On the Mixed Dirichlet – Steklov-Type and Steklov-Type Biharmonic Problems in Weighted Spaces

We studied the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we studied the unique solvability of the mixed Dirichlet–Steklov-type and Steklov-type biharmonic problems in the exterior of a compact set under the assumption that generalized solutions of these problems has a bounded Dirichlet integral with weight |x|a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

As is well known that, if Ω is an unbounded domain, one should additionally characterize the behavior of the solution at infinity.As a rule, to this end, one usually poses either the condition that the Dirichlet (energy) integral is finite or a condition on the character of vanishing of the modulus of the solution as |x| → ∞.Such conditions at infinity are natural and were studied by several authors (e.g., [1][2][3]).
The behavior of solutions of the Dirichlet problem for the biharmonic equation as |x| → ∞ is considered in [4,5], where estimates for |u(x)| and |∇u(x)| as |x| → ∞ are obtained under certain geometric conditions on the domain boundary.
Elliptic problems with parameters in the boundary conditions have been called Steklov or Steklovtype problems since their first appearance in [6].For the biharmonic operator, these conditions were first considered the authors of [7][8][9], who studied the isoperimetric properties of the first eigenvalue.
Note that standard elliptic regularity results are available in [10].The monograph covers higher order linear and nonlinear elliptic boundary value problems, mainly with the biharmonic or polyharmonic operator as leading principal part.The underlying models and, in particular, the role of different boundary conditions are explained in detail.As for linear problems, after a brief summary of the existence theory and L p and Schauder estimates, the focus is on positivity.The required kernel estimates are also presented in detail.
In [10,11], the spectral and positivity preserving properties for the inverse of the biharmonic operator under Steklov and Navier boundary conditions are studied.These are connected with the first Steklov eigenvalue.It is shown that the positivity preserving property is quite sensitive to the parameter involved in the boundary condition.Moreover, positivity of the Steklov boundary value problem is linked with positivity under boundary conditions of Dirichlet and Navier type.
In [12], the boundary value problems for the biharmonic equation and the Stokes system are studied in a half space, and, using the Schwartz reflection principle in weighted L q -space, the uniqueness of solutions of the Stokes system or the biharmonic equation is proved.
We also point out [13][14][15], in which using the methods of complex analysis the Dirichlet and Neumann problems for the polyharmonic equation are explicitly solved in the unit disc of the complex plane.The solution is obtained by modifying the related Cauchy-Pompeiu representation with the help of the polyharmonic Green function.
By developing an approach based on the use of Hardy type inequalities [1][2][3]30], in the present note, we obtain a uniqueness (non-uniqueness) criterion for a solution of the mixed Dirichlet-Steklovtype and Steklov-type problems for the biharmonic equation. Notation: We denote by H m (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 where for all open balls B 0 (R)

Definitions and Auxiliary Statements
Definition 1.A solution of the homogenous biharmonic Equation (1) in Ω is a function u ∈ H 2 loc (Ω) such that, for every function ϕ ∈ C ∞ 0 (Ω), the following integral identity holds: where P(x) is a polynomial, ord is the fundamental solution of Equation ( 1), C α = const, β ≥ 0 is an integer, and the function u β satisfies the estimate: for every multi-index γ.
Remark 1.As is known [31], the fundamental solution Γ(x) of the biharmonic equation has the form Then, the function v belongs to C ∞ (R n ) and satisfies the equation We can now use Theorem 1 of [32] since it is based on Lemma 2 of [32], which imposes no constraint on the sign of σ.Hence, the expansion holds for each a, where P(x) is a polynomial of order ord P(x) Therefore, by the definition of v, we obtain Equation (4).The proof of Lemma 1 is complete.

The Mixed Dirichlet-Steklov-Type Biharmonic Problem
Definition 2. By a solution of the mixed Dirichlet-Steklov-type problem in Equations ( 1) and ( 2) we mean , the following integral identity holds: Theorem 1.The mixed Dirichlet-Steklov-type problem in Equations ( 1) and ( 2) with the condition D(u, Ω) < ∞ has n + 1 linearly independent solutions.
Proof.For any nonzero vector A in R n , we construct a generalized solution u A of the biharmonic Equation ( 1) with the boundary conditions and the condition for A, x ∈ R n , where Ax denotes the standard scalar product of A and x.Such a solution of the problem in Equations ( 1) and ( 6) can be constructed by the variational method [31], minimizing the functional The validity of the condition in Equation ( 7) as a consequence of the Hardy inequality follows from the results in [1][2][3].Now, for any arbitrary number e = 0, we construct a generalized solution u e of Equation ( 1) with the boundary conditions and the condition The solution of the problem in Equations ( 1) and ( 8) is also constructed by the variational method with the minimization of the corresponding functional in the class of admissible functions The condition in Equation ( 9) as a consequence of the Hardy inequality follows from the results in [1][2][3].
Obviously, v is a solution of the problem in Equations ( 1) and ( 2): One can easily see that v ≡ 0 and D(v, Ω) < ∞. 1) and ( 2) with the condition D(v A , Ω) < ∞, and, moreover,

To each nonzero vector
Let A 0 , A 1 , . . ., A n be a basis in R n+1 .Let us prove that the corresponding solutions v A 0 , v A 1 , . . ., v A n are linearly independent.Let Let us show that Consequently, T = ∑ n i=0 C i A i = 0, and since the vectors A 0 , A 1 , . . ., A n are linearly independent, we obtain C i = 0, i = 0, 1, . . ., n.
Let us prove that each solution u of the problem in Equations ( 1) and ( 2) with the condition D(u, Ω) < ∞ can be represented as a linear combination of the functions v A 0 , v A 1 , . . ., v A n , i.e., Since A 0 , A 1 , . . ., A n is a basis in R n+1 , it follows that there exists constants C 0 , C 1 , . . ., C n such that We set Obviously, the function u 0 is a solution of the problem in Equations ( 1) and ( 2), and Let us show that u 0 ≡ 0, x ∈ Ω.To this end, we substitute the function ϕ(x) = u 0 (x)θ N (x) into the integral identity in Equation ( 5) for the function u 0 , where θ where By applying the Cauchy-Schwarz inequality and by taking into account the conditions D(u 0 , Ω) < ∞ and χ(u 0 , Ω) < ∞, one can easily show that J 1 (u 0 ) → 0 and J 2 (u 0 ) → 0 as N → ∞.Consequently, by passing to the limit as N → ∞ in Equation (10), we obtain Using the integral identity we find that if u 0 (x) is a solution of the homogeneous problem in Equations ( 1) and ( 2), then ∆u 0 = 0.
Theorem 2. The mixed Dirichlet-Steklov-type problem in Equations ( 1) and ( 2) with the condition D a (u, Ω) < ∞ has: (i) the trivial solution for n − 2 ≤ a < ∞, n > 4; (ii) n linearly independent solutions for n − 4 ≤ a < n − 2, n > 4; (iii) n + 1 linearly independent solutions for −n ≤ a < n − 4, n > 4; and (iv) k(r, n) linearly independent solutions for −2r The proof of Theorem 2 is based on Lemma 1 about the asymptotic expansion of the solution of the biharmonic equation and the Hardy type inequalities for unbounded domains [1][2][3].In Case (iv), we need to determine the number of linearly independent solutions of the biharmonic Equation (1), the degree of which not exceed the fixed number.
It is well know that the dimension of the space of all polynomials in R n of degree ≤ r is equal to ( r+n n ) [34].Then, the dimension of the space of all biharmonic polynomials in R n of degree ≤ r is equal to since the biharmonic equation is the vanishing of some polynomial of degree r − 4 in R n .If we denote by k(r, n) the number of linearly independent polynomial solutions of Equation ( 1) whose degree do not exceed r and by l(r, n) the number of linearly independent homogeneous polynomials of degree r, that are solutions of Equation ( 1), then Further, we prove that the mixed Dirichlet-Steklov-type problem in Equations ( 1) and (2) with the condition D a (u, Ω) < ∞ for −2r + 2 − n ≤ a < −2r + 4 − n has equally k(r, n) linearly independent solutions.
is the space obtained by the completion of C ∞ 0 (Ω) with respect to the norm ||u(x); H m (Ω)||.
is the space obtained by the completion of C ∞ 0 (Ω) with respect to the family of semi-norms