On the solvability conditions of the first boundary value problem for a system of elliptic equations that strongly degenerate at a point
© Rutkauskas; licensee Springer. 2011
Received: 11 April 2011
Accepted: 22 August 2011
Published: 22 August 2011
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 well-posed Dirichlet type problem in the class of smooth functions are given and sufficient conditions on the existence and uniqueness of the solutions are obtained.
1 Introduction and statement of the problems
is considered under the condition a(r) = O(r2α), 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 well-posed. (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.
S R = ∂∑ R , f = (f1, f2, ..., f N } and u = (u1, u2, ..., 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.
which correspond to the blocks of Jordan matrix JΛ, 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 λ0 = 0 and, for convenience, only one eigen vector corresponds to this eigen-value of Λ. Then, Re λ i < 0 for the rest , since the matrix Λ is non-negatively defined. As mentioned above, the solvability of a Dirichlet type problem under the condition Re λ i < 0 is investigated in [6, 7].
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 D2. 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
represent the particular solutions of system (9).
where M σ is some constant independent of n, and (σ = 1, 2, ...).
Proof. We prove relation (15) by induction.
with some constant M1 independent of n holds. Thus, the validity of (15) is proved for σ = 1.
Therefore, there exists a constant M k such that (15) holds for σ = k under the condition .
If , then the first integral on the right-hand side of (14) converges as r ∈ (0, R), and, evidently, . ■
under the conditions of Lemma 1.
Note that w1 ≡ 1 and w2 = r-2n-1are 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-2n-1) 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 non-negative integer. (Note that α0 = 0.)
We use below denotation in the case ψ (x) ≡ 1.
because of (16), i.e., Q n is the matrix solution of Equation 12. Evidently, equality (18) follows from (14).
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).
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 αk+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.
where c nm is arbitrary constant column vector.
3 Existence and uniqueness of the solutions of problems D1and D2
of the particular solutions obtained above. Note, if for some k0, 1 ≤ k0 ≤ s -1, then α k = 0 for all natural k ≤ k0 - 1. (Such a situation can come to exist, if α < 2.) Therefore, all the sums in (23), in which the inequality αk-1<α k is impossible, are taken to be equal to zero.
due to both (18) and (22).
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: x1 = r sin ϑcos φ, x2 = r sin ϑsin φ, and x3 = r cos ϑ.
Assume that condition (26) is fulfilled in addition to the smoothness of g. Then, , i.e., series (27) converges (component-wise) uniformly and absolutely on the sphere S R .
of the series on the right-hand 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 .
in for ∀n ≥ αk-l. 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 .
in (i = 1,2, and 3).
We prove thereby the existence of the solution of problem D1, if both α and s are related by (33).
The uniqueness of the solutions of both the problems D1 and D2 yields the following lemma.
Lemma 2. Let v = (v1, v2, ..., v s ) be a solution of problem D1 or problem D2 with the homogeneous Dirichlet condition . If relation (33) holds, then v i = 0 in .
Proof. Assume that v = (v1, v2, ..., v s ) is a solution of problem D1. Since Δv1 = 0 in and , we get that v1 ≡ 0 in because of the relation v1(x) = o(r-1), as x → 0, which holds because of the validity of condition (11). Then, it follows from system (9) that Δv2 = 0 in . Both the conditions and v2(x) = o(r-1), as x → 0, yield the identity v2 ≡ 0 in to (11). Continuing this process, we obtain that all the components in .
If v = (v1, v2, ..., v s ) is a solution of problem D2, 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 ∈ C2(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 D1, which can be represented by formulas (28)-(30). If orthogonality conditions (34) hold, then there exists a unique solution v of problem D2 with the components v i of the shape (35).
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