Abstract
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 mdimensional cone K. The second part deals with the first boundary value problem for the heat equation in the domain . Here the righthand 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.
Introduction
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
In the case , we write . If, moreover, , then we write .
For the case of smooth boundary ∂Ω (of class ), the asymptotics of solutions was obtained in our previous paper [1]. For the particular case , , we refer also to the paper [2] by Kozlov and Maz’ya, and for the case , , to the paper [3] 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 [8] 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
with a remainder . Here, is an extension of the function
Φ denotes the fundamental solution of the heat equation in . The proof of this result (Theorem 2.2) is essentially the same as in [1]. However, the proofs of some lemmas in [1] have to be modified under our weaker assumptions on ∂Ω.
At the end of the paper, we show that the extensions of the functions can be defined as
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 [[1], Corollary 4.5] to the case .
1 The Green function of the heat equation in a cone
We start with the problem
Let be the Green function for the problem (4), (5). It is defined for every as the solution of 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
This means that are the solutions of the quadratic equation . Obviously, and for .
By [[8], 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 [8] only for , but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in [8] do not depend on t, then one can use the same argument as in the proof of [[8], 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 righthand side of (6).
The following lemma will be applied in the proof of Lemma 1.2. Here and in the sequel, we use the notation and .
Lemma 1.1Letbe the Green function introduced above, and letdenote the Green function of the initialboundary value problem
Then
Proof The solution of the problem
is given by the formula
We define
Then it follows from (8) and (9) that
Furthermore,
Therefore,
Comparing this with the formula
we get (7). □
In the sequel, σ is an arbitrary real number satisfying the conditions
where
and . Here, we used the notation
We define as the weighted Sobolev space with the norm
Lemma 1.2Suppose thatσis a real number such thatandis not integer for. Furthermore, letand. Then
Proof We prove the lemma by induction in .
First, let . Then it follows from [[7], Corollary 4.1 and Theorem 4.2] (see also [[6], Theorem 3.2]) that for all , , , where . Thus, the assertion of the lemma is true for .
Suppose the assertion is proved for . Now let . We set if and if , where ε is a sufficiently small positive number. Then
By the induction hypothesis, we have
where is given by (11) (with instead of σ and instead of ), , . The coefficients in are given by (13) and satisfy the equation . Therefore,
for , . Obviously, for . Using the same equality for the Green function , we obtain
Furthermore,
Using the formula
we get
where
( 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 [[7], Theorem 4.2], we obtain
where . The coefficients are given by the formula
where . The integral in (16) is well defined, since
and , , for . The remainder and the coefficients in (15) satisfy the estimate
where
and . Obviously, for . Using (18) and the equality
we conclude that
It remains to show that the coefficients
in (15) have the form (13) for . First, note that
contain only functions with . Thus, the orthogonality of the functions implies
for . Applying Lemma 1.1, we conclude that has the form
where . Since and for all , it follows from (18) that
The function on the righthand side belongs to for all , , , while the lefthand side belongs only to if
Combining the last equality with (21), we get the representation
Inserting this into the equation , we obtain
The substitution leads to the differential equation
which has the solution
with arbitrary constants and . Consequently,
Using (6) and (17), one gets the estimate
with certain functions for . Thus, the constant in (22) must be zero. Integrating (19), we get
by means of (20). Hence,
The integral on the lefthand side is equal to . Thus, we get and
This means that the formula (13) is valid for the coefficients if . The proof of the lemma is complete. □
1.2 Point estimates for the remainder in the asymptotics of Green’s function
We are interested in point estimates for the remainder in Lemma 1.2 in the case . For this, we need the following lemma.
Lemma 1.3Suppose thatand, where. Then
with a constantcindependent ofu.
Proof Let be a point int K, and let be a ball centered at with radius . We introduce the new coordinates and set . Obviously, the point has the distance 1 from ∂K. Hence,
This implies
The result follows. □
Using the last two lemmas, we can prove the following theorem.
Theorem 1.1Suppose thatσis a real number satisfying (10). Then
where
for, , . Herefor, whileis an arbitrarily small positive real number if.
Proof Since for small positive ε, we may assume, without loss of generality, that is not integer for . We prove the theorem by induction in .
If , then the assertion of the theorem follows from [[8], 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
where is given by (11) (with instead of σ and instead of ). Since for sufficiently small δ, it follows from the induction hypothesis that
for , , . As was shown in the proof of Lemma 1.2, the remainder admits the decomposition
where
and for , , . Here . Furthermore (cf. (14)),
Let χ be a smooth cutoff function on the interval , in and on . We define for , . Then
where
Thus, by [[7], Theorem 4.1], there exists a constant c such that
for all , , . We estimate the norm of f. Using (24), we get
Since vanishes outside the region and , the estimate (24) also yields
Finally, it follows from the inequality
that
Consequently, by (25),
with a positive constant κ. Applying the estimate
for (cf. [[9], Lemma 1.2.3]), we obtain (23) for .
It remains to prove the estimate (23) for . Let be the “regularized distance” of the point to the boundary ∂K, i.e., ρ is a smooth function in K satisfying the inequalities
with positive constants and (cf. [[10], Chapter VI, § 2.1]). Moreover, ρ satisfies the inequality
We consider the function
for . It follows from the equation that
where , and . Using (24) and (27), we obtain
Let . The inequalities and yield
(see (26)). Consequently by [[7], Theorem 4.1], the function satisfies the estimate
Applying Lemma 1.3 to the function with an arbitrary multiindex α with length , we get
for , , . Since p can be chosen arbitrarily large, the estimate (23) holds in the case . The proof is complete. □
2 Asymptotics of solutions of the problem in
Now we consider the problem (1), (2) in the domain . Throughout this section, it is assumed that , where p and β satisfy the inequalities
and q is an arbitrary real number >1. Let be the Green function of the problem (4), (5). Furthermore, let
be the fundamental solution of the heat equation in . Then
is the Green function of the problem (1), (2). We consider the solution
of the problem (1), (2).
We again denote by the function (11) introduced in Section 1. In the sequel, σ is an arbitrary real number such that
and
Then . Let χ be an infinitely differentiable function on equal to one on the interval and vanishing on . We define
Obviously,
where
We also consider the decomposition
where
and
is an extension of the function
with defined by (13). Our goal is to show that both remainders v and w are elements of the space . We start with the case .
2.1 Estimates in weighted Sobolev spaces
Let be the weighted Sobolev space with the norm (3). Furthermore, let
In this subsection, we assume that , where p and β satisfy (28). First, we prove that . This was shown in [[1], Corollary 2.3] for the case . In the case , we must keep in mind that the secondorder 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 [1].
Lemma 2.1Suppose that. Thenand
Proof A simple calculation (see the proof of [[1], Corollary 1]) yields
where denotes the commutator of and . Obviously, the inequalities
are satisfied on the support of the kernel
Since, moreover, the eigenfunctions satisfy the inequality (37) for , we obtain
for . Using Hölder’s inequality, we obtain
where
and
where
This means that is a constant. This proves the lemma. □
Next, we estimate the firstorder xderivatives of the remainder v. For this, we employ the following lemma (cf. [[11], Lemma A.1]).
Lemma 2.2Letbe the integral operator
with a kernelsatisfying the estimate
where, , , . Thenis bounded on.
In the proof of the following assertion, we use another decomposition of the remainder v as in [[1], Lemma 2.4]. This allows us to apply directly the estimate in Theorem 1.1.
Lemma 2.3Letpandβsatisfy the condition (28). Furthermore, letvbe the function (33), where, . Thenforand
with a constantcindependent off. The same is true for the functionw.
Proof Obviously,
where
and
We show that the integral operators with the kernels
are bounded in for and . Using Theorem 1.1, we get
where ε is an arbitrarily small positive number. Applying Lemma 2.2 with , , , , we conclude that the integral operator with the kernel is bounded in for .
Since on the support of , the estimate (6) implies
with arbitrary real a. Thus, by Lemma 2.2, the integral operator with the kernel is bounded in for .
We consider the kernel . Since has the form
we get the representation
where
Here we used the fact that on the support of the function . The inequalities and imply
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
is bounded in for . This proves the lemma. □
Furthermore, the assertions of [[1], 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 [1]. We give here only the formulation of [[1], Theorem 2.7].
Theorem 2.1Let, wherepandβsatisfy the condition (28). Then there exists a solution of the problem (1), (2) which has the form
whereand, , are given by (12), (31) and (35), respectively. The functionsdepend only on, andtand satisfy the estimates
2.2 Weighted estimates for the remainder
We assume now that and consider the decomposition
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 [[12], Theorem 3.8].
Lemma 2.4Suppose thatis a linear operator onsatisfying the following conditions:
(ii) for alland for all functionshwith support in the layersuch that .
Then the inequality
holds for arbitraryq, . Here the constantcdepends only on, , pandq.
The condition (ii) of the last lemma can be verified in some cases by means of the following lemma (cf. [[8], Lemma 10]).
Lemma 2.5Suppose that the kernel of the integral operator (39) satisfies the estimate
for allwith support in the layer. Here, the constantcis independent ofandδ.
It is more easy to estimate the remainder , where Σ is defined by (32). For this reason, we estimate the difference first.
Lemma 2.6Let Σ andbe the functions (32) and (34), respectively. If, thenand
for allkandα, . Here, the constantsare independent off. In particular, .
Proof We have
where is given by (38). Let be the integral operator with the kernel
where . As was shown in the proof of Lemma 2.1, this operator is bounded in . Now let h be a function in with support in the layer satisfying the condition . Then
Analogously to the proof of Lemma 2.1, we obtain
for . Since and on the support of , we can append the factors
with arbitrary exponents a and b on the righthand side of (42). For and , we obviously have . Consequently,
for and , where a and b are arbitrary real numbers and ε is an arbitrarily small positive real number. Hence, by Lemmas 2.4 and 2.5, the operator is bounded in for .
We consider the operator with the kernel
It follows from the boundedness of the operator in that is bounded in , . Furthermore, one can check that
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 [[1], Theorem 3.7] under the weaker assumption on Ω of the present paper.
Theorem 2.2Let, wherepandβsatisfy the condition (28) andqis an arbitrary real number, . Then there exists a solution of the problem (1), (2) which has the form
where, are given by (12) and (35), respectively, and. The functionsare extensions of the functions (36) depending only on, andtand satisfy the estimate
Proof We have to show that the integral operator with the kernel
is bounded in for . For this is true by Theorem 2.1. Let , , and be the same functions as in the proof of Lemma 2.3 and let
Then . We show that the operators satisfy the condition (ii) of Lemma 2.4. Let h be a function in with support in the layer satisfying the condition for all x. Then
Using Theorem 1.1, we get
Thus,
for and . Applying Lemma 2.5 with , and , we conclude that
for and . Analogously, the estimate (6) yields
for and , where a is an arbitrary real number. Here, we used the fact that on the support of . Thus, by Lemma 2.5, the inequality (44) holds for and .
Analogously to the estimation of the kernel in the proof of Lemma 2.3, we obtain the estimate
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 .
In order to prove this for , we consider the adjoint operator. Let and be the integral operators with the kernels
respectively. From the boundedness of in it follows that is bounded in , . We show that
for all , and for all functions h with support in the layer such that . Let h be such a function. Then
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 [[1], 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 [1] is also correct under our assumptions on the boundary of Ω. Moreover, it was proved in [[1], Lemma 4.4], for the particular case , that the extension
satisfies the conditions (40) and (41). Here is a smooth function with support in satisfying the conditions
with certain positive constants , κ and
Furthermore, R is a smooth function with support on the cube having the form
where
with a sufficiently large integer .
We extend the result of [[1], 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
is bounded if or . Since the function depends only on the variables , , and t, it suffices to prove that the operator
This means that is the integral operator with the kernel
where and . As was shown in [1], 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
for allwith support in the layer. Here, the constantcis independent ofandδ.
Proof Obviously,
for . Consequently, it follows from our assumption on K that
Thus, we can apply Lemma 2.5. □
We will show that the operator satisfies the condition of the last lemma. This leads to the following assertion.
Lemma 3.2Suppose that, and that at least one of the conditionsoris satisfied. Furthermore, we assume that the numberin (46) is greater than. Then the operatoris bounded in.
Proof For the case , we refer to [[1], Lemma 4.4].
We consider the case . Let be an arbitrary function with support in the layer such that for all x. Then for , while
for . We verify the condition of Lemma 3.1 for the kernel of the last integral operator. To this end, we use the same decomposition
for the derivatives of as in the proof of [[1], Lemma 4.4], where
and
Here we used the notation and . Applying the estimates
and
with arbitrary positive M and certain positive . Furthermore, the estimates
and
for with certain positive κ and arbitrary positive M yield
Finally, (cf. formulas (4.7) and (4.8) in [1]), we get the estimates
and
where
If , , and s lies between and τ, we have . Consequently, it follows from (48) that
for and . This means that the kernel of the integral operator (47) satisfies the condition of Lemma 3.1 if . Hence, by Lemmas 2.4 and 3.1, the operator is bounded in if or .
In order to prove this for , we consider the adjoint operator. Let be the integral operator with the kernel
Since is bounded in under the assumptions of the lemma, the operator is bounded in , where . Suppose that is a function with support in the layer such that for all x. Then
As was shown above, the derivatives of satisfy the estimate
with the same M as before. This implies
Therefore, it follows from Lemma 3.1 that
for all with support in the layer if or . Applying Lemma 2.4, we conclude that is bounded in for if or . Consequently, the operator is bounded in for if or . The proof is complete. □
Using the last lemma, we obtain the following result which generalizes [[1], Corollary 4.5].
Theorem 3.1Let, wherepandβsatisfy the condition (28) andqis an arbitrary real number, . Then there exists a solution of the problem (1), (2) which has the form
where, are given by (12) and (36), respectively, and.
Proof By Lemma 3.2, the functions satisfy the same condition (43) as the functions in Theorem 2.2. Thus, it follows from [[1], Lemma 4.1] that
This together with Theorem 2.2 implies (49) with a remainder . □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
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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