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

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 .

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 right-hand 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 initial-boundary 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

We define for , while

where

and . Here, we used the notation

We define as the weighted Sobolev space with the norm

for and integer .

Lemma 1.2Suppose thatσis a real number such thatandis not integer for. Furthermore, letand. Then

wherefor, , .

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

Obviously, . This means that

where . Consequently,

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

since and .

Obviously, the functions and

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 right-hand side belongs to for all , , , while the left-hand 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 left-hand 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

Since and for , we obtain

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 cut-off 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

Here, . Thus,

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 multi-index α with length , we get

for , , . Since p can be chosen arbitrarily large, the estimate (23) holds in the case . The proof is complete. □

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 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 [1].

Lemma 2.1Suppose that. Thenand

forand allk.

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

The substitution , yields

i.e., . Consequently,

where

Substituting and , we obtain

This means that is a constant. This proves the lemma. □

Next, we estimate the first-order x-derivatives 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

forand

for allk, α, γ, .

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:

(i) for all,

(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, , where, , , , , . Then

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 right-hand 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

for allsuch thator.

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. □

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

is bounded if or .

We define the operator as

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, , where, , , and. Then

for allwith support in the layer. Here, the constantcis independent ofandδ.

Proof Obviously,

for and

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

for , and , we obtain

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

if . Thus,

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

for , where

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 . □

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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