In the first part of the paper, the authors obtain the asymptotics of Green’s function of the first boundary value problem for the heat equation in an m-dimensional cone K. The second part deals with the first boundary value problem for the heat equation in the domain . Here the right-hand side f of the heat equation is assumed to be an element of a weighted -space. The authors describe the behavior of the solution near the -dimensional edge of the domain.
The paper is concerned with the first boundary value problem for the heat equation
in the domain
where is a cone in , , Ω denotes a subdomain of the unit sphere, and is the -dimensional edge of . We are interested in the asymptotics of solutions in the class of the weighted Sobolev spaces . Here the space is defined for an arbitrary integer and real , , β as the set of all function on with the finite norm
For the case of smooth boundary ∂Ω (of class ), the asymptotics of solutions was obtained in our previous paper . For the particular case , , we refer also to the paper  by Kozlov and Maz’ya, and for the case , , to the paper  by de Coster and Nicaise. The goal of the present paper is to describe the asymptotics of solutions with a remainder in under minimal smoothness assumptions on the boundary. Throughout the paper, we assume that .
The paper consists of two parts. The first part (Section 1) deals with the asymptotics of the Green function for the heat equation in the cone K. We obtain the same decomposition
as in [4,5] (for the definition of , , , and , see Section 1.1). However, the proof in [4,5] does not work if ∂Ω is only of the class . We give a new proof, which is completely different from that in [4,5]. Our tools are estimates for solutions of the Dirichlet problem for the Laplace equation in a cone in weighted Sobolev spaces and asymptotic formulas for solutions of this problem which were obtained in the papers [6,7] by Maz’ya and Plamenevskiĭ. Moreover, we use the estimates of the Green function in the recent paper  by Kozlov and Nazarov. In contrast to the case , the estimates for the second order - and -derivatives of the remainder contain an additional factor with a negative exponent −ε. Here, is the distance from the boundary of ∂K.
In the second part of the paper (Section 2), we apply the results of Section 2 in order to obtain the asymptotics of solutions of the problem (1), (2) for . We show that, under a certain condition on β, there exists a solution of the form
Φ denotes the fundamental solution of the heat equation in . The proof of this result (Theorem 2.2) is essentially the same as in . However, the proofs of some lemmas in  have to be modified under our weaker assumptions on ∂Ω.
where T and R are certain smooth functions on and , respectively (see the beginning of Section 3 for their definition). This extends the result of [, Corollary 4.5] to the case .
1 The Green function of the heat equation in a cone
We start with the problem
Furthermore, if ( are defined below), and ζ is a function in equal to one in a neighborhood of the point . Here is the space of all functions on such that for . The goal of this section is to describe the behavior of the Green function for .
1.1 Asymptotics of Green’s function
Let be the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on Ω (with the Dirichlet boundary condition) counted with their multiplicities, and let be an orthonormal (in ) sequence of eigenfunctions corresponding to the eigenvalues . Furthermore, we define
By [, Theorem 3],
for , . Here denotes the distance of the point from the boundary ∂K. Furthermore, is defined as zero for , while is an arbitrarily small positive real number if . Actually, the estimate (6) is proved in  only for , but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in  do not depend on t, then one can use the same argument as in the proof of [, Theorem 3] when treating the derivatives along the edge of the domain . This argument shows that the kth derivative with respect to t will bring only an additional factor to the right-hand side of (6).
Proof The solution of the problem
is given by the formula
Then it follows from (8) and (9) that
Comparing this with the formula
we get (7). □
In the sequel, σ is an arbitrary real number satisfying the conditions
By the induction hypothesis, we have
Using the formula
( for ). Let χ be a smooth function with compact support on such that for . Using the notation , the function χ can be also considered as a function in K. Since for , we have and for all , . Consequently, and
Applying [, Theorem 4.2], we obtain
we conclude that
It remains to show that the coefficients
Combining the last equality with (21), we get the representation
which has the solution
Using (6) and (17), one gets the estimate
by means of (20). Hence,
1.2 Point estimates for the remainder in the asymptotics of Green’s function
with a constantcindependent ofu.
The result follows. □
Using the last two lemmas, we can prove the following theorem.
Theorem 1.1Suppose thatσis a real number satisfying (10). Then
If , then the assertion of the theorem follows from [, Theorem 3]. Suppose that , , and that the theorem is proved for . We set if . In the case , let be an arbitrary real number satisfying the inequalities and . By the induction hypothesis, we have
Thus, by [, Theorem 4.1], there exists a constant c such that
Finally, it follows from the inequality
Consequently, by (25),
with a positive constant κ. Applying the estimate
for (cf. [, Lemma 1.2.3]), we obtain (23) for .
with positive constants and (cf. [, Chapter VI, § 2.1]). Moreover, ρ satisfies the inequality
We consider the function
(see (26)). Consequently by [, Theorem 4.1], the function satisfies the estimate
is the Green function of the problem (1), (2). We consider the solution
of the problem (1), (2).
We also consider the decomposition
is an extension of the function
In this subsection, we assume that , where p and β satisfy (28). First, we prove that . This was shown in [, Corollary 2.3] for the case . In the case , we must keep in mind that the second-order derivatives of the eigenfunctions must not be bounded. Then we have the estimate
for , where for and is an arbitrarily small positive real number if . However, this requires only a small modification of the proof in .
Proof A simple calculation (see the proof of [, Corollary 1]) yields
are satisfied on the support of the kernel
Next, we estimate the first-order x-derivatives of the remainder v. For this, we employ the following lemma (cf. [, Lemma A.1]).
In the proof of the following assertion, we use another decomposition of the remainder v as in [, Lemma 2.4]. This allows us to apply directly the estimate in Theorem 1.1.
with a constantcindependent off. The same is true for the functionw.
We show that the integral operators with the kernels
we get the representation
It is no restriction to assume that in addition to (30) and (31). Therefore, we can apply Lemma 2.2 with , and to the integral operator with the kernel . It follows that the integral operator with the kernel is bounded in for . Consequently, the integral operator with the kernel
Furthermore, the assertions of [, Lemmas 2.5, 2.6, Theorem 2.7] are also valid if ∂Ω is only of the class . The proof under this weaker assumption on Ω does not require any modifications of the method in . We give here only the formulation of [, Theorem 2.7].
of the solution (29), where is defined by (34). Our goal is to show that if p and β satisfy the condition (28). For the proof, we will use the next lemma which follows directly from [, Theorem 3.8].
Then the inequality
The condition (ii) of the last lemma can be verified in some cases by means of the following lemma (cf. [, Lemma 10]).
Lemma 2.5Suppose that the kernel of the integral operator (39) satisfies the estimate
Proof We have
Analogously to the proof of Lemma 2.1, we obtain
with arbitrary a and b. Thus, as in the first part of the proof, we conclude that (and therefore also the adjoint operator of ) is bounded in for . This means that is bounded in for all . The lemma is proved. □
By means of Lemma 2.5, it is also possible to prove the assertion of [, Theorem 3.7] under the weaker assumption on Ω of the present paper.
Using Theorem 1.1, we get
by means of (37). We may assume, without loss of generality, that in addition to (30) and (31). Then we conclude from Lemma 2.5 that (44) is valid for and . Hence, by Lemma 2.4, the operator is bounded in for if .
By means of 1.1, we obtain
Analogously, the estimate (6) implies
where a is an arbitrary real number, since on the support of the function . Applying Lemma 2.5, we obtain (45) for and . Using the representation for , the estimate (37), and the fact that on the support of , we obtain
We may assume again that in addition to (30) and (31). Then it follows from Lemma 2.5 that (45) is valid for and . Therefore, by Lemma 2.4, the operator is bounded in for if . This means that is bounded in for all q if . The proof of the theorem is complete. □
3 Another representation for the coefficients
As was proved [, Lemma 4.1], the functions in Theorem 2.1 can be replaced by other extensions of the functions provided these extensions also satisfy the conditions (40) and (41). Note that the proof of this assertion in  is also correct under our assumptions on the boundary of Ω. Moreover, it was proved in [, Lemma 4.4], for the particular case , that the extension
We extend the result of [, Lemma 4.4] to the case . First, note that , where is the integral operator
with the kernel
Our goal is to show that the operator
where and . As was shown in , the operator is bounded in if or . In order to prove the boundedness in for , we verify the condition (ii) of Lemma 2.4. For this, we apply the following lemma.
Lemma 3.1Suppose that the kernel of the integral operator (39) satisfies the condition
Thus, we can apply Lemma 2.5. □
Proof For the case , we refer to [, Lemma 4.4].
for the -derivatives of as in the proof of [, Lemma 4.4], where
Finally, (cf. formulas (4.7) and (4.8) in ), we get the estimates
with the same M as before. This implies
Therefore, it follows from Lemma 3.1 that
Using the last lemma, we obtain the following result which generalizes [, Corollary 4.5].
Proof By Lemma 3.2, the functions satisfy the same condition (43) as the functions in Theorem 2.2. Thus, it follows from [, Lemma 4.1] that
The authors declare that they have no competing interests.
The authors achieved the key results of the paper during a research stay of JR in Linköping in October 2012. Both authors read and approved the final manuscript.
The paper partially arose during the stay of J. Rossmann in Linköping in October 2011. The second author thanks the Department of Mathematics at the University of Linköping for the hospitality.
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