## Abstract

Applying the spectral function techniques, developed by Croisille and Lebeau, we prove the existence of solutions to problems of plane and cylindrical waves diffraction by an elastic wedge with stress-free boundary conditions. We also formulate radiation conditions, under which the uniqueness holds. The latter implies absolute continuity of the spectrum of the Lamé operator in a wedge domain with stress-free boundary.

## 1. Introduction

The problem of diffraction by an elastic wedge has a significant history. A detailed review of the works in this direction is given by Croisille & Lebeau (1999) and Kamotski *et al*. (submitted); thus we will restrict ourselves to brief comments only.

The problem under consideration seems very unlikely to have a solution in a closed form, and up to the mid-1990s most of the authors were interested in the computational aspects of the problem. Several numerical approaches were developed (see Fujii 1994; Budaev & Bogy 1996; Gautesen 2002; Kamotski *et al*. submitted and references therein). The question of well posedness of the problem was never addressed, until the spectral functions techniques were introduced in Lebeau (1995) and Croisille & Lebeau (1999). Investigating a different problem (an immersed elastic wedge) Croisille & Lebeau (1999) proved existence theorem for the plane wave diffraction problem.

The aim of this paper is twofold: in the first part we apply the spectral function techniques to prove the existence of solutions to problems of plane and cylindrical waves diffraction by an elastic wedge with stress-free boundary. Limits of applicability of an inherent to the approach uniqueness result motivated the second part of the work: we formulate radiation conditions under which we prove a stronger uniqueness theorem.

### (a) Notation

We will adopt the notation of Croisille & Lebeau (1999). Suppose that a wedge of angle *φ*∈(0,2*π*),(1.1)is occupied by an elastic medium. We introduce two Cartesian coordinate systems (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}) associated with wedge faces (see figure 1), and associated with these systems polar angles *θ*_{1} and *θ*_{2}. We have(1.2)(1.3)In the sequel we follow the convention that we omit index of a coordinate if and only if it is equal to 1.

We consider linearized equations of isotropic elasticity with suppressed time-harmonic factor: the displacement field *u*=(*u*^{1},*u*^{2})^{t} satisfies dynamic Lamé system in the wedge,(1.4)and stress-free boundary conditions on its faces(1.5)Here , are Lamé constants, *ρ* is the density of the medium, *ω* is the frequency of the process, *ϵ* is the deformation tensor given in arbitrary Cartesian coordinates (*z*_{1}, *z*_{2}) by and *n* is the inward normal.

As usual we suppose that the total field *u* is the sum of an incident and scattered parts(1.6)*u*^{inc} is an incident longitudinal or transversal plane wave.

We change variables to dimensionless: we introduce function *v* defined by(1.7)where is the longitudinal velocity. Then equations (1.4)–(1.6) are equivalent to the system(1.8)The cases where *=L, T correspond to problems of longitudinal and transversal wave diffraction. Here the dimensionless elasticity operator is and the normal stress operator is(1.9)The dimensionless incident wave is given by(1.10)for the case of longitudinal wave incidence, and by(1.11)for the transversal wave. We have the following relations of dimensionless values with physical ones(1.12)(1.13)where *c*_{R} is the Rayleigh wave velocity. Since , we have , *λ*+2*μ*=1.

### (b) Strategy and structure of the paper

Usually when dealing with a diffraction problem, in order to obtain well posedness, one requires not only fulfilment of the equations of motion, but assumes some radiation conditions to be satisfied by an unknown function as well. These conditions are well known for scalar problems of diffraction both for bounded and wedge-like scatterers (e.g. Courant & Hilbert 1962). For systems admitting surface waves in domains with angular outlets at infinity these conditions are not so well studied.

In the cases of scatterers with piece-wise smooth boundary one also has to add physically consistent local regularity conditions (energy finiteness) in vicinities of angular or conical points. For the two-dimensional Lamé system with natural boundary conditions local finiteness of energy is equivalent to regularity of the displacement field (e.g. Kozlov *et al*. 2001).

Thus, following the strategy of Croisille & Lebeau (1999) to select a proper solution to (*S*^{*}) we suppose that it satisfies a version of the limiting absorption principle. We will consider the problem (*S*^{*}) as a special case (*ϵ*=0) of the problem(1.14)Investigating this problem for *ϵ*∈(0,*π*) we will see that it possesses a unique solution, which can be represented as a sum of two ‘single layer’ potentials, corresponding to two faces(1.15)here functions are defined in the entire by(1.16)where *δ*_{j} is an integration measure along the face *Γ*_{j}, given by its action on arbitrary test function by(1.17)Densities *α*_{j}(*x*_{j}) and *β*_{j}(*x*_{j}) in equation (1.16) are supposed to belong to a special class of distributions supported on the positive semi-axis, defined as follows.

*We say that f*∈ *if* *and*(1.18)(1.19)

Here we used the notation for the Fourier transform of the function *f*:(1.20)Combining results of Lebeau (1995, §2.3, proposition 3), Croisille & Lebeau (1999, lemma 2.2) and general theory of elliptic problems in angular domains (e.g. Kondrat'ev 1963; Nazarov & Plamenevski 1994) we obtain the following regularity description for potentials .

*Assume α*_{j}*, β*_{j}∈, *j*=1, 2. *Then the two tempered distributions* *defined by equation (**1.16**) converge in* *when ϵ*→0 *to two tempered distributions* *which solve*(1.21)*Moreover, the following regularity holds for any ϵ*∈[0,*π*):

*are continuous in*;*in polar coordinates*(*θ*_{j},*r*)*we have*

(1.22)

This lemma justifies the following.

*We call v an outgoing solution of* (*S*^{*}) *if v is a solution of the form*(1.23)*with*(1.24)*where α _{j}*,

*β*∈,

_{j}*j*=1, 2.

This concept of an outgoing solution appears to be extremely suitable for diffraction problems by the wedge-like scatterers. It fits perfectly to resolve difficulties both at infinity and in the wedge tip: as we demonstrate below it selects a unique solution, which locally (in vicinity of the wedge tip) belongs to the energy space (lemma 1.2) and ‘at infinity’ satisfies a sort of a limiting absorption principle.

In §2 we derive an integral representation for *v*^{ϵ} in terms of Fourier transforms of *α*_{j}, *β*_{j}. The pair of these Fourier transforms will be called in the sequel the spectral function. Next we substitute these representations into boundary conditions and reduce the problem (*S*^{*}) to an integral equation for the spectral function. Further, using coercivity of the form associated with *E*+e^{−2iϵ} and regularity results for elliptic problems in domains with angular points, we establish unique solvability of this integral equation in class , the class of Fourier transforms of functions from . We conclude §2 with a calculation of the far-field asymptotics of diffracted field in terms of the spectral function.

The price to pay for the perfect match of the above outgoing solutions in the wedge problem is the ‘instability’ of this concept with respect to perturbations of the problem: for example one will have to face significantly more elaborate problems trying to adjust this idea even for analytical perturbations of the boundary or the operator. Another disadvantage is inapplicability of the approach to the direct analysis of eigenfunctions existence. If there were an eigenfunction it just would not be an outgoing solution in the above sense: obviously a non-trivial eigenfunction would not possess assumed analytical continuation properties.

The rest of the paper is devoted to construction of both physically and mathematically more transparent conditions at infinity. In §3 using the same approach we first establish existence of Green's tensor for the elastic wedge and then calculate its far-field asymptotics. Its properties play a crucial role in §4 for deriving the radiation conditions (4.12) and (4.13). We suggest a form of radiation conditions that allows us to modify them for formulation and proof of uniqueness results in more general problems involving asymptotically angular outlets at infinity and surface waves. We finish the paper by the *L*^{2} uniqueness result and consequently obtaining the absence of the discrete spectrum for the Lamé system in a wedge with stress-free boundary.

## 2. Plane wave diffraction

In this section, we briefly introduce the spectral function techniques in application to the problem of plane longitudinal (or transversal equally) wave diffraction by an elastic wedge (a more detailed exposition of the techniques is given in Croisille & Lebeau 1999). The most significant, but not crucial difference between the following and the treatment of Croisille & Lebeau (1999) is not that we consider different problems, but that we do not impose the restriction *φ*<*π*. We show that if hypothesis that neither incident, nor any of multiply reflected waves (which may arise in the case when *φ*<*π*) is an incoming grazing one, is satisfied, than there exists a unique outgoing solution to the problem (*S*^{*}).

### (a) Integral system for spectral function

Rewriting equation (1.16) in terms of Fourier transforms of the right-hand side and slightly deforming contour of integration (just as the class permits to) we obtain an expression for valid for *ϵ*∈[0,*π*) (cf. Croisille & Lebeau 1999, equation (2.57))(2.1)here *Γ*_{0} is the contour depicted on figure 2, is the branch of continuous on the contour *Γ*_{0} and possessing non-negative imaginary part on the real axis. Matrices are given by(2.2)(2.3)and *Σ*_{j} is the Fourier transform of (*α*_{j}, *β*_{j})(2.4)The pair (*Σ*_{1}, *Σ*_{2}) is called spectral function in sequel. Substitution of with represented by equation (2.1) into the normal stress boundary operator *B* given in equation (1.9), results in the following (cf. Croisille & Lebeau 1999, lemma 3.1).

*For* , Im *ξ*<0 *the Fourier transform of B*(*v*^{ϵ}) *on the face Γ*_{1} *is given by*(2.5)*where DM*^{ϵ} *and TM*^{ϵ} *are the integral operators defined for all* *by*(2.6)(2.7)*The kernels are given by*(2.8)*with*(2.9)*where denoting* , *=L, T *we have*(2.10)(2.11)(2.12)*and*(2.13)*with*(2.14)*and the rank 1 matrices tm*

_{L}

*and*

**tm**_{T}

*are given by*(2.15)(2.16)

The proof of the lemma is analogous to Croisille & Lebeau (1999). Now we are ready to formulate the problem for the spectral function to satisfy.

*For ϵ*∈[0,*π*) *the function v*^{ϵ} *is a solution to the problem* *if and only if the spectral function Σ*=(*Σ*_{1}, *Σ*_{2}) *is a solution to the system*(2.17)*where*(2.18)(2.19)(2.20)(2.21)

Indeed, the right-hand side in (see equation (1.14)) is given by(2.22)(2.23)for the case of longitudinal wave incidence, i.e. *=L. Taking the Fourier transform of equations (2.22) and (2.23) we deduce from equation (2.5) identities (2.17)–(2.19). Calculations for the transversal wave follow the same scheme. ▪

### (b) Properties of integral and translation operators

Before proceeding with investigation of solvability of equation (2.17), we formulate some useful concepts and results, which can be deduced following the strain of thought of Croisille & Lebeau (1999):

*Properties of the matrix dm*

**dm**is analytical in*, bounded on each line z*=*ξ*e^{iϵ}*, with*,*ϵ*∈(0,*π*)*and non-singular for z*≠±*ν*_{R}*, where*,*ν*_{R}>*ν*_{T}*is the only positive root of the Rayleigh equation*det(*dm**ξ*)=0,*dm*^{−1}(*ξ*)*is bounded on V outside a neighbourhood of*±*ν*_{R}.

*H*^{+} *is a space of all analytic in* *functions h, such that*(2.24)The space *H*^{+} is in fact the Fourier transform of . The left-hand side in equation (2.24) defines a norm in *H*^{+}, such that the Fourier transform is an isometry of onto *H*^{+} up to a factor of 2π.

*For all ϵ*∈(0,*π*) *operator DM*^{ϵ} *is bounded as an operator in* (*H*^{+})^{2}.

*For all ϵ*∈(0,*π*) *operator TM*^{ϵ} *is bounded in* (*H*^{+})^{2}. *Moreover there exist c*_{0}(*ϵ*), *c*_{1}(*ϵ*), *c*_{2}(*ϵ*)>0 *such that for all f*∈(*H*^{+})^{2} *its image TM*^{ϵ}*f possesses an analytical continuation into the domain* *and for all α*∈(−*c*_{2}, *c*_{2}) .

Following Croisille & Lebeau (1999) we define translation operators(2.25)and the domains(2.26)In construction of the solution to the plane wave diffraction problem we will also need the sets(2.27)defined for arbitrary following the recursive procedure: subsets —the ‘generation *l*’—are given by(2.28)and initial sets are(2.29)The sets we are interested in are generated by *ξ*_{1}=cos *θ*_{inc} and *ξ*_{2}=cos(*φ*−*θ*_{inc}). Those are the sets of poles of spectral function, corresponding to (multiply) reflected bulk waves. It may be shown (cf. Croisille & Lebeau 1999, lemma 2.4), that these sets are finite. This fact illustrates a ‘physically obvious’ phenomena: any incident on a wedge ray after several reflections becomes an outgoing one, i.e. it does not hit the wedge faces anymore. (Here ‘reflection’ means generation of both longitudinal and transversal transmitted rays with non complex directions.)

The generic hypothesis we inherit from Croisille & Lebeau (1999) under which we will prove existence of the solution to the plane wave diffraction problem is(2.30)It means, that neither incident, nor any of the reflected waves is an incoming grazing one.

### (c) Isomorphism theorem

Let us denote the operator corresponding to the entire problem(2.31)The next result is the central one in the spectral function approach, it is obtained following the procedure rigorously presented by Croisille & Lebeau (1999).

*For all ϵ*∈(0,*π*) *operator* ^{ϵ} *is an isomorphism of the space* (*H*^{+})^{2}⊕(*H*^{+})^{2} *onto itself.*

Before proceeding to the proof let us introduce some new notation and recall auxiliary results: first we define a space *E*(*Ω*) of distributions in the wedge , *φ*∈(0,2*π*).

*f*∈*E*(*Ω*) *if it satisfies*(2.32)*where I*=(0, *φ*).

We will also need the following lemma (Croisille & Lebeau 1999, lemma 4.4).

*Suppose ϵ*∈(0,*π*). *The Neumann elasticity problem*(2.33)*and the Dirichlet elasticity problem*(2.34)*do have unique solutions* .

The core of this result is of course the regularity statement.1 In comparison with more obvious *H*^{1} existence and uniqueness, this lemma provides us with possibility to prove the surjectivity part of the isomorphism theorem.

Let us check injectivity first. Suppose and comprise an (*H*^{+})^{2}⊕(*H*^{+})^{2} solution to homogeneous equation(2.35)We define(2.36)(2.37)where are the inverse Fourier transforms of , and the integration measures *δ*_{j} are given by equation (1.17). Next we consider a tempered distribution —the solution to(2.38)Note that obviously for all *γ*>0 and moreover *u*∈*E*(*Ω*) (Lebeau 1995, §2.3, proposition 4). According to lemma 2.2 we see that equation (2.35) implies that *v*=*u*|_{Ω} is a solution to the homogeneous Neumann problem in the wedge(2.39)Thus lemma 2.9 allows us to conclude, that *v*=0. Next, due to continuity of *u* and uniqueness for the Dirichlet problem we have , and therefore the normal traction jumps over *Γ* are zeros as well, so *α*_{j}=*β*_{j}=0, *j*=1, 2.

Let us address now surjectivity of ^{ϵ}. For any we have representations(2.40)with , *j*=1, 2. It is enough to demonstrate, that there exists a single layer potential, which has normal traction traces on *Γ*_{j} from inside of the wedge equal to (*α*_{j}, *β*_{j})^{t}. To this end we perform in some sense the above argument in the reverse order: consider the Neumann problem inside the wedge(2.41)It has a solution according to lemma 2.9. Next we consider the Dirichlet problem outside the wedge(2.42)and construct a continuous function in entire space (2.43)To finish the proof it is enough to notice that the above function is a single layer potential with an *L*^{2} density: indeed, this density on the face *Γ*_{j} is equal to normal traction jump . We have(2.44)since owing to lemma 2.9 *v*_{+}|_{Γ}∈*H*^{1}(*Γ*), and as *v*_{−} is a solution to equation (2.42), we therefore deduce *Bv*_{−}|_{Γ}∈*L*^{2}(*Γ*). ▪

Recalling that is the Fourier transform of class (see definition 1.1) we move on to an important corollary.

*Suppose* *is an analytic function in* *with some δ*>0. *Suppose as well, that for all ϵ*∈(0,*δ*) *we have* . *Then there exists a unique* *solution to the system*(2.45)

First note that according to the isomorphism theorem for all *ϵ*∈(0,*δ*) there exists a solution to the system(2.46)Further, it is enough to demonstrate, that is indeed a proper solution to equation (2.45). Simple change of variables in the definition of the operators *DM* and *TM* shows us that the definition of *X* is correct for all *ϵ*∈(0, *δ*). Stated in the theorem growth properties of *X* can be deduced from the properties of the integral operators *TM* and *DM*.

The uniqueness part bases on possibility of analytical continuation into the domain , *γ*>0 small, of any solution to the problem(2.47)Indeed, we may rewrite this equation as follows(2.48)here *g*_{j}=*TMΣ*_{j}, *j*=1, 2 possess better regularity properties then *Σ*_{j}. Assuming we have for small *ϵ*_{0}>0 and *γ*∈(0, *ϵ*_{0}). Next due to ‘almost invertibility’ of *DM* (in the sense that an solution *f* to *DM* *f*=*g* possesses no worse regularity then *g*) we deduce that . The latter implies in its turn that . A finite iteration of this argument results in (compare with a detailed argument of Croisille & Lebeau 1999, Section 4.2). ▪

### (d) Existence

Recall, that the plane wave diffraction problem is equivalent to the integral system(2.49)where *z*_{1}=*ν*_{*} cos *θ*_{inc}, *z*_{2}=*ν*_{*} cos(*φ*−*θ*_{inc}) and we use notation *DM*=*DM*^{0}, *TM*=*TM*^{0}. We will need the following.

*is a space of analytical in* *functions h, such that for all ϵ*∈(0,*π*) *h*(e^{iϵ}.)∈*H*^{+}.

First we introduce the spectral function decomposition. The following result can be obtained adapting the arguments of Croisille & Lebeau (1999).

*There exist two functions y*_{1}, *y*_{2} *being the finite sums of simple poles*(2.50)*(where* ^{j}=*Z*^{j}(*z*_{1}, *z*_{2})*) such that*(2.51)

This lemma and the isomorphism theorem allow us to deduce existence of an outgoing solution to the problem (*S*^{*}) given in equation (1.8). Next, deforming the contour of integration *Γ*_{0} in the system (2.49) into a contour running form +∞−i0, turning round *ν*_{L} and ending at +∞ one obtains a ‘functional equation’. This equation links linearly values of one of *Σ*_{j} in *ξ* with values of the other component of spectral function in , *=L, T. This provides a possibility to investigate regularity of the boundary values of the spectral function on (cf. Croisille & Lebeau 1999, §§3.5 and 4.3). Resuming the above arguments we obtain the following result.

*Under the hypothesis* (*H*) *(see equation* *(2.30)**) there exists a unique outgoing solution to the problem* (*S*^{*}) *(see equation* *(1.8)**). Corresponding to the solution spectral function is of the form*(2.52)*with X*_{j}∈*, such that its boundary values X*_{j}(*ξ*−i0), *are analytical for* *, where* −*ν*_{R} *is the simple pole, corresponding to the Rayleigh wave, and* *is the set of branching points, corresponding to the head wave and its reflections, given by*(2.53)*Any ξ*_{0}∈*C*^{j} *is a square-root type branching point of the spectral function.*

### (e) Far-field asymptotics of diffracted field

To calculate asymptotics of *v*=*v*_{1}+*v*_{2}, where *v*_{j} are defined by equation (2.1) with *ϵ*=0, we change variable of integration *ξ*=*ν*_{*} cos *β*, to obtain representation(2.54)where *t*=sgn sin *θ* and (orthogonal for real *β*) projectors *P*_{L} and *P*_{T} are(2.55)(2.56)Contour *C*_{0} is presented on figure 3. For *θ*<*π*, i.e. *t*=1 we have(2.57)here we assume *P*_{*}(*β*)=*P*_{*}(*β*, 1). In the case *θ*>*π* we have(2.58)we have used here identity *P*_{*}(*β*, −1)=*P*_{*}(−*β*). Now we note that *Σ*_{1}(*ν*_{*} cos *β*) is an even function and change variable *β*=−*β*′ to obtain the following representation:(2.59)the contour *C*_{1} is pictured on figure 4.

To calculate asymptotics of integrals in equations (2.57) and (2.59) we deform contours *C*_{0} and *C*_{1} into stationary ones—contours *γ*_{θ} are pictured for both cases on figures 3 and 4.

While deforming we might cross some singularities of the spectral function. According to theorem 2.13 those can only be simple poles (giving rise to plane bulk waves and surface Rayleigh waves) and branching points (corresponding to head waves, generated by interaction of the cylindrical longitudinal wave with the boundary). For example the pole −*ν*_{R} of *Σ*_{1} produces a Rayleigh wave travelling along the face *Γ*_{1}:(2.60)We address structure of head waves and their contribution in §3*c*.

The most important for applications are diffracted by the vertex cylindrical waves, which are given by the contribution of the stationary point *β*=*π*−*θ* in equation (2.59). So we find(2.61)In the same way we obtain contribution of *Σ*_{2} in diffracted field, we have(2.62)recall, that *θ*_{2}=*φ*−*θ*. The total cylindrical wave, scattered by the tip one finds by taking the sum of expressions(2.63)Now we change variables to polar ones in these expressions to obtain separately main terms of transversal and longitudinal cylindrical waves diffracted by the tip. We will use the following notation(2.64)where **e**_{r}, **e**_{θ} are unit coordinate vector fields, and (*v*^{(d),r}, *v*^{(d),θ}) are coordinates of displacement in this basis. In this notation we find, that the longitudinal wave scattered by the tip is given by(2.65)and the transversal is(2.66)

## 3. Green's tensor

In this section we deduce appropriate representation of the free space Green's tensor, then calculate its stress on the boundary of the wedge. Next we define the notion of Green's tensor in the wedge and calculate the right-hand sides of an integral equation for the corresponding spectral function to satisfy. After the analysis of the right-hand sides' regularity we establish existence of Green's tensor for the wedge. Finally we calculate the Green's tensor far-field asymptotics.

### (a) Free-space Green's tensor

The time-harmonic elastodynamics free-space Green's tensor _{f}(*x*,*y*,*x*^{0},*y*^{0}) is tensor whose *km* component represents the *k*th component of the displacement caused by a unit line force acting at (*x*^{0},*y*^{0}) along the *m*-axis.2 Being represented in the matrix form, it satisfies the following equation(3.1)where is the unit matrix.

*The free space Green's tensor admits the following representation:*(3.2)*where matrices M*_{*} *are defined by equations* *(2.2) and (2.3)* *with ϵ*=0.

This lemma can be obtained in the same way, as one does the computations for representation (2.1) of single layer potential via spectral function. ▪

One can derive exact formulae for _{f} (e.g. Kupradze 1965), but representation (3.2) is more convenient for the sequel.

### (b) Existence of Green's tensor for an elastic wedge

*Green's tensor for an elastic wedge is the sum* =_{f}+_{0}*, where* _{0} *is an outgoing solution to the problem*(3.3)

*For arbitrary* (*x*^{0},*y*^{0})∈*Ω there exists a unique outgoing solution to equation* *(3.3)**, which can be represented, as in the case of plane wave diffraction, as a sum of two terms, corresponding to faces Γ*_{j}:(3.4)*where*(3.5)

To prove this theorem it is enough to demonstrate that there exists a spectral function *Σ*=(*Σ*_{1},*Σ*_{2}) satisfying an integral equation with the same operator as we have deduced for the plane wave diffraction problem(3.6)here −*R*_{j} is the Fourier transform of the free-space Green's tensor normal stress on the face *Γ*_{j}(3.7)Thus, due to theorem 2.10 to establish solvability (or unsolvability) of equation (3.6) it will be enough to investigate regularity of its right-hand side (if necessary and possible) to get rid of its singularities and finally to apply the isomorphism theorem. So, simple calculations show us that(3.8)where(3.9)(3.10)and(3.11)is a change of variables (*x*_{1},*y*_{1}) to (*x*_{2},*y*_{2}) matrix. This non-symmetry in equation (3.8) is due to non-invariance of the Green's tensor. Evaluating the Fourier transform of equation (3.8) we finally obtain(3.12)As one can see from this formula, *R*_{j} are holomorphic functions in and moreover we have(3.13)where is the polar angle of . Indeed, one can treat equation (3.12) as an action of Hilbert projector on(3.14)Due to the choice of branches of *ζ*_{*}, can be extended analytically into domain and, what is important, for and it admits an estimate(3.15)So and thus, due to Hilbert projector properties we obtain property (3.13).

Now we apply theorem 2.10 with and obtain existence and uniqueness of solution to equation (3.6). ▪

We should also mention that, the only singularities of *Σ*_{j}(*z*−i0), are the Rayleigh pole −*ν*_{R} and branching points of *C*^{j}, given in equation (2.53).

### (c) Green's tensor far-field asymptotics

We will restrict ourselves to evaluation only of the first term of the asymptotics of (*x*,*y*,*x*^{0},*y*^{0}), as .

We apply the same change of variable in representation (3.5) of _{j} as we have done in calculations of diffraction coefficient: *ξ*=*ν*_{*} cos *β*. We arrive at evaluation of asymptotics of integrals(3.16)where *x*_{j}=*r* cos *θ*_{j}, *y*_{j}=*r* sin *θ*_{j} and *P*_{*} are defined by equation (2.55) with *t*=1.

We see that asymptotics of _{j} is the sum of contributions of a stationary point in representation (3.16), singularities of *C*^{j} and the Rayleigh pole. These terms correspond respectively to the scattered by the tip cylindrical, (multiply reflected) head and Rayleigh waves. Let us write down all these terms; according to above-mentioned structure of the wave, we introduce the following decomposition: . We have(3.17)(3.18)(3.19)here , and *Χ*_{n}(*θ*) is a characteristic function of the interval of existence of the head wave:(3.20)with *H* being the Heaviside function.

As we see the head wave is negligible in comparison with the cylindrical waves almost everywhere, i.e. for all admissible angles of observation *θ* with exception for a finite number of critical directions *π*−*θ*_{n}, which correspond to collision of fronts of a head wave and a cylindrical transversal wave (for *=L the corresponding term in representation of ^{(H)} decays exponentially, due to . Moreover in these directions the above asymptotics becomes inapplicable. To investigate the behaviour of _{j} close to these directions recall that we have the following decomposition of *Σ*_{j}(3.21)where *a* and *b* are regular functions in small neighbourhood of *θ*_{nT}=*ν*_{T} cos *θ*_{n}. The contribution of *a*(*ν*_{T} cos *β*) in the integral representation (3.16) will give the classical type of asymptotics, while to treat the singular part we turn to uniform stationary phase method (e.g. Borovikov 1994), which results in the following type of asymptotics: in assumption *a*=0 we have uniformly with respect to |*θ*_{j}−(*π*−*θ*_{n})|(3.22)where *D*_{α} is a parabolic cylinder function of order *α*. This asymptotics matches of course the above asymptotics of _{j} for |*θ*_{j}−(*π*−*θ*_{n})|>0. In the sequel, when the need of asymptotics of will arise in most cases we will neglect the head wave, as well, as singular behaviour of diffraction coefficient in critical directions, but of course we will illustrate legitimacy of this simplification of arguments.

Now, consider the free space Green's function. The main term of the asymptotics of _{f} does not depend on the location of the source (*x*^{0},*y*^{0}), so we can suppose (*x*^{0},*y*^{0})=0 in equation (3.2). The same change of variable will bring us to the integral(3.23)which we evaluate with the stationary phase method and finally arrive at(3.24)

## 4. Uniqueness in the elastic wedge

### (a) Introduction

In this section we will make use of previously constructed Green's tensor to deduce radiation conditions which insure uniqueness of solution to the Lamé system in the wedge with stress-free boundary.

Recall the Green's formula for elasticity operator: for any bounded domain *D* and two arbitrary *u* and *v*, satisfying regularity properties needed for integration by parts, one has(4.1)where (.,.) is the bilinear pairing and . Interchanging *u* and *v* and then subtracting equation (4.1) from the result we arrive at(4.2)Suppose is an arbitrary solution to elasticity equation (*E*+1)*u*=0 in the wedge, satisfying stress-free boundary conditions on its faces.3 Applying formula (4.2) to *u*(.,.) and to Green's tensor in the wedge (.,., *x*^{0}, *y*^{0}) in domain *Ω*_{R}=*Ω*∩*B*_{R} we obtain the following identity:(4.3)where ∂_{R}*Ω*=*Ω*∩*S*_{R}. So, if we knew that the integral in the right-hand side tends to 0 as *R*→∞, we could conclude that *u*(*x*^{0},*y*^{0})=0; and if we knew that this is true in some domain, we would conclude that *u*=0 globally. Generally speaking, this assumption (that the right-hand side of equation (4.3) vanishes) is the zero approximation of the radiation condition. Its disadvantages are that function is not available in explicit form and that it supposes fulfilment of too many conditions—for each (*x*^{0},*y*^{0}) in some domain, etc. Thus, to obtain a uniqueness theorem one should follow the plan: replace with its asymptotics in equation (4.3), and assuming *u* to satisfy suitable growth restrictions as *R*→∞, neglect weak terms of asymptotics. Then, properly grouping different terms obtained in calculating asymptotics of *B* with the ones already derived for , deduce radiation conditions in the form of demand of some integrals' tendency to 0. These become physically consistent and mathematically transparent assumptions for a theorem of uniqueness.

### (b) Uniqueness theorem and radiation conditions

Before proceeding to formulation of the main result let us introduce for *ϵ*>0 the following covering of the arch ∂_{R}*Ω*(4.4)where the first is the inner part(4.5)and the later are close to *Γ*_{j} parts(4.6)(4.7)We denote , intervals perpendicular to faces *Γ*_{j}.

*Suppose* *satisfies Lamé system in the wedge Ω and homogeneous stress-free boundary condition on Γ:*(4.8)(4.9)*Assume also that for some ϵ*∈(0,1) *it satisfies growth restrictions as R*→∞(4.10)(4.11)*internal radiation conditions*(4.12)*and surface radiation conditions*(4.13)*then u*=0.

The above conditions may seem a bit unusual. Radiation conditions in integral form were first introduced in the work of Magnus (1949) for an acoustic problem of diffraction by a bounded obstacle. Those conditions assumed integration over expanding spheres. In our case we have two different types of bulk waves propagating within the wedge and surface waves travelling along its faces. Conditions (4.12) and (4.13) take into account this structure of the field. One can try to reformulate them in a more traditional form as Sommerfeld radiation conditions, demanding quantities similar to integrands in equations (4.12) and (4.13) to tend to zero uniformly. To obtain a uniqueness theorem on this way, one will still have to distinguish the inner and close to boundary parts of the wedge. One will also face difficulties formulating surface radiation conditions in the latter part, due to presence of both bulk and Rayleigh waves which have different orders in *R*. We believe that the most natural way to resolve this problem is to formulate radiation conditions in an integral form the way we do.

This theorem is optimal in the following sense: on one hand conditions (4.10)–(4.13) admit physically consistent solutions: an outgoing solution to a plane wave diffraction problem upon extraction of an explicit part containing all multiply reflected bulk waves satisfies them, and so does the scattered part of Green's tensor _{0}. On the other hand strengthening conditions (4.10)–(4.13) looks likely to make admissible class too narrow to include the above solutions. Also, dropping any of the radiation conditions leads to non-uniqueness: for example dropping the surface radiation condition one admits a solution to the Rayleigh wave incidence diffraction problem, which comprises both incoming and outgoing waves.

Assuming *u* to satisfy conditions of the theorem we obtain representation (4.3). We now start to implement the above plan: we will show that in assumptions of the theorem the right-hand side of equation (4.3) tends to 0, as *R*→∞.

Note that normal stress operator *B* on ∂_{R}*Ω* is given by the formula(4.14)and it acts on _{j} in representation (3.16) as multiplication of the integrands by the matrices(4.15)Thus, the first terms of asymptotics of *B*_{j} and _{j} will differ only by the matrix i*K*_{*}(*θ*, *π*−*θ*) for cylindrical wave term and by for Rayleigh wave; multiplier of the same kind i*K*_{*}(*θ*, *π*−*θ*) will differ the first term of asymptotics of *B*_{f} from the one of _{f}(4.16)(4.17)(4.18)We have introduced here the following notation:(4.19)(4.20)(4.21)(4.22)

We note here that, like in case of itself, one should replace these asymptotics with uniform ones in vicinities of critical directions which will differ from expression (3.22) by the same multiplier as we have for the cylindrical waves. We omit this step as purely technical.

Now we substitute all these asymptotic expressions into the right-hand side of equation (4.3): we have(4.23)where the main terms of cylindrical and Rayleigh waves contributions are(4.24)(4.25)here we have used a smooth partition of unity corresponding to covering (4.4),(4.26)and the notation(4.27)(4.28)for cylindrical and Rayleigh wave components of the Green's function. To deduce this, we benefit from the identity while changing the variables in expressions for _{2} and *B*_{2}. Note also, that ; therefore moving symmetric matrix to the other argument of pairing we re-write *Z*_{S} in the following way:(4.29)Let us have a closer look at the second argument of pairing in equation (4.29). To do so, we now change variables to polar ones: we will use the following notation,(4.30)where **e**_{r}, **e**_{θ} are unit coordinate vector fields and (*u*^{r}, *u*^{θ}) are the displacement coordinates in this basis. In this representation projectors and are given by(4.31)and the normal stress operator *B* on ∂_{R}*Ω* can be rewritten as follows(4.32)so, we have(4.33)(4.34)Right-hand sides of these expressions are not exactly of the type we see in formulations of radiation conditions, and indeed it turns out that the third and the fourth terms in equations (4.33) and (4.34) are negligible. Doubtlessly the terms *u*^{r}/*R* and *u*^{θ}/*R* are weak ones in assumption of boundedness of , and the terms with tangent to ∂_{R}*Ω* derivatives can be neglected as well. Let us deal with . On this example we will also illustrate possibility to neglect considerations of critical directions. Note, that we can consider contribution of this term on earlier stage, rather then in equation (4.29): indeed it is enough to demonstrate that the integral(4.35)tends to zero. So, we integrate by parts to obtain(4.36)Let us consider the contribution of vicinity of singular direction *π*−*θ*_{n} in the integral on the right-hand side of equation (4.36): differentiating the integral representation of with respect to *θ* we obtain a uniform estimate,(4.37)We estimate *D*_{−1/2} with a constant and taking into account that in regular directions is of order *R*_{−1/2} we finally estimate(4.38)So, we conclude(4.39)Now, after getting rid of weak terms we apply Cauchy inequality to the right-hand side of equation (4.29) and finally evaluate asymptotically(4.40)Now consider contribution equation (4.25) of the boundary layer. We treat it in a similar way: just like in the case of cylindrical waves we extract the strongest term and obtain an estimate analogous to (4.40). First, we get rid of the cut-off function *Χ*^{(b,j)} (due to exponential decay of *C*_{j,*} at distance from the face), next we transform the arch of integration into the interval perpendicular to the face . Further, recalling that *c*_{j,*}(*x*^{0},*y*^{0}) is projector onto the eigenvector of ** dm**(−

*ν*

_{R}) it is obviously enough to consider just the Rayleigh wave

*w*

_{−}, travelling away from the tip, instead of

*c*

_{j,*}; so consider(4.41)Now we note that(4.42)and after proper grouping different terms in equation (4.41) we find(4.43)where the first term

*α*

_{R}(

*w*

_{−},

*u*) involves normal to the face

*y*-derivatives and is given by(4.44)and the second is(4.45)For the Rayleigh wave we have ; hence we estimate(4.46)where(4.47)is the norm involved in the surface radiation condition (4.13), thus

*β*

_{R}→0 as

*R*→∞. On the other hand

*α*

_{R}(

*w*,

*u*) can be estimated by the same quantity. Indeed, applying Green's formula in a curved trapezoid we find(4.48)Thus we conclude that

*α*

_{R}(

*w*

_{−},

*u*) has a constant limit.

Now let us consider : up to an exponentially decaying term we have

A similar to the above argument shows, that in assumption of tendency to 0 the derivative tends to a constant as well, and this constant obviously must be 0. So, we have asymptotically(4.49)Thus, recalling representation (4.44) we conclude, that if , as *R*→∞, then(4.50)So, to establish tendency to 0 of *α*_{R}(*w*_{−}, *u*) it is enough to estimate and with *N*_{R}(*u*), and this result can be deduced via standard elliptic estimates.

*Suppose v is a solution to the Lamé system (E+1)v=0 in the domain Π _{2ρ}≔{|x|≤2ρ, 0≤y≤4ρ}, satisfying the stress-free boundary condition on {y=0}, then the following estimate holds:*(4.51)

*where*

*j*

_{r}={

*x*=

*r*,

*y*≤2

*ρ*},

*and*

*i*

_{0}={

*x*=0,

*y*≤

*ρ*}.

*The constant*

*C*

*does not depend on*

*ρ*.

It follows from this lemma that(4.52)Combining the latter with equations (4.46) and (4.50) we estimate . Now recalling equations (4.23) and (4.40) we conclude, that as *R*→∞. This implies the theorem, since equation (4.3) now reads *u*(*x*^{0},*y*^{0})=0 for all (*x*^{0},*y*^{0})∈*Ω*. ▪

Let us rescale to new variables *X*=*ρ*^{−1}*x*, *Y*=*ρ*^{−1}*y* and introduce *V*(*X*,*Y*)=*v*(*x*,*y*), *Π*_{1}=*ρ*^{−1}*Π*_{ρ}, *I*_{0}=*ρ*^{−1}*i*_{0}. We estimate: for arbitrary *γ*>0(4.53)The second inequality is an elliptic estimate, as (*E*+*ρ*^{2})*V*=0. Now, changing back to (*x*,*y*) variables we find(4.54)now considering *ρ* to be big rather than small and choosing *γ*=*ρ*^{−1} in the last inequality we obtain the desired estimate. ▪

### (c) *L*^{2} uniqueness

The aim of this subsection is as follows.

Due to homogeneity of the problem, this theorem implies absence of discrete spectrum for Lamé operator in the wedge.

We will show that *u* satisfies conditions of theorem 4.1. As we have already mentioned the growth restrictions in equations (4.10) and (4.11) and radiation conditions in equations (4.12) and (4.13) are not very confining: an arbitrary *H*^{1}(*Ω*) function satisfies them on a sequence *R*→∞. We use the following result, which we establish in the end of the section.

*In assumptions of* *theorem* 4.3 *u*∈*H*^{1}(*Ω*).

Let us first note that this lemma implies an even stronger then equation (4.10) estimate(4.55)Now in order to obtain the theorem it suffices to observe that almost without modification of the argument of lemma 4.2 one obtains(4.56)with *δ*∈(0,1). The latter equation together with equation (4.55) suggests that *u* satisfies all conditions of theorem 4.1. ▪

It is enough to demonstrate, that ‖*u*; *H*^{1}(*Ω*_{R})‖ is bounded, as *R*→∞. In order to deal with local estimates let us change variables to slow ones: *x*=*Ry*, then function *v*(*y*)=*u*(*x*) satisfies(4.57)(4.58)Let us introduce a radial cut-off function , which is equal to 1 if its argument is less than 1, and 0 if the argument is greater, than 2. We have(4.59)where [*E*, *ψ*]=*Eψ*−*ψE* is a first order operator, supported in *Ω*_{2}\*Ω*_{1}. Thus, applying Green's formula (4.1) to *ψv* and we obtain(4.60)Here [*B*, *ψ*] is a *C*^{∞} matrix, supported in *Γ*_{1,2}=*Γ*∩{1<*r*<2}. We estimate the left-hand side of equation (4.60) from below via the Korn inequality (e.g. Duvaut & Lions 1972)(4.61)We integrate by parts the second term in the right-hand side of equation (4.60) and apply Cauchy inequality with small parameter, so we obtain(4.62)Next we estimate the last term in equation (4.60) first by square of the *L*^{2}(*Γ*_{1,2}) norm, then via the Sobolev compact imbedding estimate and finally recalling that *v* satisfies stress-free boundary condition equation (4.58) we apply the standard elliptic estimate(4.63)Here *VΓ*_{1,2} is a vicinity of *Γ*_{1,2}. Now combining equations (4.61)–(4.63) with identity (4.60) and choosing *ϵ*_{1}=*C*_{K}/2 and *ϵ*_{2}=*R*^{−2} we finally estimate(4.64)Changing the variables back to fast ones we obtain(4.65)which implies the lemma. ▪

## Acknowledgements

V.V.K. was supported by INTAS YSF 01-198 and RFBR 01-01-00251 grants while performing this work.

## Footnotes

↵In fact this result can be improved for smooth data: general theory of elliptic problems in angular domains (e.g. Kondrat'ev 1963; Nazarov & Plamenevski 1994) allows to justify complete asymptotic expansions of solutions in the vicinity of the wedge tip. These tip asymptotics are series in powers of

*r*and log*r*: one has for solution of Lamé system (*E*+e^{2iϵ})*u*=0,*ϵ*∈[0,*π*) in the wedge with stress-free boundary conditions the following expansion (see ch. 3 and 4 in Kozlov*et al*. 2001)here*q*_{m}are solutions of special transcendental equations such that Re*q*_{m}≥0, and*N*_{m}≤1.↵This is a traditional definition, not an invariant one.

↵Here and below we assume a weak solution, when referring to ‘a solution of a boundary-value problem’. Since we deal only with the Lamé operator in a wedge with stress-free boundary, regularity issues concern only behaviour at the tip. Indeed, due to the general theory of elliptic boundary-value problems, outside a vicinity of the tip all our solutions are analytical, and thus belong to .

- Received May 9, 2005.
- Accepted August 3, 2005.

- © 2005 The Royal Society