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 polyhedral angles, in weighted Sobolev spaces.

1. Introduction

The Dirichlet problem for polyharmonic equations in bounded domains of 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., [4–7]).

In particular, in [7], the author obtains a uniqueness result for the Dirichlet problem for polyharmonic operators of order 2m in polyhedral angles of . 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 second- and fourth-order linear elliptic differential equations with discontinuous coefficients in the polyhedral angle , 0 ≤ l ≤ n − 1, n ≥ 2, in weighted Sobolev spaces.

The first problem we consider is the following:

(1)

where, for

and

denotes a weighted Sobolev space where the weight is a power of the distance from the origin,

is the closure of

, and

; see Section 2 for details.

The second problem we study is

(2)

In both cases the coefficients a_ij 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].

2. Preliminary Results

Let Ω be an open subset of

with n ≥ 2, whose boundary contains x = 0. For

and

denotes the space of all functions

such that |x|^sD^αu ∈ L²(Ω) for |α | ≤ k, normed by

(3)

From [24] and Propositions 6.3 and 6.5, we get the following.

Proposition 1. If G is a bounded open subset in with 0 ∈ ∂G, then

(4)

Furthermore, for each q ∈ [1,2[ there exists ϵ₀ = ϵ₀(q) > 0 such that

(5)

In the present paper we use the following notation:

(i)
is a cone with vertex in the origin of coordinates;
(ii)
B_R, R > 0, is the open ball of center in the origin and radius R;
(iii)
V_R = V∩B_R;
(iv)
for every l ∈ {0, …, n − 1},
(6)
is the “polyhedral angle” with vertex in the origin;
(v)
is the half-space;
(vi)
.

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]).

Lemma 2 (generalized Hardy’s inequality). Let p > 1 and be such that r + n − p ≠ 0. Assume that for a sufficiently smooth function g the following condition is fulfilled:

(7)

where ▽g = (∂g/∂x₁, …, ∂g/∂x_n) is the gradient of the function g and 0 < R₁ < R₂. Then, there exist two constants M, K > 0 such that

(8)

where K does not depend on the function g, R₁, and R₂. If, in addition, g(0) = 0 then M = 0.

Remark 3. We remark that there are always important restrictions on the dimension n of the space, the order of “singularity” r, and the summability exponent p (see, e.g., [23, 25–29], where different variants of Hardy or Caffarelli-Kohn-Nirenberg type inequalities are proved).

3. Dirichlet Problem for Second-Order Elliptic Equations

We consider the following differential operator in divergence form in the polyhedral angle

, 0 ≤ l ≤ n − 1:

(9)

where the coefficients a_ij are measurable functions such that there exist two positive constants λ and μ such that

(10)

We study the Dirichlet problem

(11)

where

Definition 4. We say that a function u is a generalized solution of problem (11) if it satisfies the integral identity

(12)

for any R > 0 and any function

Now we prove our first uniqueness result.

Theorem 5. Let be a generalized solution of problem (11), with f = 0. Then there exists ϵ₀ > 0 such that if s ≤ ϵ₀/2 and s ≠ (2 − n)/2 one has u ≡ 0 in .

Proof. Let Θ(t) be an auxiliary function in defined by

(13)

where θ(t) is such that 0 ≤ θ(t) ≤ 1. Let us also assume that there exists a positive constant K₀ such that

(14)

Set, for any R > 0,

(15)

Note that the function Θ_R is such that, for any j = 1, …, n, one has

(16)

Let be a generalized solution of problem (11), with f = 0. We put

(17)

Clearly, by definition of Θ_R and as a consequence of our boundary condition, one has that

Thus, using v_R as test function in (12), we get

(18)

From (10), (16), and (18) we deduce that there exists a positive constant K₁ = K₁(n, μ) such that

(19)

where u_x denotes the modulus of the gradient of u.

By applying Young’s inequality one gets that for any ϵ > 0

(20)

Thus, taking into account (14) and applying the generalized Hardy’s inequality (8) (with p = 2 and r = 2s) to the second term in the right-hand side of (20), we deduce that if s ≠ (2 − n)/2,

(21)

From the ellipticity condition in (10) and for ϵ = λ/K₁K₀, we have

(22)

where the constant K₂ = K₂(n, λ, μ, K₀, K).

Thus for any P > 0 and for any R > P we obtain

(23)

Since u is a generalized solution of problem (11), with f = 0, and the constant K₂ does not depend on the radius R and on the solution u, the right-hand side of (23) tends to zero when R → +∞ and then

(24)

This implies that

(25)

therefore

(26)

By Proposition 1 we deduce that if the solution

with s ≤ 0, then u ∈ W^1,2(Q_P), for any P > 0. On the other hand, if s > 0 for any q ∈ [1,2[ there exists ϵ₀ = ϵ₀(q) > 0 such that if 0 < s ≤ ϵ₀/2, then u ∈ W^1,q(Q_P) for any P > 0. Thus, by (26) the function u(x) is a constant in

, and since

one concludes that u = 0 in

4. Dirichlet Problem for 4th-Order Elliptic Equations

Let us now consider the following differential operator of 4th order in the polyhedral angle

, 0 ≤ l ≤ n − 1,

(27)

where a_ij are measurable symmetric coefficients and there exist two positive constants λ and μ such that

(28)

We want to prove a uniqueness result for the solution of the Dirichlet problem

(29)

where

Definition 6. We say that a function u is a generalized solution of problem (29) if it satisfies the integral identity

(30)

for any R > 0 and any function

The result is the following.

Theorem 7. Let be a generalized solution of problem (29), with f = 0. Then there exists ϵ₀ > 0 such that if s ≤ ϵ₀/2 and s ≠ (2 − n)/2, (4 − n)/2 one has u ≡ 0 in .

Proof. We shall rely on the methods developed in [5, 7]. We consider the function Θ_R(x) defined in (13) and satisfying (14). Furthermore, we assume that there exists a positive constant K₁ such that

(31)

Note that the function Θ_R is such that, for any i, j = 1, …, n, one has (16) and

(32)

where δ_ij denotes the Kronecker delta.

Again we put

(33)

where

is a generalized solution of problem (29), with f = 0.

Observe that the definition of Θ_R together with the boundary condition satisfied by u gives that . Hence, by the symmetry of a_ij, if we take v_R as test function in (30) we get

(34)

From (28) and (34) we deduce that

(35)

By applying (16), (32), and Young’s inequality one gets that there exist two positive constants K₂ = K₂(n, λ, μ, K₀) and K₃ = K₃(n, λ, μ, K₀, K₁) such that for any ε, ε₁ > 0

(36)

Thus, applying repeatedly the generalized Hardy’s inequality (8) (with p = 2 and r = 2s to the third integral on the right-hand side and with p = 2 and r = 2s − 2 to the last integral on the right-hand side and then again with p = 2 and r = 2s), we deduce that if s ≠ (2 − n)/2, (4 − n)/2,

(37)

where the constant K₄ = K₄(n, λ, μ, K₀, K₁, K).

Thus for any P > 0 and for any R > P we obtain

(38)

Now, arguing as in the proof of Theorem 5, since u is a generalized solution of problem (29), with f = 0, the right-hand side of (38) tends to zero when R → +∞ and then

(39)

This implies that

(40)

therefore

(41)

In view of Proposition 1 we obtain that if the solution with s ≤ 0, then u ∈ W^2,2(Q_P), for any P > 0, while if s > 0 for any q ∈ [1,2[ there exists ϵ₀ = ϵ₀(q) > 0 such that if 0 < s ≤ ϵ₀/2, then u ∈ W^2,q(Q_P) for any P > 0. Thus, by (41) the function u_x is constant a.e. in Q_P, and since one concludes that u_x = 0 a.e. in Q_P, for any P > 0. The thesis follows then as the one of Theorem 5.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Uniqueness Results for Higher Order Elliptic Equations in Weighted Sobolev Spaces

Abstract

1. Introduction

2. Preliminary Results

3. Dirichlet Problem for Second-Order Elliptic Equations

4. Dirichlet Problem for 4th-Order Elliptic Equations

Conflicts of Interest

References

Citing Literature

References

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Uniqueness Results for Higher Order Elliptic Equations in Weighted Sobolev Spaces

Abstract

1. Introduction

2. Preliminary Results

3. Dirichlet Problem for Second-Order Elliptic Equations

4. Dirichlet Problem for 4th-Order Elliptic Equations

Conflicts of Interest

References

Citing Literature

References

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