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Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary
Boundary Value Problems volume 2012, Article number: 142 (2012)
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 . 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
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 . However, the proofs of some lemmas in  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 [, 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 [, 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 k th 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 1.1 Let be the Green function introduced above, and let denote the Green function of the initial-boundary value problem
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
We define for , while
and . Here, we used the notation
We define as the weighted Sobolev space with the norm
for and integer .
Lemma 1.2 Suppose that σ is a real number such that and is not integer for . Furthermore, let and . Then
where for , , .
Proof We prove the lemma by induction in .
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
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
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,
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 1.3 Suppose that and , where . Then
with a constant c independent of u.
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,
Since and for , we obtain
The result follows. □
Using the last two lemmas, we can prove the following theorem.
Theorem 1.1 Suppose that σ is a real number satisfying (10). Then
for , , . Here for , while is 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 [, 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
and for , , . Here . Furthermore (cf. (14)),
Let χ be a smooth cut-off function on the interval , in and on . We define for , . Then
Thus, by [, 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
Consequently, by (25),
with a positive constant κ. Applying the estimate
for (cf. [, 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. [, 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 [, 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. □
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
Then . Let χ be an infinitely differentiable function on equal to one on the interval and vanishing on . We define
We also consider the decomposition
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 [, 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 .
Lemma 2.1 Suppose that . Then and
for and all k.
Proof A simple calculation (see the proof of [, 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
The substitution , yields
i.e., . Consequently,
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. [, Lemma A.1]).
Lemma 2.2 Let be the integral operator
with a kernel satisfying the estimate
where , , , . Then is bounded on .
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.
Lemma 2.3 Let p and β satisfy the condition (28). Furthermore, let v be the function (33), where , . Then for and
with a constant c independent of f. The same is true for the function w.
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
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 [, 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].
Theorem 2.1 Let , where p and β satisfy the condition (28). Then there exists a solution of the problem (1), (2) which has the form
where and , , are given by (12), (31) and (35), respectively. The functions depend only on , and t and satisfy the estimates
for all k, 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 [, Theorem 3.8].
Lemma 2.4 Suppose that is a linear operator on satisfying the following conditions:
for all ,
for all and for all functions h with support in the layer such that .
Then the inequality
holds for arbitrary q, . Here the constant c depends only on , , p and q.
The condition (ii) of the last lemma can be verified in some cases by means of the following lemma (cf. [, Lemma 10]).
Lemma 2.5 Suppose that the kernel of the integral operator (39) satisfies the estimate
for , , where , , , , , . Then
for all with support in the layer . Here, the constant c is independent of and δ.
It is more easy to estimate the remainder , where Σ is defined by (32). For this reason, we estimate the difference first.
Lemma 2.6 Let Σ and be the functions (32) and (34), respectively. If , then and
for all k and α, . Here, the constants are independent of f. 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 [, Theorem 3.7] under the weaker assumption on Ω of the present paper.
Theorem 2.2 Let , where p and β satisfy the condition (28) and q is 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 functions are extensions of the functions (36) depending only on , and t and satisfy the estimate
for all such that or .
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
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 [, 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
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
with a sufficiently large integer .
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
is bounded if or . Since the function depends only on the variables , , and t, it suffices to prove that the operator