Abstract
A system of elliptic equations which are irregularly degenerate at an inner point is considered in this article. The equations are weakly coupled by a matrix that has multiple zero eigenvalue and corresponding to it adjoint vectors. Two statements of a wellposed Dirichlet type problem in the class of smooth functions are given and sufficient conditions on the existence and uniqueness of the solutions are obtained.
Keywords:
systems of elliptic equations; degenerate elliptic equations; boundary value problems; Dirichlet type problem1 Introduction and statement of the problems
The first results in the area of boundary value problems for an elliptic equation with degeneracy at an inner point of the considered domain are obtained in [1]. In that study, the Dirichlet problem for a weakly (regularly) degenerating elliptic equation with the main part of Laplace's operator is studied. These results are developed in [2], where the degenerate elliptic operator is generalized and, over and above, the second boundary value problem is investigated. In [3], the existence of a weak solution to the Dirichlet problem for an elliptic equation degenerating at isolated points in the class of Hölder functions is proved. In the case of the strong (irregular) degeneracy, can new effects emerge which influence the wellposedness of the boundary value problems. For instance, in [4], it is shown that in a wellposed Dirichlet type problem the asymptotic of the solution near the degeneracy point is supposed to be known. Many more difficulties come into being in the investigation of the systems of degenerate elliptic equations. Some results for weakly related degenerate elliptic systems are obtained in [57]. Particularly, these articles deal with Dirichlet type problems for the elliptic system
where r = x, a is a continuous function such that a(r) = o(1) as r → 0, and a(r) > 0 for r > 0, x = 0 is an inner point of domain D, Δ is Laplace's operator, B_{i}(x) and C(x) are diagonal and square matrices, consequently, which are smooth enough in . In [5,6], the Dirichlet problem in the class of vector functions u bounded in D_{0 }= D\{x = 0} is solved under the assumption that elements of the matrices B_{i}(x) tend to zero, as x → 0, fast enough. In [7], a weighted Dirichlet problem with supplementary weighted condition of the shape
is considered under the condition a(r) = O(r^{2α}), α >1, as r → 0. In the same study, Ψ (x) is some matrix entries of which are decreasing as x → 0, and h is a given vector function smooth on the unit sphere. It is noteworthy that the matrix C(x) is assumed to be negatively definite in D, i.e., it does not have any zero eigenvalue. Moreover, C(0) should be a normal matrix for the weighted Dirichlet problem to be wellposed. (If coefficients B_{i}(x) have the main influence to the asymptotic of the solutions of system (1), then the last requirement is dispensable [8,9]). Therefore, it is important to consider the case where C(0) has multiple zero eigenvalue and corresponding to it adjoint vectors.
Hence, the present article deals with a particular case of system (1) of the shape
in the ball ∑_{R }= {x : x <R}⊂ R^{3 }with the Dirichlet condition
In this article, Λ is a real constant nonnegative definite N × N matrix having the eigenvalue λ = 0, q is scalar continuous function positive for r ≠ 0 and such that
S_{R }= ∂∑_{R}, f = (f_{1}, f_{2}, ..., f_{N}} and u = (u_{1}, u_{2}, ..., u_{N}) are the given and unknown vector functions, respectively. (Condition (5) means with respect to system (1) that a(r) vanishes as r → 0 not faster than any power of r.) Hence, the order of system (3) is strongly degenerate at the point x = 0 because of α > 1.
Let S be a nondegenerate matrix such that
is the canonical Jordan form of Λ with m_{i }× m_{i }lower blocks
Multiplying both (3) and (4) from the left by S, we get the system
and the Dirichlet condition
where v = Su, and g = Sf. Therefore, system (6) and Dirichlet condition (7) can be split into p + 1 separate systems
and Dirichlet conditions
which correspond to the blocks of Jordan matrix J_{Λ}, where both v^{i }and g^{i }are m_{i}dimensional vector functions. If Λ is a matrix of simple structure, then all m_{i }= 1, i.e., (6) splits into N separate equations, obviously.
Let λ_{0 }= 0 and, for convenience, only one eigen vector corresponds to this eigenvalue of Λ. Then, Re λ_{i }< 0 for the rest , since the matrix Λ is nonnegatively defined. As mentioned above, the solvability of a Dirichlet type problem under the condition Re λ_{i }< 0 is investigated in [6,7].
The main aim of this article is to give a wellposedness of the Dirichlet type problems to the system
which is in accordance with eigenvalue λ_{0 }= 0 of Λ. In order to avoid the complicated notations, instead of (8), we consider the system
where v = (v_{1}, v_{2}, ..., v_{s}) and L_{s}(0) is a s × s lower Jordan block with zero diagonal entries. It is easily seen that
is the outspread form of (9).
Denote . Let v be the Euclidian norm of a vector v. We propose the two following statements of the Dirichlet type problem to system (9).
Problem D_{1}. Find a solution of Equation 9 that satisfies Dirichlet condition
and relation
Problem D_{2}. Find a solution of Equation 9, such that it satisfies Dirichlet condition (11) and is bounded in .
2 The properties of particular solutions of Equation 8
Let be mth the harmonic of a homogeneous harmonic polynomial of degree n,^{a }i.e., and . Then, (here ω = x /r) is the mth spherical harmonic of order n continuous on the unit sphere S_{1}. Let c_{nm }be any constant vector, and let Q_{n}(r) be a matrix solution of ODEs system
where
and w is an unknown sdimensional vector function. Then, the functions
represent the particular solutions of system (9).
We seek for a solution Q_{n}(r) of system (12), which satisfies condition Q(R) = E, where E is the unit matrix. To this end, on the set of functions ψ bounded on the interval (0, R), we consider the integral operator
and its integer powers
where by definition K^{0}(ψ)(x) ≡ ψ (x). Obviously, according to this definition
Lemma 1. Let relation (5) hold. If , then
where M_{σ }is some constant independent of n, and (σ = 1, 2, ...).
Proof. We prove relation (15) by induction.
Since ψ (r) is bounded on (0, R), inequality (15) holds for σ = 0 with some constant M_{0}. It follows from relation (5) that 0 <q(r) ≤ Mr^{2α }∀r ∈ (0, R), where M is a positive constant. Then, ∀t ∈ (0, R). Assuming that , we obtain that
i.e., the estimate
with some constant M_{1 }independent of n holds. Thus, the validity of (15) is proved for σ = 1.
Let (15) be valid for σ = k 1 under the condition . Then,
i.e., the first integral in expression (14) converges, if , and
Therefore, there exists a constant M_{k }such that (15) holds for σ = k under the condition .
If , then the first integral on the righthand side of (14) converges as r ∈ (0, R), and, evidently, . ■
It is easy to verify that
under the conditions of Lemma 1.
Note that w_{1 }≡ 1 and w_{2 }= r^{2n1 }are linearly independent solutions of the differential equation l_{n }(w) = 0. Thus, if w is the solution of this equation such that w(r) = o(r^{2n1}) as r → 0, then w(r) ≡ const.
Denoting, as usual, by [a] the integer part of the real number a, we introduce the integer , where k is a nonnegative integer. (Note that α_{0 }= 0.)
We use below denotation in the case ψ (x) ≡ 1.
Theorem 1. Let relation (5) hold. If n ≥ α_{s1}, then there exists a unique matrix solution Q_{n}(r) = {q_{nij}(r)} of Equation 12 such that
and
Proof. Let the condition n ≥ α_{s1 }be valid. Then, according to Lemma 1, the functions , , are continuous on the interval (0, R). Introduce the s × s matrix Q_{n}(r) = {q_{nij}(r)} by the formula
Note that estimate (15) yields the relations
This implies the validity of condition (17), because 2n+1 > 2(s 1)(α  1) ≥ 2(i  j)(α  1), for . Moreover,
because of (16), i.e., Q_{n }is the matrix solution of Equation 12. Evidently, equality (18) follows from (14).
It remains to prove the uniqueness of the solution of problems (12), (17), and (18). Let be a matrix solution of system (12) continuous on (0, R), and satisfying both conditions , as r → 0, and , where Θ is zero matrix. Then, the equalities
on the interval (0, R) hold. Since as r → 0, we obtain that on the interval (0, R). Then, the condition yields the identity because of the continuity of the function on (0, R). In such a case, the elements of the second row of matrix satisfy the equation . For the same reason as above, we obtain that on (0, R). Further, continuing this process, we get that on (0, R). Hence, on (0, R). This yields the uniqueness of the solution of problems (12), (17), and (18).
What is the structure of the solutions of system (12) that increase slower than r^{2n1 }in the case where n does not satisfy the condition n ≥ α_{s1}? In order to get the answer to this question, we introduce s × s matrices with entries for s  k + 1 ≤ i ≤ s, and for all rest i and j. (Note that E_{s }= E according to this definition.) Let us compose the matrixes
where Q_{n }is matrix elements of which are given by (19). It is easily seen that , and the elements of rest matrixes are defined by following formula
If n ≥ α_{k1}, then the powers exist for all i and j such that s  k + 1 ≤ j ≤ i ≤ s, and according to (20), the relation
holds. Moreover, we obtain by direct calculation that
for due to the definition of matrix . Hence, there holds the following
Theorem 2. Let relation (5) hold, and let natural k, 1 ≤ k ≤ s  1, be such that α_{k+1 }≤ n α_{k}. Then, there exists a unique matrix solution of Equation 12 such that relation (21) holds, and boundary value condition (22) is satisfied.
The uniqueness of the matrix solution can be proved in the same way as that of the matrix solution Q_{n}. In this case, condition (21) is essential, just similar to condition (17) in Theorem 1.
Hence, we obtain to system (9) the following set of particular solutions (see (13)):
where c_{nm }is arbitrary constant column vector.
3 Existence and uniqueness of the solutions of problems D_{1 }and D_{2}
Let us compose the superposition
of the particular solutions obtained above. Note, if for some k_{0}, 1 ≤ k_{0 }≤ s 1, then α_{k }= 0 for all natural k ≤ k_{0 } 1. (Such a situation can come to exist, if α < 2.) Therefore, all the sums in (23), in which the inequality α_{k1 }<α_{k }is impossible, are taken to be equal to zero.
Evidently, if the series (23) converges and its sum v is twice differentiable in the spherical layer with arbitrarily small δ, then this series satisfies system (9) in the ball . Note that
due to both (18) and (22).
Assume that the boundary vector function g = (g_{1}, g_{2}, ..., g_{s}) (see (10)) is twice differentiable on unit sphere S_{1}. Thus, it can be expressed on the sphere S_{R }by Laplace series [10]:
which converge (componentwise) uniformly and absolutely according to the assumed smoothness of the vector function g. The coefficients in (25) can be calculated as follows ^{b}:
where h_{i }(φ, ϑ) = g_{i}(x), for x = R and , φ, ϑ (0 ≤ φ ≤ 2π, 0 ≤ ϑ ≤ π) are spherical coordinates which are introduced by the rule: x_{1 }= r sin ϑcos φ, x_{2 }= r sin ϑsin φ, and x_{3 }= r cos ϑ.
It is easily seen that series (24) coincides with series (25), if c_{nm }= a_{nm }for n ≥ α_{s 1}, and E_{k}c_{nm }= a_{nm }for α_{k1 }≤ n <α_{k}, , i.e., if components h_{1}, h_{2}, ..., h_{s1 }of vector function h satisfy the following orthogonality conditions
on sphere S_{R}. Let us consider series (23), in which c_{nm }= a_{nm}:
Assume that condition (26) is fulfilled in addition to the smoothness of g. Then, , i.e., series (27) converges (componentwise) uniformly and absolutely on the sphere S_{R}.
We shall prove that series (27) converges uniformly and absolutely in the spherical layer with arbitrarily small δ. Note that components v_{i } of the vector function v = (v_{1}, v_{2}, ..., v_{s}) in (27) can be formally represented in the form
where
The terms
of the series on the righthand side of (29) are harmonic functions in ∑_{R}. Since these series converge uniformly on the sphere S_{R}, they also converge uniformly in ∑_{R}, and their sums w_{k}(r, ω), , are harmonic functions in ∑_{R }because of Harnack's theorem [11].
Further, according to Lemma 1 estimates
hold, where n ≥ α_{kl }and M_{kl }is a constant independent of n. Consequently,
in for ∀n ≥ α_{kl}. Note that the constants M_{k}, , do not depend on n as well as on δ. Evidently, they yield the uniform and absolute convergence of series (30) in .
Let G_{δ}(x, ξ) be the Green function of the Dirichlet problem to Laplace equation in , and let w_{kn}(x) and W_{kn}(x) be the nth partial sum of corresponding series (29) and (30). Since
hold, where dσ_{ξ }is a volume element of . These yield the equalities
Owing to the uniform and absolute convergence in of sequences {w_{kn }(r, ω)} and {W_{kn }(r, ω)}, as n → ∞, we obtain, from (31) and (32), coherently, that the functions w_{k}(r, ω) and W_{k}(r, ω), defined by (29) and (30), are twice differentiable and
Hence, the vector function v = (v_{1}, v_{2}, ..., v_{s}) with the components v_{i }defined by (28)(30) is from class , and it satisfies system (9) in and the Dirichlet condition , only if orthogonality conditions (26) hold. Besides, it follows from Lemma 1 that
Therefore, rv(x) = o(1) as x → 0, if
Note that this inequality holds, if, for instance,
We prove thereby the existence of the solution of problem D_{1}, if both α and s are related by (33).
If the coefficients in (28) and (29) are such that
i.e., the components of the vector function h satisfy the orthogonality conditions
then the solution v of system (8), given by (28)(30), is bounded in and continuous in . Thus, under ortogonality conditions (34), we obtain the solution v = (v_{1}, v_{2}, ..., v_{s}) of problem D_{2 }of the shape
where
The uniqueness of the solutions of both the problems D_{1 }and D_{2 }yields the following lemma.
Lemma 2. Let v = (v_{1}, v_{2}, ..., v_{s}) be a solution of problem D_{1 }or problem D_{2 }with the homogeneous Dirichlet condition . If relation (33) holds, then v_{i }= 0 in .
Proof. Assume that v = (v_{1}, v_{2}, ..., v_{s}) is a solution of problem D_{1}. Since Δv_{1 }= 0 in and , we get that v_{1 }≡ 0 in because of the relation v_{1}(x) = o(r^{1}), as x → 0, which holds because of the validity of condition (11). Then, it follows from system (9) that Δv_{2 }= 0 in . Both the conditions and v_{2}(x) = o(r^{1}), as x → 0, yield the identity v_{2 }≡ 0 in to (11). Continuing this process, we obtain that all the components in .
If v = (v_{1}, v_{2}, ..., v_{s}) is a solution of problem D_{2}, then it satisfies (11), too. This implies the identity v ≡ 0 in , without doubt.
One can summarize the reasoning given above as follows:
Theorem 3. Let g ∈ C^{2}(S_{R}), and let relation (5) hold. If orthogonality conditions (26) are fulfilled, and the parameters α and s satisfy inequality (33), then there exists a unique solution v of problem D_{1}, which can be represented by formulas (28)(30). If orthogonality conditions (34) hold, then there exists a unique solution v of problem D_{2 }with the components v_{i }of the shape (35).
Endnotes
^{a}One can express the spherical function in Cartesian coordinates x = (x_{1}, x_{2}, x_{3}) by formula [12]:
where is adjoint Legendre's function, and i is the imaginary unit. ^{b}Our opinion is that spherical coordinates are more convenient than Cartesian in the calculation of the coefficients a_{nm }of series (25). The matter is such that spherical functions have quite a simple expresion:
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