Asymptotics and Uniqueness of Solutions of the Elasticity System with the Mixed Dirichlet–Robin Boundary Conditions

: We study properties of generalized solutions of the Dirichlet–Robin problem for an elasticity system in the exterior of a compact, as well as the asymptotic behavior of solutions of this mixed problem at inﬁnity, with the condition that the energy integral with the weight | x | a is ﬁnite. Depending on the value of the parameter a , we have proved uniqueness (or non-uniqueness) theorems for the mixed Dirichlet–Robin problem, and also given exact formulas for the dimension of the space of solutions. The main method for studying the problem under consideration is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions.


Introduction
Dedicated to the blessed memory of my parents who went to heaven this year.
General boundary value problems for elliptic systems in domains with smooth boundaries were studied in [1][2][3][4]. Boundary value problems for the elasticity system in bounded domains are quite well studied. A presentation of the basic facts of this theory can be found in Fichera's monograph [5]. In [6][7][8], Kondratiev and Oleinik established generalizations of Korn's inequality and Hardy's inequality for bounded domains and a large class of unbounded domains, and applied these to investigate the main boundary value problems for the elasticity system, which were also considered in [9,10]. The paper [10] uses Korn's inequality and Hardy's inequality to study the uniqueness and stability of generalized solutions of mixed boundary value problems for the elasticity system in an unbounded domain provided that E(u, Ω) is finite.
In [11,12], shells of variable thickness are considered in three-dimensional Euclidean space around surfaces that have a limited principal curvature. Here the author derives the Korn interpolation inequality, the inequality also introduced in [13], and the second Korn inequality in domains in which no boundary or normalization conditions are imposed on the vector function u. The constants in the estimates are asymptotically optimal in terms of the thickness of the region. Note that this is the first paper that defines the asymptotic behavior of the optimal constant in the classical Korn second inequality for shells over the thickness of the domain in almost complete generality, and the inequality holds for almost all thin domains. In [14], the author extends the L 2 Korn interpolation inequality, as well as the second Korn inequalities, in thin domains, proved in [12], to the space L p for any 1 < p < ∞. Note the paper [15], in which the authors prove asymptotically sharp weighted Korn and Korn-type inequalities in thin domains with singular weights. The choice of weights is based on some factors; in particular, the spatial case arises when transforming Cartesian variables to polar change of variables in two dimensions.
In [16], a regularity result is proved for a system of linear elasticity theory with mixed boundary conditions on a curved polyhedral domain in weighted Sobolev spaces, for which the weight is determined by the distance to the set of edges. These results are then extended to other strongly elliptic systems and higher dimensions.
In [17,18] methods are proposed that allow one to construct the asymptotics of solutions of the Laplace and poly-Laplace equations in a neighborhood of singular points, which are zero and infinity, as well as the asymptotics of these equations on manifolds with singularities. In [19], asymptotics were constructed for the solution of the Laplace equation on manifolds with a beak-type singularity in a neighborhood of the singular point.
We also note the papers [20][21][22], in which the basic boundary value problems and problems with mixed boundary conditions for the biharmonic (polyharmonic) equation are studied. In particular, the existence and uniqueness of solutions in the ball were established, and necessary and sufficient conditions for the solvability of boundary value problems for the biharmonic (polyharmonic) equation, including those with a polynomial right-hand side, were obtained.
It is well known that if Ω is unbounded, then one must also characterize the behavior of a solution at infinity. This is usually done by requiring that the Dirichlet integral D(u, Ω) or the energy integral E(u, Ω) be finite, or a condition on the nature of the decay of the modulus of a solution as |x| → ∞.
In this paper we study the properties of generalized solutions of the mixed Dirichlet-Robin problem for the elasticity system in an unbounded domain Ω with a finiteness condition of the weighted energy integral: E a (u, Ω) ≡ Ω |x| a n ∑ i,j=1 Imposing the same constraint on the behavior of the solution at infinity in various classes of unbounded domains, the author [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] studied the uniqueness (non-uniqueness) problem and found the dimensions of the spaces of solutions of boundary value problems for the elasticity system and the biharmonic (polyharmonic) equation.
The main research method for constructing solutions to the mixed Dirichlet-Robin problem is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions. Further, using Korn's and Hardy's-type inequalities [6][7][8], we obtain a criterion for the uniqueness (or non-uniqueness) of solutions to this problem in weighted spaces.
This article contains proofs of the results announced in [36]. Notation: C ∞ 0 (Ω) is the space of infinitely differentiable functions in Ω with compact support in Ω.
We denote by H 1 (Ω, Γ), Γ ⊂ Ω the Sobolev space of functions in Ω obtained by the completion of C ∞ (Ω) vanishing in a neighborhood of Γ with respect to the norm (Ω) is the space of functions in Ω obtained by the completion of C ∞ 0 (Ω) with respect to the norm ||u; H 1 (Ω)||; • H 1 loc (Ω) is the space of functions in Ω obtained by the completion of C ∞ 0 (Ω) with respect to the family of semi-norms We set ∂ α u = ∂ |α| u/∂x α 1 1 . . . ∂x α n n , with α = (α 1 , . . . , α n ), where α i ≥ 0 are integers, and |α| = α 1 + · · · + α n . Let By the cone K in R n with vertex at x 0 we mean a domain such that if x − x 0 ∈ K, then λ(x − x 0 ) ∈ K for all λ > 0. We assume that the origin x 0 = 0 lies outside Ω.

Definitions and Auxiliary Statements
Definition 1. A solution of the system (1) in Ω is a vector-valued function u ∈ H 1 loc (Ω) such that for any vector-valued function ϕ ∈ C ∞ 0 (Ω) the following integral identity holds Before proceeding to the consideration of the boundary value problem (1), (2), we establish two auxiliary lemmas. Lemma 1. Let u be a solution of the system (1) in Ω such that E a (u, Ω) < ∞. Then where P(x) is a polynomial, ord P(x) ≤ m = max{1, 1 − n/2 − a/2}, Γ(x) is the fundamental solution of the system (1), C α = const, β 0 = 1 − n/2 + a/2, β ≥ 0 is an integer, and the function u β satisfies the estimate

Remark 1.
It is known [38], that there exists a fundamental solution Γ(x), which for n > 2 has the following estimate For n = 2 the fundamental solution has a representation Γ(x) = S(x) ln |x| + T(x), where S(x) and T(x) are square matrices of order 2 whose entries are homogeneous functions of order zero [39].
We extend v to R n by setting v = 0 on G = R n \ Ω. Then the vector-valued function v belongs to C ∞ (R n ) and satisfies the system It is easy to see that E a (v, R n ) < ∞. If a + n = 0, then Korn's inequality ( [7], § 3, inequality (1)) implies that v(x) = w(x) + Ax, where A is a constant skew-symmetric matrix and w satisfies D a (w, Ω) < ∞.
We can now use Theorem 1 of [40] since it is based on Lemma 2 of [40], which imposes no constraints on the sign of σ . Hence the expansion Hence, by the definition of v, we obtain (3) with P(x) = P 0 (x) + Ax. Now let a + n = 0. Then for each δ > 0, By Korn's inequality ( [7], § 3, inequality (1)), there exists a constant skew-symmetric matrix A such that where the constant C is independent of v(x). Hence, using Theorem 1 of [40], we get which proves the Lemma for a = −n.

Lemma 2.
Let u be a solution of the system (1) in Ω such that E a (u, Ω) < ∞ for some a ≥ 0. Then for all x ∈ Ω equality (3) holds with u β , satisfying an estimate similar to that in Lemma 1; in addition, P(x) = Ax + B, where A is a constant skew-symmetric matrix and B is a constant vector.
Proof. Let u be a solution of the system (1) in Ω. Then by Lemma 1, we have where P(x) is a polynomial, ord P(x) ≤ 1, and Let us prove that P(x) = Ax + B, where A is a constant skew-symmetric matrix and B is a constant vector. Obviously, if E a (u, Ω) < ∞ and a ≥ 0, then E(u, Ω) < ∞.
Let P(x) = Ax + B, that is, P i (x) = ∑ n j=1 a ij x j + b i . Then where the a ij are the entries of A. The last integral converges if and only if a ij = −a ji , that is, A is a constant skew-symmetric matrix.

Definition 2.
A solution of the mixed Dirichlet-Robin problem (1), (2) is a vector-valued function Let Ker 0 (L) be the space of generalized solutions of the mixed Dirichlet-Robin problem (1), (2), that have a finite energy integral, that is, We set by definition Let dim Ker 0 (L) and dim Ker a (L) be dimensions of Ker 0 (L) and Ker a (L), respectively. We shall calculate the values of dim Ker a (L) in their dependence on the parameter a.

Proof.
Step 1. Let n ≥ 3. For any constant skew-symmetric matrix A, we construct a generalized solution u A of the Dirichlet-Robin problem for the system (1) in Ω with the boundary conditions satisfying the conditions E(u A , Ω) < ∞, D(u A , Ω) < ∞, and Such a solution may be constructed by the variational method, minimizing the functional The boundedness of the Dirichlet follows from Korn's inequality ( [7]; §3, inequality (43)). Condition (6) is a consequence of Hardy's inequality [7]. Let A 1 , . . . , A p , p = (n 2 − n)/2, be linearly independent constant skew-symmetric n × n -matrices. We consider the solutions u A 1 , . . . , u A p .
Step 2. Now in the same way, for any constant vector e = e k = 0, e k = (e 1 k , . . . , e n k ), e j k = 1, k = j, we construct a generalized solution u e k of the Dirichlet-Robin problem for the system (1) with the boundary conditions u e k | Γ 1 = e k , σ(u e k ) + τu e k Γ 2 = 0 and with conditions E(u e k , Ω) < ∞, D(u e k , Ω) < ∞, Ω |u e k | 2 |x| −2 dx < ∞.
Step 3. The solutions u A 1 − A 1 x, . . . , u A p − A p x, u e 1 − e 1 , . . . , u e n − e n are linearly independent. Indeed, if where B is a constant vector. Since the u e i satisfy (7), we have B = 0. The vectors e i , i = 1, . . . , n are linearly independent, and therefore c i = 0, i = 1, . . . , n.
Thus, we have proved that the homogenous Dirichlet-Robin problem has at least n(n + 2)/2 linearly independent generalized solutions.
Step 4. Let us now prove that any generalized solution u of the homogenous Dirichlet-Robin problem with the condition E(u, Ω) < ∞ is a linear combination of the constructed solutions. According to Korn's inequality ( [7]; §3, inequality (43)), there is skew-symmetric matrix A such that Let us show that w ≡ 0. We substitute in the integral identity (4) for w the vector-valued function We claim that the right-hand side of (8) approaches zero as N → ∞. Indeed, the Cauchy-Schwartz inequality yields that Using the integral identity we find that if w is a solution of the homogeneous problem (1), (2), then w = A 1 x + B 1 . The set of points where A 1 x + B 1 = 0 is a linear manifold of dimension less than n − 1, since the rank of the matrix A 1 is ≥ 2 if A 1 ≡ 0. Consequently, w = 0. This conclusion follows from the fact that The theorem is proved for n ≥ 3. Let now n = 2. For a nontrivial constant skew-symmetric matrix A, we construct a generalized solution u A of the Dirichlet-Robin problem for the system (1) in Ω with the boundary conditions (5), minimizing the corresponding functional Φ(v) in the class of functions where N >> 1 is such that G ⊂ {x : |x| < N}.
We prove further that any generalized solution u of the homogeneous Dirichlet-Robin problem (1), (2) has the form u = c 0 (u A − Ax), where c 0 = const, A is a skew-symmetric matrix, and A = 0.
Let us prove that w = 0. Substituting in the integral identity (4) for w the function Further, as above, we obtain that w ≡ 0. This concludes the proof.
Consider now the case when −n ≤ a < 0 and n > 2.
where the constant C is independent of u.
For the function v = u − Ax we have D a (v, Ω) < ∞ and E a (v, Ω) < ∞. Moreover, v is a solution of (1) in Ω. Hence, by Lemma 1, it has the form (3): Let us prove that ord P(x) = 0. First, establish the inequality D a (P(x), Ω) < ∞. We have D a (v, Ω) < ∞, and it is easy to verify that D a (R(x), Ω) < ∞ for −n ≤ a < 0. Hence D a (P(x), Ω) < ∞ by the triangle inequality..
The last integral converges if and only if a + 2k − 2 + n < 0. Hence k < 1 and, therefore, ord P(x) = 0 and P( where A is a constant skew-symmetric matrix and B is a constant vector. It is easy to verify that E(R(x), Ω) < ∞ and E(Ax + B, Ω) = 0. Hence E(u, Ω) < ∞, that is, u ∈ Ker 0 (L). We obtain the embedding Ker a (L) ⊂ Ker 0 (L). In addition, it is obvious that Ker 0 (L) ⊂ Ker a (L) for a < 0.
Step 1. Now let n = 2. For a non-trivial constant skew-symmetric matrix A, we construct a generalized solution u A (x) of the mixed Dirichlet-Robin problem for the system (1) in Ω with boundary conditions where N >> 1 and G ⊂ {x : |x| < N}.
Step 2. In this same way we obtain generalized solutions of the mixed Dirichlet-Robin for the system (1) in Ω with the boundary conditions Since |Ax| ≥ |x| and |Γ(x) d| ≤ C ln |x|, it follows that d 0 = 0. Hence Γ(x) d = 0, and applying the elasticity operator to this equation, we obtain where I is the 2 × 2 unit matrix, δ(x) is the Dirac function. Hence it follows that Hence the Dirichlet-Robin problem (1), (2) has at least three linearly independent solutions satisfying E a (u, Ω) < ∞.
Step 3. We claim that each generalized solution u of the Dirichlet-Robin problemn (1), (2) with condition E a (u, Ω) < ∞ is a linear combination of the solutions constructed above. By Korn's inequality ( [7]; §3, inequality (43)), there is a constant skew-symmetric matrix A 1 such that where the constant C is independent of u. For the functionv 1 Since v 1 is a solution of the system (1) in Ω, it follows by Lemma 1 that In a similar way to the above we can show that ord P(x) = 0 and P( Obviously, w is a solution of the system (1) It is easy to see that D(w, Ω) < ∞ and Ω |w| 2 |x| −2 | ln |x|| −2 dx < ∞. Thus, w(x) is a solution of the following problem (w): Let us prove that the solution w(x) of problem (w) is unique, that is, w(x) ≡ 0, x ∈ Ω. To this end, we write the integral identity (4) for the vector-valued function We claim that the right-hand side of (11) approaches zero as N → ∞.
Indeed, the Cauchy-Schwartz inequality yields that Using the integral identity we find that if w is a solution to the homogeneous problem (1), (2), then w = A 2 x + B 2 . The set of all x such that A 2 x + B 2 = 0 is a linear manifold whose dimension is less than n − 1, since the rank of the matrix implies that w ≡ 0 on a set of positive measure on ∂Ω, and therefore, w(x) ≡ 0, x ∈ Ω. The theorem is proved.
Such a solution is constructed by the variational method. We minimize the corresponding functional over the class of admissible functions {v ∈ C ∞ (Ω), v e | Γ 1 = e, v has compact support in Ω}. The boundedness of the Dirichlet integral follows from Korn's inequality ( [7]; §3, inequality (43)). Condition (13) follows from Hardy's inequality ( [7]; §3, inequality (27)). By Lemma 2 the solution u e (x) takes the form u e (x) = P e (x) + R e (x), where P e (x) is a polynomial, P e (x) = Ax + B, with A being a constant skew-symmetric matrix and B being a constant vector, and We claim that P e (x) ≡ 0. Assume that P e (x) ≡ 0. Then in the interior of a certain cone K we have |P e (x)| > C and This contradiction shows that P e (x) ≡ 0. Thus, Let us prove that For a proof we consider a ball Q R = {x : |x| < R} with centre at the origin suck that G ⊂⊂ Q R .
There exists a sequence of domains Ω R k such that Ω R k ⊂ Ω R k+1 ⊂ · · · ⊂ Ω R n ⊂ R n and ∪ k Ω R k = Ω.
We claim that the integrals in the right-and left-hand sides of (17) converge. Indeed, the Cauchy-Schwartz inequality yields that because Ω R ⊂ Ω and Ω |∇u e | 2 dx < ∞.
We now claim that = const R 2−n → 0 as R → ∞ for n > 2. The constants c and C are independent of R.
Letting R in (17) tend to infinity, we obtain the required Equation (16): and bearing in mind that u e | Γ 1 = e, (σ(u e ) + τu e )| Γ 2 = 0, we get where σ(u e ) ≡ a ij kh ∂u ej ∂x h ν k . We claim that the constant C 0 is non-zero in (15). Indeed, if C 0 = 0, then |u e | ≤ C|x| 1−n and σ(u e )| ≤ C|x| −n . Taking the scalar product of the system (1) and 1 and integrating over Ω R , we obtain and by (18) we obtain Using the integral identity we get σ(u e ) = 0, since e = 0. By [5], it follows that u e = Ax + B, where A is a constant skew-symmetric matrix and B is a constant vector.
Step 2. Let e = 0 be an arbitrary vector in R n . We consider the solution u e such that u e | Γ 1 = e, (σ(u e ) + τu e )| Γ 2 = 0, and u e (x) = C 0 Γ(x) + u 0 e (x), where C 0 = 0. We can associate with each vector e in R n the corresponding vector C 0 in R n , thus obtaining a transformation S : R n → R n such that S : e → C 0 , where e = 0, C 0 = 0. It is easy to verify that the transformation S is linear and non-degenerate.
Let e = {e 1 , . . . , e n } be a basis in R n . For arbitrary linearly independent vectors C 0 = {C 01 , . . . , C 0n } there exists a unique linear transformation (matrix) S such that C 0 = S e. Then Step 3. Consider now the elasticity system (1) in Ω with boundary conditions where A is a constant skew-symmetric matrix. For every such matrix A we construct a generalized solution of the system (1) with the boundary conditions (20) and the properties Such a solution can be constructed using the variational method and minimizing the corresponding functional over the set of admissible functions {u : u ∈ H 1 (Ω), u | Γ 1 = Ax, u has compact support in Ω}. The Dirichlet integral is bounded by Korn's inequality ( [7]; §3, inequality (43)). Condition (21) follows from Hardy's inequality [7]. By Lemma 2 we have As before, we can show that P A (x) ≡ 0. Hence, Step 4. Consider now the difference where e = S −1 C 0 , and u e and u A are defined by (15) and (23) respectively. Obviously, v is a solution of (1) in Ω and v | Γ (15) and (23) that We now claim that v ≡ 0. For let v ≡ 0, that is, On the other hand, |Ax| → ∞ as |x| → ∞, Ax = 0. This contradiction shows that v ≡ 0. Let us prove that if A 1 , . . . , A p is a basis in the space of skew-symmetric matrices, then v A 1 , . . . , v A p are linearly independent solutions, i.e., from the equality where T = ||t ij || n×n . Then Ω |x| a |∇Tx| 2 dx < ∞, because Tx = W 1 and Ω |x| a |∇W 1 | 2 dx < ∞. On the other hand, ∞ > Ω |x| a |∇Tx| 2 dx = Ω |x| a |t ij | 2 dx, and the integral on the right-hand side is finite if and only if t ij = 0, that is, Thus, the mixed Dirichlet-Robin problem (1), (2) has at least p = (n 2 − n)/2 linearly independent generalized solutions.
It is easy to see that D(Z, Ω) < ∞ and Ω |Z| 2 |x| −2 dx < ∞. Thus, we obtain problem (Z b ): By construction, we have problem (e): We shall now prove the uniqueness of a solution of problem (e). Let u e and u e be solutions such that Then the function u 0 = u e − u e satisfies We claim that u 0 ≡ 0 in Ω. Indeed, consider the integral identity (4) for u 0 and put where E (u 0 ) ≡ a ij kh ∂u 0j ∂x h ∂u 0i ∂x k . In the same way as in (11) (Theorem 2, case n = 2), we can show that the right-hand side of (25) tends to zero as N → ∞. Hence,

Using the integral identity
we find that if u 0 is a solution to the homogeneous problem (1), (2), then u 0 = A 0 x + B 0 . The set of all x such that A 0 x + B 0 = 0 is a linear manifold whose dimension is less than n − 1, since the rank of the matrix A 0 is ≥ 2 if A 0 ≡ 0. Therefore, u 0 = 0. The relation implies that u 0 ≡ 0 on a set of positive measure on ∂Ω, and therefore, u 0 ≡ 0, x ∈ Ω. Thus, the solution to problem (e) is unique.
We now claim that E a (u e , Ω) = Ω |x| a |ε(u e )| 2 dx = ∞ for e = 0. (26) First of all we show that if C 0 = 0 in (15), then By the properties of the fundamental solution of the elasticity system [38], if Γ(x) = |x| 2−n U(x), where U(x) is a homogeneous function of order zero, then |x| a |ε(C 0 Γ(x))| 2 is a homogeneous function of order a + 2(1 − n), that is, where f 0 (x) is a homogeneous function of order zero. We fix a point x 0 such that f 0 (x 0 ) = 0.
By continuity, f 0 (x) = 0 in a neighborhood Q δ (x 0 ) of x 0 . We consider a cone K with vertex at the origin such that Q δ (x 0 ) ⊂ K. Then Applying the triangle inequality to the Formula (15) of the type It is easy to verify that E a (R 1e (x), Ω) < ∞. Hence E a (u e , Ω) = ∞ for e = 0, and we obtain the problem (e a ): By Formula (26), .
This contradiction shows that b = 0, that is, e = 0. Hence W = Z is a solution of the following problem (z 0 ): By the unique solubility of the problem (e), we have Z ≡ 0 in Ω. Hence W = 0, and since W | Γ 1 = 0, (σ(W) + τW)| Γ 2 = 0, it follows that W ≡ 0 in Ω. This proves the theorem for n ≥ 3.
The proof in the case n = 2 is carried out in a similar way. For a non-trivial constant skew-symmetric matrix A, we construct a generalized solution u A (x) of the mixed Dirichlet-Robin problem for the system (1) in Ω with the boundary conditions (20) by minimizing the corresponding functional over the class of admissible functions {u : u ∈ H 1 (Ω), u | Γ 1 = Ax, u has a compact support in Ω}. This solution satisfies E(u A , Ω) < ∞ and D(u A , Ω) < ∞. By Hardy's inequality [6] we obtain where N >> 1 and G ⊂ {x : |x| < N}.
Let us prove that each generalized solution u of the problem (1), (2) satisfying the condition E a (u, Ω) < ∞ has the following form: where A is a constant skew-symmetric matrix. By Lemma 2, the solution of the system (1) has the form (22) with P A (x) = Ax + B, where A is a constant skew-symmetric matrix and B is a constant vector. We claim that A = 0. For assuming that A = 0, we can write (22) in the following form: for each R. It is easy to see that Hence, ρ<|x|<R |∇R 1 (x)| 2 dx < C 2 for any R. Since Γ(x) is a fundamental solution of (1), Γ(x) = S(x) ln |x| + T(x), where S(x) and T(x) are (2 × 2)-matrices whose entries are homogeneous functions of order zero (see [39]), and so |∇(Γ(x)C 0 )| ≤ C(C 0 )| ln |x|||x| −1 . It follows that By (22) and the triangle inequality, we have On the other hand, Hence C 3 R 2 ≤ C 1 + C 2 + C (C 0 )(ln R) 3 for each R >> 1. This contradiction shows that A = 0 and P A (x) = B. Hence, u A = B + Γ(x)C 0 + R 1 (x).
Proof. Consider the case n ≥ 3. Let a = n. We shall prove the theorem by contradiction. Assume that dim Ker a (L) > 0. Then there is a u such that u ∈ Ker a (L) and u ≡ 0. Since a = n, we have u ∈ Ker n (L) ⊂ Ker n−2 (L). Hence by Theorem 3 we obtain where e = S −1 C 0 (see (19)) and C 0 is defined by Formula (23). Substituting (15) and (23) in (27), we obtain where We claim thatC 1 = 0 in (28). Indeed, we assume thatC 1 = 0. Taking the scalar product of (1) with u and integrating over Ω R , we obtain We claim that |x|=R u σ(u) ds → 0 for R → ∞.
By the properties of the fundamental solution of the system (1) (see [38]) we have Γ(x) = |x| 2−n U(x), where U(x) is a homogeneous function of order zero. Hence |x| n |ε((C 1 ∇)Γ(x))| 2 is a homogeneous function of order (−n), that is, where f 0 (x) is a homogeneous function of order zero. We fix a point x 0 such that f 0 (x 0 ) = 0.
By continuity, f 0 (x) = 0 in a neighborhood Q δ (x 0 ) of x 0 . We consider a cone K with vertex at the origin such that Q δ (x 0 ) ⊂ K. Then This contradiction shows that u ≡ 0. This completes the proof for n ≥ 3.
Consider now the case n = 2. It sufficient to show that dim Ker a (L) = 0 for a = 2. Assume that dim Ker a (L) > 0, that is, there exists u such that u ∈ Ker a (L) and u ≡ 0. Since a = 2, it follows that u ∈ Ker a (L) ⊂ Ker 0 (L). Hence by Theorem 3 we obtain u = u A − Ax and for u A Lemma 2 yields a representation (22), that is, where P A (x) = Ax + B, A is a constant skew-symmetric matrix and B is a constant vector. Substituting the expansion of u A (x) in the representation of u(x), we obtain We prove that C 0 = 0 by contradiction. Indeed, assume that C 0 = 0. Then we have E a (R(x), Ω)=E a (u, Ω)<∞.

In view of the boundary conditions
Since |u| ≤ C and |σ(u)| ≤ C|x| −2 ln |x|, it follows that Passing to the limit as R → ∞ in equality (30), we obtain Using the obtained integral identity, we conclude that if u is a solution to the homogeneous problem (1), (2), then u = A 1 x + B 1 , where A 1 is a constant skew-symmetric matrix, B 1 is a constant vector. Hence, In view of boundary conditions (2), B = 0 and u ≡ 0, so that dim Ker a (L) = 0 for n = 2 and 2 ≤ a < ∞. The theorem is proved.
Proof. Assume that n > 2. To prove the theorem, we need to determine the number of linearly independent polynomial solutions of a system (1), the degree of which does not exceed the fixed number. Let P = (P 1 , . . . , P n ) be a polynomial solution of the system (1) of degree r. Then the degree of the polynomial P i does not exceed r, and P can be represented in the following form: where P (s) = (P (s) 1 , . . . , P (s) n ) is a homogeneous polynomial of degree s, satisfying the system (1) (see [38]).
The space of polynomials in R n of degree at most r has dimension (r + n)!/r!n! (see [41]). Hence the dimension of the space of vector-valued polynomials in R n of degree at most r is equal to n(r + n)! r!n! = (r + n)! r!(n − 1)! .
Polynomials of this kind solving the elasticity system form a space of dimension because each equation of the elasticity system is equivalent to the vanishing of some polynomial of degree (r − 2). We denote by k(r, n) the number of linearly independent polynomial solutions of (1) whose degree is at most r, and let l(r, n) be the number of linearly independent homogeneous polynomials of degree r that are solutions of (1). Using representation (P) we obtain k(r, n) = r ∑ s=0 l(s, n), where l(s, n) = n s + n − 2 n − 2 + s + n − 3 n − 2 for s ≥ 1, l(0, n) = n.
(i) Let w 1 , . . . , w k be a basis in the space of polynomial solutions of (1) whose degrees do not exceed r. Since ord w i ≤ r, it follows that E a (w i , Ω) < ∞ for −2r − n ≤ a < −2r − n + 2. For each w i , i = 1, . . . , k we consider the solution v i of the system (1) Such a solution we can construct by the variational method, minimizing the corresponding functional over the class of admissible functions {v : v ∈ H 1 (Ω), v | Γ 1 = w, v has compact support in Ω}.
Consider next the difference: Let us prove that z i , i = 1, . . . , k, are linearly independent. Indeed, if Hence, |W| 2 = |V| 2 , |∇W| 2 = |∇V| 2 and By Lemma 1, the solution V of the system (1) in Ω has the following form: where P(x) is a polynomial, and It is easy to verify that By the triangle inequality, We claim that P(x) ≡ 0. Indeed, assume that ord P(x) = r. Then in the interior of a certain cone This integral converges only when r < 0. Therefore, P(x) ≡ 0.
and by the estimates (31) we obtain Since w i is a basis in the space of polynomial solutions of (1) whose degrees do not exceed r, it follows that c i = 0, i = 1, . . . , k. Hence the problem has at least k(r, n) linearly independent solutions.
(ii) Let us prove that each solution u of the system (1) with boundary conditions u | Γ 1 = 0, (σ(u) + τu)| Γ 2 = 0 and E a (u, Ω) < ∞ can be represented as a linear combination of the solutions z i , i = 1, . . . , k, z i = w i − v i . By Lemma 1, every solution of the system (1) in Ω may be written as Since −2r − n ≤ a < −2r − n + 2, it follows that −n/2 − a/2 ≤ r < 1 − n/2 − a/2 and, therefore, We claim that P(x) is a solution of the system (1). Indeed, Since LP(x) is a polynomial and LP(x) = −LR(x) → 0 as |x| → ∞, it follows that LP(x) ≡ 0, that P(x) is a polynomial solution of the system (1). Hence it is represented as a linear combination of the functions w i , i = 1, . . . , k: By our construction of the solutions, after elementary transformations we obtain Let us prove that u 0 ≡ 0. Indeed, u 0 is a solution, that is, Lu 0 = 0 in Ω, u 0 | Γ 1 = 0, and (σ(u 0 ) + τu 0 )| Γ 2 = 0. By the construction of the solutions v i we have Moreover, it is easy to verify that Hence, Since u e (x) is a unique solution of problem (e) in Theorem 3, it follows that u 0 ≡ 0. This proves the theorem for n > 2.
For each w i , i = 3, . . . , k we consider a solution v i of the system (1) , Ω) < ∞ and, by Hardy's inequality [6], we have Such a solution may be constructed by the variational method, by minimizing the corresponding functional over the class of admissible functions. In the same way, we can construct solutions of (1) with boundary conditions By Hardy's inequality [6] we obtain where N >> 1 and G ⊂ {x : |x| < N}.
We claim that P(x) is a solution of (1). Indeed, we have where LR(x) → 0 as |x| → ∞.

Conclusions
The problem of studying boundary value problems for the system of elasticity theory began to be dealt with at the beginning of the 20th century. One of the first papers initiating the systematic investigation of these problems was Fredholm's classical paper [42], in which the first boundary value problem for the linear elasticity system in the case of an isotropic homogeneous body was studied by the method of integral equations. The second boundary value problem for the elasticity system in the case of a bounded domain was studied by Korn [43], who was the first to establish inequalities between the Dirichlet integral D(u, Ω) of the solution and the energy E(u, Ω) of the system, which are now known as Korn's inequalities. Friedrichs's paper [44] played a major role in the analysis of the mathematical aspects of the stationary elasticity theory. In that paper Korn's inequalities are proved and the first and the second boundary value problems of the elasticity theory are analyzed in a bounded domain by the variational method. Here we also note Fichera's monograph [5], who used Korn's inequalities and functional methods to study various boundary value problems for the elasticity system. For a wide class of unbounded domains Kondratiev and Oleinik [6][7][8] established generalizations of Korn's and Hardy's inequalities and used them for the analysis of the main boundary value problems for the elasticity system. In particular, they investigated the existence, uniqueness and stability of solutions of boundary value problems with a finite energy integral.
This article considers the boundary value problem for the elasticity system in the exterior of a compact set with the mixed boundary conditions: the Dirichlet condition on one part of the boundary and the Robin condition on the other; and also with the condition of boundedness of the energy integral E a (u, Ω) with the weight |x| a , which characterizes the behavior of the solution of this problem at infinity. Depending on the value of the parameter a, for each interval, we determine the dimension of the kernel of the operator of the theory of elasticity. The main research method for constructing solutions to the mixed Dirichlet-Robin problem is the variational principle, which assumes the minimization of the corresponding functional in the class of admissible functions. Further, using Korn's and Hardy's-type inequalities, we obtain a criterion for the uniqueness (or non-uniqueness) of solutions to this problem in weighted spaces. These results find their practical application in the field of shell theory, mechanics of deformable solids, as well as in the study of some problems in the theory of scattering, optics, applied and astrophysics.
Note that a new inequality called the Korn's interpolation inequality (since it interpolates between the first and second Korn's inequalities) was applied to study shells. An asymptotically exact version of the interpolation estimate was proved by Harutyunyan (see [12], and other papers) for practically any thin domains and any vector field.
Further, this theory has found its development in many papers in the field of mathematical physics and applied mathematics; some of them are given in the bibliography.

Application
As an application, we note the book [45], in which astronomical optics and the elasticity theory give a very complete and comprehensive description of what is known in this field. After extensive introduction to optics and elasticity, this book discusses a multimode deformable mirror of variable curvature, as well as in-depth active optics, its theory, and fields of application.