekua lemma (for details see [22]).
Lemma 2.1. Letw∈Som(Ω−) be a solution of(2.5)inΩ− and let
R→+∞lim =n Z
ΣR
[ Λ(∂x, n(x))w(x) ] [w(x) ]dΣR
o= 0,
where ΣR is the sphere centered at the origin and radius R, and Λ(∂, n) denotes the co-normal differentiation
Λ(∂x, n(x)) :=akjnk(x)∂j. Then w= 0 inΩ−.
Note that, ifw is a solution of the homogeneous equation (2.5), thenw is an analytic function of the real variablexin the domain Ω−. Moreover, if, in addition, w ∈ C1(Ω−)∩Som(Ω−) and the boundary surface S =
∂Ω± is sufficiently smooth (C1, α smooth say), then the following integral representation formula holds (cf. [42], [23])
Z
S
γ(x−y, ω)[Λ(∂, n)w(y)]−dSy− Z
S
[Λ(∂y, n(y))γ(y−x, ω)] [w(y)]−dSy=
=
w(x) for x∈Ω−,
0 for x∈Ω+, (2.9)
where
γ(x, ω) =−exp{i ω√%2(ea−1x·x)1/2}
4π|ea|1/2(ea−1x·x)1/2 , |ea|= detea, (2.10) is a radiating fundamental solution to the equation (2.5) (see, e.g., Lemma 1.1 in [23]), the symbols [·]± denote limits onS from Ω±.
Here and throughout the papern(y) stands for theoutward unit normal vectortoS at the pointy∈S.
For sufficiently large|x|we have the following asymptotic representation γ(x−y, ω) =c(ξ)exp{i ξ · (x−y)}
|x| +O(|x|−2), c(ξ) =− |ea ξ|
4πω(%2|ea|)1/2,
(2.11)
whereyvaries in a bounded subset ofR3 andξ =ξ(η)∈Sωcorresponds to the directionη=x/|x|; the asymptotic formula (2.11) can be differentiated any times with respect tox andy (see [23]).
From (2.9) with the help of (2.11) we get the asymptotic representation (for sufficiently large|x|) of a radiating solution to the equation (2.5)
w(x) =w∞(ξ)exp{i ξ · x}
|x| +O(|x|−2), (2.12) where
w∞(ξ) =c(ξ) Z
S
e−i ξ·y [Λ(∂, n)w(y)]−+i(ea ξ·n(y)) [w(y)]− dSy
withξ and c(ξ) as in (2.11); w∞(ξ) is the so-calledfar-field pattern of the radiating solutionw(cf. [9]).
2.3. Formulation of direct and inverse interaction problems. We recall that any Lipschitz surfaceS satisfies the uniform cone condition and vice versa [18], i.e., each point x ∈S is the vertex of two truncated cones γ(±)(x) with common axis that are congruent to a fixed cone
{x= (x1, x2, x3)∈R3 : 0≤x3≤h, q
x21+x22≤c∗(h−x3)}, c∗>0, h >0, and such that all points of these cones except x lie in the respective do- mainsγ(±)(x)⊂Ω±. Usually, these conesγ(±)(x) are callednontangential approach regionsand are subjected to some regularity conditions described, e.g., in [43]. Note that the exterior normal vectorn(x) exists almost every- where onS and belongs to the spaceL∞(S).
In what follows the boundary values [·]± on the surfaceS are taken in the sense ofpoint-wise nontangential convergenceatalmost every pointwith
respect to the surface measure (if not otherwise stated). In particular, [u(x)]±= lim
γ(±)(x)3y→x∈Su(y), [w(x)]± = lim
γ(±)(x)3y→x∈Sw(y), [T(∂x, n(x))u(x)]±= lim
γ(±)(x)3y→x∈ST(∂y, n(x))u(y), [Λ(∂x, n(x))w(x)]± = lim
γ(±)(x)3y→x∈SΛ(∂y, n(x))w(y), for almost allx∈S.
Further, we denote byM±(v) thenontangential maximal functionsonS corresponding to a functionv
M±(v)(x) = sup
y∈γ(±)(x)|v(y)| for almost all x∈S (for details see [43], [11]).
Remark 2.2. Denote by Ha, ω(Ω±) the subspace ofC2(Ω±) consisting of functions w that satisfy the homogeneous equation (2.5) in Ω± and such that the nontangential boundary values [w]± and [Λ(∂, n)w]± exist almost everywhere on S, and the maximal nontangential functions M±(w) and M±(∂jw) (j = 1,2,3) are in L2(S). Then automatically, [w]± ∈ H21(S), [Λ(∂, n)w]± ∈L2(S), andw∈H232(Ω+) andw∈H2, loc32 (Ω−).
Analogously, let HA, ω(Ω±) be the subspace of [C2(Ω±)]3 consisting of vectorsu that satisfy the homogeneous equation (2) in Ω± and such that the nontangential boundary values [u]± and [T(∂, n)u]± exist almost ev- erywhere on S, and the maximal nontangential functions M±(uk) and M±(∂juk) (k, j= 1,2,3) are inL2(S). Then automatically, [u]±∈[H21(S)]3, [T(∂, n)u]±=!∈[L2(S)]3, andu∈[H232(Ω+)]3andu∈[H2, loc32 (Ω−)]3.
Note that for solutionswanduof the homogeneous equations (2.5) and (2), respectively, the following equivalences hold
M+(w)∈L2(S) ⇔ w∈H212(Ω+), M−(w)∈L2(S) ⇔ w∈H2, loc12 (Ω−), M+(u)∈[L2(S)]3 ⇔ w∈[H212(Ω+)]3, M−(u)∈[L2(S)]3 ⇔ w∈[H2, loc12 (Ω−)]3, M+(∂jw)∈L2(S) ⇔ w∈H232(Ω+), M−(∂jw)∈L2(S) ⇔ w∈H2, loc32 (Ω−), M+(∂ju)∈[L2(S)]3 ⇔ w∈[H232(Ω+)]3, M−(∂ju)∈[L2(S)]3 ⇔ w∈[H2, loc32 (Ω−)]3, wherej= 1,2,3 (for details see [11], [32], [33], [1], [2]).
Note that for functions of the classHa, ω(Ω+) (respect. HA, ω(Ω±)) there hold standard Green’s formulas where the boundary limiting values on the boundarySare understood in the above described point-wise nontangential convergence sense.
First we set the direct fluid-structure interaction problem.
Let a total wave field in Ω− is represented as a sum of incident and scattered fields
wtot(x) =winc(x) +wsc(x),
where the incident fieldwinc is taken in the form of a plane wave
winc(x) =winc(x;d) = exp{i x · d}, x∈R3, d∈Sω, (2.13) while the scattered field (scattered acoustic pressure) wsc(x) = wsc(x;d) is a radiating solution of equation (2.5); here d = (d1, d2, d3) denotes the direction of propagation of the plane wave.
Problem P(dir). Find a vector u = (u1, u2, u3)> ∈ HA, ω(Ω+) and a radiating function wsc ∈ Ha, ω(Ω−)∩Som(Ω−), satisfying the following (kinematic and dynamic) coupling conditions in the sense of point-wise non- tangential convergence at almost every point onS:
[u(x)·n(x)]+=b1[Λ(∂, n)wtot(x)]−=b1[Λ(∂, n)wsc(x)]−+f0(x), (2.14) [T(∂, n)u(x)]+=b2[wtot(x)]−n(x) =b2[wsc(x)]−n(x) +f(x), (2.15) where T(∂, n)u is the stress vector given by formula (2.4), Λ(∂, n)w = apqnp∂qwis the co-normal derivative,n(x) denotes the unit outward normal vector toS at the pointx∈S,
b1= [%2ω2]−1, b2=−1. (2.16) Remark that all the arguments below are valid ifb1andb2are given complex constants satisfying the conditionsb1b26= 0 and=[b1b2] = 0.
Here the boundary scalar functionf0 and the vector-valued functionf are defined as follows:
f0(x) =f0(x;d) =b1Λ(∂, n)winc(x;d), (2.17) f(x) = (f1(x), f2(x), f3(x))>=f(x;d) =b2winc(x;d)n(x). (2.18) As it follows from the above statement, in the direct problem the domains Ω+and Ω−are fixed and we look for the displacement vectoruand the ra- diating scalar function wsc (scattered field). The inverse fluid-structure acoustic interaction problem consists in finding the surfaceS(i.e., the scat- terer Ω+) if the corresponding far-field patternw∞sc(·;d) is known for several or all direction vectorsd∈Sω. More rigorous mathematical formulation of the inverse problem considered in this paper reads as follows.
Problem P(inv). Find an elastic scatterer Ω+ with a compact, con- nected, Lipschitz boundary surfaceS provided that the conditions of Prob- lemP(dir)are satisfied onS and the far-field patternw∞sc(ξ; d) is a known
function ofξ onSω
w∞sc(ξ;d) =G(ξ;d)
for several (or all) direction vectorsd∈Sω; hereG(·;d) is a given function ofξ onSω andξ corresponds to the vectorη=x/|x|(see (2.7)).
In the both problems the oscillation parameterω is an arbitrarily fixed positive number. The investigation of the inverse problem becomes compli- cated due to the fact that, in general, the direct interaction problem for arbitrary scatterer Ω+ is not unconditionally solvable for allω. For excep- tional values of the parameterω, i.e., for those values of ω for which the corresponding homogeneous direct problem possesses nontrivial solutions, the boundary data f0 and f, involved in the equations (2.14) and (2.15), have to satisfy special compatibility (necessary) conditions. However, as we shall show below for functions given by (2.17) and (2.18) these necessary conditions are fulfilled automatically and ProblemP(dir)is always solvable.
Moreover, the scalar field wsc is defined uniquely in Ω− for all ω, while the elastic field u is defined modulo Jones modes, in general (see Section 3). This makes meaningful and justifies the above setting of the inverse problem with arbitraryω.
We shall study the above problems by the layer potentials (boundary integral equations) method. The properties of the corresponding potential type operators partly can be found in [43], [10], [11], [41], [31], [32], [33], but for the readers convenient we bring together needed material in the forthcoming subsection.
2.4. Scalar potentials. Steklov-Poincar´e type relations. Let us introduce the single and double layer scalar potentials related to the operatora(∂, ω):
Va, ω(g)(x) = Z
S
γ(x−y, ω)g(y)dSy, x∈R3\S,
Wa, ω(g)(x) = Z
S
[Λ(∂y, n(y))γ(y−x, ω)]g(y)dSy, x∈R3\S, whereg is a scalar density function.
For a solutionw∈Ha, ω(Ω+) of equation (2.5) in Ω+we have the follow- ing integral representation
Wa, ω [w]+
(x)−Va, ω [Λw]+ (x) =
( w(x) for x∈Ω+,
0 for x∈Ω−. (2.19) The similar representation holds also for a radiating solution w ∈ Ha, ω(Ω−)∩SK(Ω−) to the equation (2.5) in Ω−,
Va, ω [Λw]−
(x)−Wa, ω [w]− (x) =
( w(x) for x∈Ω−,
0 for x∈Ω+. (2.20)
These representations can be derived by standard arguments from the corre- sponding Green’s formulas which hold for functions of the classesHa, ω(Ω+) and Ha, ω(Ω−)∩Som(Ω−) (cf. Lemma 5.1 in [41], Proposition 6.6 in [1], and Proposition 5.5 in [2]).
In what follows we will essentially use the following properties of the layer potentials.
Lemma 2.3. Letg∈L2(S),h∈H21(S), and−12 ≤r≤12. Then (i)the potentials Va, ω(g), Wa, ω(g), and Wa, ω(h)are radiating solutions of equation(2.5)inR3\S and
M±(Wa,ω(g))∈L2(S), M±(∂jWa,ω(h))∈L2(S), j= 1,2,3, M±(Va,ω(g))∈L2(S), M±(∂jVa,ω(g))∈L2(S), j= 1,2,3, Wa,ω(g)∈H212(Ω+), Wa,ω(g)∈H2,loc12 (Ω−)∩Som(Ω−), Wa,ω(h)∈Ha,ω(Ω±)∩Som(Ω−), Va,ω(g)∈Ha,ω(Ω±)∩Som(Ω−);
(ii)the following jump relations hold onS for almost all z∈S [Va, ω(g)(z)]±=
Z
S
γ(z−y, ω)g(y)dSy=:Ha, ωg(z),
[Λ(∂, n)Va, ω(g)(z)]±=∓2−1g(z) + Z
S
[Λ(∂z, n(z))γ(z−y, ω)]g(y)dSy=:
=:h
∓2−1I+K(1)a, ω
ig(z), (2.21)
[Wa, ω(g)(z)]±=±2−1g(z) + Z
S
[Λ(∂y, n(y))γ(y−z, ω)]g(y)dSy=:
=:h
±2−1I+K(2)a, ω
ig(z), (2.22)
[Λ(∂, n)Wa, ω(h)(z)]+= [Λ(∂, n)Wa, ω(h)(z)]−=:La, ωh(z), (2.23) whereI stands for the identical operator;
(iii)the operators
Ha, ω : H2−12+r(S)→H212+r(S), Ka, ω(2) : H212+r(S)→H212+r(S), Ka, ω(1) : H−
1 2+r
2 (S)→H−
1 2+r 2 (S), La, ω : H212+r(S)→H2−12+r(S), are continuous;
(iv)the operators
Ha, ω : H2−12+r(S)→H212+r(S),
±2−1I+K(1)a, ω, ±2−1I+K(2)a, ω : L2(S)→L2(S), La, ω : H21(S)→L2(S), are bounded Fredholm operators with zero index;
(v) the following operator equations
Ha, ωK(1)a, ω=K(2)a, ωHa, ω, La, ωHa, ω=−4−1I+ K(1)a, ω2
, La, ωK(2)a, ω=K(1)a, ωLa, ω, Ha, ωLa, ω=−4−1I+
K(2)a, ω2
,
(2.24) hold in appropriate function spaces.
Proof. The proof of items (i)-(iv) (except the relations involving the operatorLa, ω) based on the harmonic analysis technique can be found in the reference [43] for ω = 0 (see also [10] where a variational approach is used and the analogous results are obtained for−12 < r <12 with the help of duality and interpolation arguments based on the Sobolev trace theorem).
The estimates
|γ(x, ω)−γ(x,0)|< ω C0(λ, µ),
|∂l[γ(x, ω)−γ(x,0) ]|< ω2C1(λ, µ),
∂l∂m[γ(x, ω)−γ(x,0) ] =O(|x|−1),
(2.25)
show that the potential and boundary operators corresponding to ω 6= 0 andω= 0 differ by smoothing (compact) operators. Therefore, the results obtained in [43] can be extended to the operators corresponding to arbitrary ω (for details see, e.g., [41]).
The properties of the normal derivative of the double layer potential Wa, ω and the operatorLa, ωare studied in detail in [41]. However, we give here a simpler proof of (2.23) which does not require invertibility of any boundary operator and can be extended to more general cases (e.g., to the case of elastic double layer vector potential). Since the double layer potential Wa, ω(h) withh∈H21(S) belongs to the classHa, ω(Ω±)∩Som(Ω−), we can write the integral representation formulas (2.19) and (2.20) with Wa, ω(h) forw. Add termwise these formulas and apply the jump relations (2.22) to obtain
Va, ω [ΛWa, ω(h)]+−[ΛWa, ω(h)]−
= 0 in R3\S.
Whence we arrive at (2.23) by jump relations (2.21).
To prove the item (v) let us remark that the representation formula (2.19) implies
−2−1I+K(2)a, ω
[w]+=Ha, ω[Λw]+, La, ω[w]+=
2−1I+K(1)a, ω
[Λw]+. (2.26)
The operator equalities (2.24) can be then obtained by substitution into (2.26) single and double layer potentials with densities from the spaces L2(S) andH21(S), respectively. It is evident that the first and second equa- tions in (2.24) originally hold in L2(S) but they can be continuously ex- tended to the spaceH2−1(S) due to the mapping properties of the operators involved; analogously, the third and fourth equations which originally hold inH21(S), can be continuously extended to the spaceL2(S).
To obtain a boundary integral formulation equivalent to the basic oscil- lation problems in exterior domains we need the following
Lemma 2.4. Letg∈H21(S)and
w(x) =Wa, ω(g)(x)−i Va, ω(g)(x), x∈Ω−. (2.27) If wvanishes in Ω−, theng= 0on S.
Proof. If the function (2.27) vanishes in Ω− due to the jump properties of the layer potentials we conclude
[Λw]+−i[w]+= 0 on S. (2.28) Since g ∈H21(S) we have w∈Ha, ω(Ω+) and there holds Green’s formula (cf. [41])
Z
Ω+
[akj∂jw ∂kw−%2ω2|w|2]dx= Z
S
[ Λw]+[w]+dS . (2.29) With the help of (2.6) and (2.28) we conclude from (2.29) that [w]+ = 0
which yields [w]+−[w]−=g= 0 onS.
Further, let us introduce the boundary operators Da, ωg :=
−2−1I+K(2)a, ω
−iHa, ω, (2.30) Na, ωg := La, ω −i
2−1I+K(1)a, ω
. (2.31)
These operators are generated by the limiting values onS (from Ω−) of the superposition of potentials (2.27) and its co-normal derivative.
Lemma 2.5. (i)The operators
Da, ω : L2(S)→L2(S), (2.32) : H21(S)→H21(S), (2.33) Na, ω : H21(S)→L2(S) (2.34) are isomorphisms.
(ii)The exterior Dirichlet BVP
a(∂, ω)w(x) = 0 in Ω−, w∈Som(Ω−), [w(z)]−=ϕ(z) on S, ϕ∈L2(S), M−(w)∈L2(S),
is uniquely solvable and the solution is representable in the form w(x) = (Wa, ω−i Va, ω) (Da, ω−1 ϕ)(x), x∈Ω−.
Moreover, w∈H2, loc12 (Ω−) and for arbitrary R there is a positive constant C0(R)independent ofw andϕsuch that
kwk
H
1 2
2(Ω−R)≤C0(R)kϕkL2(S)
withΩ−R:= Ω−∩BR, whereBR is the ball centered at the origin and radius R.
Ifϕ∈H21(S)thenM−(∂jw)∈L2(S) (j= 1,2,3),w∈H2, loc32 (Ω−), and kwk
H
3 2
2(Ω−R)≤C1(R)kϕkH12(S). (iii)The exterior Neumann BVP
a(∂, ω)w(x) = 0 in Ω−, w∈Som(Ω−), [Λ(∂, n)w(z)]−=ψ(z) on S, ψ∈L2(S), M−(∇w)∈L2(S),
is uniquely solvable and the solution is representable in the form w(x) = (Wa, ω−i Va, ω) (Na, ω−1ψ)(x), x∈Ω−.
Moreover, w∈H2, loc32 (Ω−) and for arbitrary R there is a positive constant C2(R)independent ofw andψsuch that
kwk
H
32
2(Ω−R)≤C2(R)kψkL2(S).
(iv)If two functionsg∈H21(S)andh∈L2(S)are related by the equation Na, ω−1 h=D−1a, ωgonS, thengandhare Cauchy data onSof some radiating solutionwof the homogeneous equation (2.5)inΩ−, namely,g= [w]− and h= [Λ(∂, n)w]− on S. Consequently, the Dirichlet and Neumann data for an arbitrary radiating solution w of the equation (2.5)are related on S by the following generalized Steklov-Poincar´e type relation
Na, ω−1[Λ(∂, n)w]−=Da, ω−1 [w]−.
Proof. First we show the invertibility of the operator (2.32). Due to the results in [43] the operatorDa(0):=−2−1I+K(2)a,0 : L2(S)→L2(S) is Fredholm with index zero. In accordance with (2.10) and (2.25) the operator Da, ω− Da(0) : L2(S) → L2(S) is compact. Therefore it remains only to prove the injectivity of (2.32). To this end we show that the null space of the corresponding adjoint operator (without complex conjugation) is trivial.
Letψ∈L2(S) be a solution to the homogeneous adjoint equation −2−1I+Ka, ω(1)
ψ−iHa, ωψ= 0 on S.
It then follows that the single layer potentialVa, ω(ψ)∈Ha, ω(Ω+) solves the homogeneous Robin type problem with boundary condition (2.28). There- fore, Va, ω(ψ) = 0 in Ω+. This implies [Va, ω(ψ) ]− = 0 on S. Thus Va, ω(ψ) solves the homogeneous exterior Dirichlet BVP. Write Green’s for- mula (2.29) for the functionw=Va, ω(ψ) and for the region Ω−R= Ω−∩BR
whereBRis the ball centered at the origin and radiusR, and∂BR= ΣR, Z
Ω−R
[akj∂jVa, ω(ψ)∂kVa, ω(ψ)−%2ω2|Va, ω(ψ)|2]dx=
= Z
ΣR
[ ΛVa, ω(ψ) ] [Va, ω(ψ) ]dS.
Evidently
=n Z
ΣR
[ Λ(∂, n)Va, ω(ψ) ] [Va, ω(ψ) ]dΣR
o= 0,
and in accordance with Lemma 2.1 we get Va, ω(ψ) = 0 in Ω−. Therefore [ Λ(∂, n)Va, ω(ψ) ]−−[ Λ(∂, n)Va, ω(ψ) ]+=ψ= 0 from which the injectivity and, consequently, the invertibility of the operator (2.32) follows.
As it is shown in the references [43] and [41] the operator Da(0) = : H21(S) → H21(S) is Fredholm with index zero as well. Therefore the in- vertibility of the operator (2.33) follows from its injectivity.
Analogously it can be shown that the operator (2.34), as a compact perturbation of an invertible operator, is Fredholm with zero index (cf.
[41]). On the other hand with the help of the same arguments as above we easily derive that the null space of the operatorNa, ω is trivial. Therefore (2.34) is an isomorphism.
Proof of the items (ii) and (iii) are quite similar to the proofs of Theorem 6.2 and Proposition 6.8 in [1] (see also the proof of Lemma 4.1 in [16] and Theorem 5.6 in [41]).
The item (iv) immediately follows from the item (i) and the uniqueness theorems for the exterior Dirichlet and Neumann boundary value problems.
2.5. Special Robin type problem. Properties of plane waves.
Let us consider the interior Robin type BVP
a(∂, ω)w(x) = 0 in Ω+, w∈Ha, ω(Ω+), (2.35) [Λ(∂, n)w(z)−i w(z)]− =ψ on S, ψ∈L2(S). (2.36) If we look for a solution in the form of a single layer potential w(x) = Va, ω(g)(x), we arrive at the integral equation onS
Pa, ωg:=h
−2−1I+K(1)a, ω−iHa, ωi g=ψ,
where Pa, ω : L2(S)→L2(S) is a Fredholm operator with zero index due to Lemma 2.2.(iv).
Lemma 2.6. (i)The BVP (2.35)-(2.36)is uniquely solvable.
(ii)The operatorPa, ω : L2(S)→L2(S)is invertible.
(iii)An arbitrary solutionw∈Ha, ω(Ω+)of the equation(2.35)is uniquely representable in the form
w(x) =Va, ω Pa, ω−1 [Λ(∂, n)w−i w]+
(x), x∈Ω+. Moreover, for Ω+0 ∈Ω+ there holds the uniform estimate
|w(x)| ≤C δ−1||[Λ(∂, n)w−iw]+||L2(S) for all x∈Ω+0,
whereCis a positive constant independent ofwandδ. Hereδis the distance between Ω+0 andS.
Proof. The items (i) and (ii) have been shown as intermediate steps in the proof of Lemma 2.5. The item (iii) is then a direct consequence of the
invertibility of the operatorPa, ω.
From Lemma 2.6 it follows that the plane wave exp{i d·x}, whered∈Sω, can be uniquely represented in the form
ei d·x=Va, ω
Pa, ω−1
(Λ(∂, n)−i)ei d·ζ+
S
(x), x∈Ω+.
Note, that exp{i d·x}withd∈Sω is a non-radiating solution to the homo- geneous equation (2.5) inR3. Let
P(S) :=
p(x;d)≡(Λ(∂x, n(x))−i)ei d·x, x∈S : d∈Sω , Psp(S) :=nXm
q=1
cqp(x;d(q)), x∈S : p(x;d(q))∈P(S), cq ∈C, d(q)∈Sω, m∈No
, Psp(R3) :=nXm
q=1
cqei d(q)·x, x∈R3 : cq ∈C, d(q)∈Sω, m∈No
; hereNandCare the sets of all natural and complex numbers, respectively.
Lemma 2.7. The set P(S)is complete inL2(S).
Proof. Letf ∈L2(S) and Z
S
(Λ(∂y, n(y))−i)ei d·y
f(y)dSy= 0 (2.37) for alld∈Sω.
Let us consider the function
w(x) = (Wa, ω−i Va, ω)(f)(x), x∈R3\S.
Clearly, in view of (2.11) we have w(x) =c(ξ)exp{i ξ·x}
|x| Z
S
(Λ(∂y, n(y))−i)e−i ξ·y
f(y)dSy+O(|x|−2)
as|x| →+∞, whereξ ∈Sωcorresponds to ˆx andc(ξ) is defined by (2.11).
By (2.37) we concludew(x) =O(|x|−2), which impliesw(x) = 0 in Ω− due to Lemma 2.1. Therefore, we obtain [w(x)]−=Da, ωf = 0 on S.By Lemma 2.5.(i) then we havef = 0 onS. This completes the proof.
Lemma 2.8. Let Ω+ be a bounded Lipschitz domain such that Ω− = R3\Ω+be connected and letw∈Ha, ω(Ω+)be a solution to the homogeneous equation (2.35)inΩ+.
Then there exists a sequence vm ∈ Psp(R3) such that vm → w and
∂βvm → ∂βw as m → ∞ uniformly on compact subsets of Ω+ (β = (β1, β2, β3)is an arbitrary multi-index and∂β=∂1β1∂2β2∂3β3).
Proof. From Lemma 2.7 it follows that there exists inPsp(S) a sequence of type
Xm q=1
cq(Λ(∂x, n(x))−i)ei d(q)·x, x∈S,
which converges (in theL2-sense) to the function [ (Λ(∂, n)−i)w]+∈L2(S).
We set
vm(x) = Xm q=1
cqei d(q)·x, x∈Ω+.
Hence, (Λ(∂, n)−i)vm(x) →[(Λ(∂, n)−i)w(x)]+ in L2(S). By Lemma 2.6 the functionsvm andwcan be represented in the form
vm(x) =Va, ω Pa, ω−1[ (Λ(∂, n)−i)vm]+
(x), x∈Ω+, w(x) =Va, ω Pa, ω−1[ (Λ(∂, n)−i)w]+
(x), x∈Ω+.
Now, let Ω+0 ⊂Ω+ andx∈Ω+0. Denote byδ the distance between Ω+0 and S =∂Ω+. The above representations of vm and w together with Lemma 2.6 then imply
|∂βw(x)−∂βvm(x)| ≤
≤C δ−1|| Pa, ω−1[(Λ(∂, n)−i)vm]+−Pa, ω−1[ (Λ(∂, n)−i)w]+||L2(S)≤
≤C1δ−1||[ (Λ(∂, n)−i)vm]+−[ (Λ(∂, n)−i)w]+||L2(S)→0 asm→+∞(uniformly in Ω+0) for arbitrary multi-indexβ.
Corollary 2.9. Letx06∈Ω+. Then there exists a sequencevm∈Psp(R3) such that(for arbitrary multi-indexβ)
∂βvm(x)→∂βγ(x−x0, ω) uniformly inΩ+, i.e.,
||vm(x)−γ(x−x0, ω)||Ck(Ω+)→0 as m→ ∞ for arbitrary integer k≥0.
2.6. Vector-valued potential operators of the theory of steady state elastic oscillations. Denote by Γ(x, ω) the fundamental matrix (Kupradze matrix) of the steady state elastic oscillation operatorA(∂, ω), i.e.,A(∂, ω) Γ(x, ω) =I3δ(x) (for details see [28], Ch. II),
Γ(x, ω) = [ Γkj(x, ω) ]3×3, Γkj(x, ω) =
X2 l=1
(δkjαl+βl∂k∂j)exp{i kl|x|}
|x| , (2.38)
where
k12=%1ω2(λ+ 2µ)−1, k22=%1ω2µ−1,
αl=−δ2l(4π µ)−1, βl= (−1)l+1(4π %1ω2)−1.
Note that the principal singular part of Γ(x, ω) in a vicinity of the origin is the fundamental matrix Γ(x) (Kelvin’s matrix) of the operator A(∂) of elastostatics,
Γ(x) = [ Γkj(x) ]3×3, Γkj(x) =λ0δkj|x|−1+µ0xkxj|x|−3, (2.39) where
λ0=−(λ+ 3µ) [8π µ(λ+ 2µ)]−1, µ0=−(λ+µ) [8π µ(λ+ 2µ)]−1. It is easy to show that in a vicinity of the origin (|x|<1 say)
|Γkj(x, ω)−Γkj(x)|< ω C0(λ, µ),
|∂l[ Γkj(x, ω)−Γkj(x) ]|< ω2C1(λ, µ),
∂l∂m[ Γkj(x, ω)−Γkj(x) ] =O(|x|−1),
(2.40)
where k, j, l, m = 1,2,3, and C0(λ, µ) and C1(λ, µ) are positive constants depending only on the Lam´e parameters.
A vectoru belongs to the class SK(Ω−) if u∈[C1(Ω−)]3 and for suffi- ciently large|x| the following relations hold:
u(x) =u(p)(x) +u(s)(x),
∆u(p)(x) +k21u(p)(x) = 0, ∂ju(p)(x)−iˆxjk1u(p)(x) =O(|x|−2),
∆u(s)(x) +k22u(s)(x) = 0, ∂ju(s)(x)−ixˆjk2u(s)(x) =O(|x|−2), where ˆx = x/|x| and j = 1,2,3. These conditions are the Sommerfeld- Kupradze type radiation conditions in the elasticity theory (for details see [28]).
From (2.38) it follows that each column of the matrix Γ(·, ω) belongs to SK(R3\ {0}).
The analogue of Rellich’s lemma in the elasticity theory reads as follows (for details see [28], [34]).
Lemma 2.10. Letu∈SK(Ω−)be a solution of (2)inΩ− and let
R→+∞lim =n Z
ΣR
T(∂, n)u· u dΣR
o= 0,
whereΣR is the same as in Lemma2.1.Then u= 0in Ω−.
This lemma implies that the basic exterior homogeneous BVPs of steady state elastic oscillations (with given zero displacements or stresses on the boundary) have only the trivial solution (see [28]).
Further, we construct the single and double layer vector potentials, VA, ω(g)(x) =
Z
S
Γ(x−y, ω)g(y)dSy,
WA, ω(g)(x) = Z
S
[T(∂y, n(y)) Γ(y−x, ω) ]>g(y)dSy.
For a solutionu∈HA, ω(Ω+) of equation (2) in Ω+we have the following integral representation
WA, ω [u]+
(x)−VA, ω [T u]+ (x) =
( u(x) for x∈Ω+,
0 for x∈Ω−. (2.41) The similar representation holds also for a radiating solution u ∈ HA, ω(Ω−)∩SK(Ω−) to the equation (2) in Ω−,
VA, ω [T u]−
(x)−WA, ω [u]− (x) =
( u(x) for x∈Ω−, 0 for x∈Ω+. These representations follow from the corresponding Green’s formulas which hold for functions of the classesHA, ω(Ω+) andHA, ω(Ω−)∩SK(Ω−) (cf. Ch.
VII in [28], Proposition 6.6 in [1], and Proposition 5.5 in [2]).
Further, we introduce the boundary operators on S generated by the above vector potentials
HA, ωg(z) :=
Z
S
Γ(z−y, ω)g(y)dSy, (2.42)
K(1)A, ωg(z) :=
Z
S
[T(∂z, n(z)) Γ(z−y, ω) ]g(y)dSy, (2.43)
K(2)A, ωg(z) :=
Z
S
[T(∂y, n(y)) Γ(y−z, ω) ]>g(y)dSy,
LA, ωg(z) := [T(∂z, n(z))WA, ω(g)(z) ]±.
Note that the potential and boundary operatorsVA, ω,WA, ω,HA, ω,K(1)A, ω, KA, ω(2) , andLA, ω have quite the same jump and mapping properties as the corresponding scalar operators considered in Subsection 2.4 (see Lemma 2.3).
