Uniqueness Results for Higher Order Elliptic Equations in Weighted Sobolev Spaces

TheDirichlet problem for polyharmonic equations in bounded domains of R has been studied, among the first, by Sobolev in [1]. The problem was developed in various directions. For instance, Vekua in [2, 3] considers different boundary value problems in not necessarily bounded domains for harmonic, biharmonic, andmetaharmonic functions. Successively, analogous problems in more general cases, for what concerns domains and operators, have been studied with different methods by many authors (see, e.g., [4–7]). In particular, in [7], the author obtains a uniqueness result for the Dirichlet problem for polyharmonic operators of order 2m in polyhedral angles ofR.This result has been later on generalized, in [5], to the case of operators in divergence form of order 2m with discontinuous bounded measurable elliptic coefficients. In [6] the authors study a boundary value problem for biharmonic functions in presence of nonregular points on the boundary of the domain. It is well known that in the neighborhood of these singular points (corners or edges) the solution of the problem presents a singularity that can be characterized by the presence of a suitable weight. Uniqueness results for different Dirichlet problems in weighted Sobolev spaces for different classes of weights can be found in [8–12]. Studies of Dirichlet problems in the framework of weighted Sobolev spaces and in the case of unbounded domains can be found in [13–22]. In this paper, we extend the results of [5, 7] to the case of weighted Sobolev spaces. More precisely, we prove some uniqueness results for the solution of two kinds of Dirichlet boundary value problems for secondand fourth-order linear elliptic differential equations with discontinuous coefficients in the polyhedral angle Rnl , 0 ≤ l ≤ n − 1, n ≥ 2, in weighted Sobolev spaces. The first problem we consider is the following:


Introduction
The Dirichlet problem for polyharmonic equations in bounded domains of R  has been studied, among the first, by Sobolev in [1].
The problem was developed in various directions.For instance, Vekua in [2,3] considers different boundary value problems in not necessarily bounded domains for harmonic, biharmonic, and metaharmonic functions.Successively, analogous problems in more general cases, for what concerns domains and operators, have been studied with different methods by many authors (see, e.g., [4][5][6][7]).
In particular, in [7], the author obtains a uniqueness result for the Dirichlet problem for polyharmonic operators of order 2 in polyhedral angles of R  .This result has been later on generalized, in [5], to the case of operators in divergence form of order 2 with discontinuous bounded measurable elliptic coefficients.
In [6] the authors study a boundary value problem for biharmonic functions in presence of nonregular points on the boundary of the domain.It is well known that in the neighborhood of these singular points (corners or edges) the solution of the problem presents a singularity that can be characterized by the presence of a suitable weight.
In this paper, we extend the results of [5,7] to the case of weighted Sobolev spaces.More precisely, we prove some uniqueness results for the solution of two kinds of Dirichlet boundary value problems for second-and fourth-order linear elliptic differential equations with discontinuous coefficients in the polyhedral angle R   , 0 ≤  ≤  − 1,  ≥ 2, in weighted Sobolev spaces.
The first problem we consider is the following: where, for  ∈ N 0 and  ∈ R,  ,2  (Ω) denotes a weighted Sobolev space where the weight is a power of the distance from the origin, In both cases the coefficients   belong to some weighted Sobolev spaces.The main tool in our analysis is a generalization of the Hardy's inequality proved by Kondrat' ev and Olènik in [23].
In the present paper we use the following notation: (i)  ⊂ R  is a cone with vertex in the origin of coordinates; (ii)   ,  > 0, is the open ball of center in the origin and radius ; (iii)   =  ∩   ; (iv) for every  ∈ {0, . . .,  − 1}, is the "polyhedral angle" with vertex in the origin; To prove our main results, consisting in two uniqueness theorems, we will use the following inequality.We observe that this is a slightly modified version of a generalized Hardy's inequality that was proved by Kondrat' ev and Olènik in [23], adapted to our needs (see also [5]).

Dirichlet Problem for Second-Order Elliptic Equations
We consider the following differential operator in divergence form in the polyhedral angle R   , 0 ≤  ≤  − 1: where the coefficients   are measurable functions such that there exist two positive constants  and  such that We study the Dirichlet problem where Definition 4. We say that a function  is a generalized solution of problem (11) if it satisfies the integral identity for any  > 0 and any function Now we prove our first uniqueness result.
Proof.Let Θ() be an auxiliary function in where () is such that 0 ≤ () ≤ 1.Let us also assume that there exists a positive constant  0 such that Set, for any  > 0, Note that the function Θ  is such that, for any  = 1, . . ., , one has (R   ) be a generalized solution of problem (11), with  = 0. We put Clearly, by definition of Θ  and as a consequence of our boundary condition, one has that ( 2 ).Thus, using V  as test function in (12), we get From ( 10), (16), and (18) we deduce that there exists a positive constant  1 =  1 (, ) such that where   denotes the modulus of the gradient of .
By applying Young's inequality one gets that for any  > 0 Thus, taking into account (14) and applying the generalized Hardy's inequality (8) (with  = 2 and  = 2) to the second term in the right-hand side of ( 20), we deduce that if From the ellipticity condition in (10) and for  = / 1  0 , we have where the constant  2 =  2 (, , ,  0 , ).
Thus for any  > 0 and for any  >  we obtain Since  is a generalized solution of problem (11), with  = 0, and the constant  2 does not depend on the radius  and on the solution , the right-hand side of ( 23) tends to zero when  → +∞ and then This implies that therefore By Proposition 1 we deduce that if the solution  ∈  1,2  (  ) with  ≤ 0, then  ∈  1,2 (  ), for any  > 0. On the other hand, if  > 0 for any  ∈ [1, 2[ there exists  0 =  0 () > 0 such that if 0 <  ≤  0 /2, then  ∈  1, (  ) for any  > 0. Thus, by (26) the function () is a constant in R   , and since  ∈

Dirichlet Problem for 4th-Order Elliptic Equations
Let us now consider the following differential operator of 4th order in the polyhedral angle R   , 0 ≤  ≤  − 1, where   are measurable symmetric coefficients and there exist two positive constants  and  such that We want to prove a uniqueness result for the solution of the Dirichlet problem where  ∈  2 − (R   ).
Definition 6.We say that a function  is a generalized solution of problem (29) if it satisfies the integral identity for any  > 0 and any function The result is the following.

2 International
Journal of Differential Equations ) Thus, applying repeatedly the generalized Hardy's inequality (8) (with  = 2 and  = 2 to the third integral on the righthand side and with  = 2 and  = 2 − 2 to the last integral on the right-hand side and then again with  = 2 and  = 2), we deduce that if  ̸ = (2 − )/2, (4 − ) ) ( Observe that the definition of Θ  together with the boundary condition satisfied by  gives that V  ∈ Hence, by the symmetry of   , if we take V  as test function in (30) we get           (Θ  )   ()          (Θ  )     ()  = 0.               (Θ  )   () (Θ  )     ()      .