Abstract and Applied Analysis
Volume 2014 (2014), Article ID 260287, 19 pages
http://dx.doi.org/10.1155/2014/260287

Research Article

Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

Nedyu Popivanov,¹ Todor Popov,¹ and Allen Tesdall²

¹Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria
²Department of Mathematics, City University of New York, College of Staten Island, Staten Island, NY 10314, USA

Received 28 February 2014; Accepted 11 May 2014; Published 14 October 2014

Academic Editor: Donal O’Regan

Copyright © 2014 Nedyu Popivanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex of the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to . Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

1. Introduction

In the present paper, boundary value problems for the wave equation in

with points , are studied in the domain

bounded by the two characteristic cones

and the ball , centered at the origin , . The following BVPs were proposed by Protter [1].

Problem P2. Find a solution of the wave equation (1) in which satisfies the boundary conditions

and its adjoint problem.

Problem P . Find a solution of the wave equation (1) in which satisfies the adjoint boundary conditions

Protter [1] formulated in 1952 some versions of and in (i.e., in (2+1)-D case) as a multidimensional analogue of the planar Darboux problems with boundary data prescribed on one characteristic and on the noncharacteristic segment. Initially the expectation was that such BVPs are classical solvable for very smooth right-hand side functions. However, soon it became clear that contrary to this traditional belief, unlike the plane Darboux problem, Protter’s problems are not well posed. The reason is that the homogeneous adjoint Problem has an infinite number of nontrivial classical solutions (Tong [2], Popivanov and Schneider [3], and Khe [4]). It is known from [5] that for each there exists a right-hand side function of the wave equation, for which the uniquely determined generalized solution of Problem has a strong power-type singularity like at the origin .

In the present paper we examine the exact behavior of the singular solutions of Problem . In the case when the right-hand side function is harmonic polynomial, the Problem is Fredholm and we find the asymptotic expansion at of the unique generalized solution. On the other hand, in the general case when , the problem is not Fredholm because it has an infinite dimensional cokernel. We show that there are an infinite number of necessary conditions for the existence of bounded solutions. We discuss the semi-Fredholm solvability of Problem and for we prove that the necessary conditions for the existence of bounded solutions are also sufficient.

In a historical perspective, Protter studied Problems and in connection with BVPs for mixed type equations that model transonic flow phenomena. In fact, in [1], he also proposes a multidimensional analogue to the two-dimensional Guderley-Morawetz problem for the Gellerstedt equation of hyperbolic-elliptic type. The Guderley-Morawetz problem describes flows around airfoils and is well studied. The existence of weak solutions and the uniqueness of the strong ones were first established by Morawetz [6] by reducing the problem to a first-order system. Lax and Phillips [7] established that these weak solutions are strong. A survey for the classical 2D mixed-type BVPs and their transonic background can be found in [8]. The domain of Protter’s analogue could be constructed by rotation in of a symmetric planar domain for Guderley-Morawetz problem around the axis of symmetry. As a result the set forms the hyperbolic part of the domain. Although it was expected that the multidimensional mixed-type problems would be similar to the two-dimensional BVPs, for the Protter hyperbolic-elliptic problems a general understanding of the situation is still not at hand. Even the question of well posedness is surprisingly subtle and not completely resolved. One has uniqueness results for quasiregular solutions, a class of solutions introduced by Protter, but there are real obstructions to existence in this class. The Protter problems in the hyperbolic part of the domain illustrate some of the difficulties and differences between the planar BVPs and the multidimensional analogues.

In order to construct the solutions of the homogenous Problem we need the spherical functions in . Traditionally, are defined on the unit sphere (see [9]). For convenience in the discussions that follows, we keep the same notation for the radial extension of the spherical function to ; that is, for . For the definition and properties of the spherical functions see Section 3. For define the functions

where the coefficients are

with and . Then the functions

are classical solutions of the homogeneous adjoint Protter problem.

Lemma 1 (see [10]). The functions are classical solutions from of the homogeneous Problem for .

A necessary condition for the existence of classical solution for the Problem is the orthogonality with respect to the inner product, of the right-hand side function to all functions from Lemma 1. To avoid an infinite number of necessary conditions in the framework of classical solvability, we introduce generalized solutions for the Problem (see the similar definition for the (2+1)-D case in [5]).

Definition 2 (see [10]). A function is called a generalized solution of the Problem in , if the following conditions are satisfied:

(1), ;(2)the identity

holds for all such that and in a neighborhood of .

This definition allows the generalized solution to have singularity at the origin and there is a uniqueness result (see Theorem 18). Without any additional conditions imposed on the right-hand side function , it is known (see [3, 10]) that the generalized solution may have power type singularity. Alternatively, we will prove the following necessary conditions for the existence of bounded solutions.

Theorem 3. Suppose that there is a bounded generalized solution of the Protter Problem with right-hand side function . Then

for all , , .

The proof of Theorem 3 is given in Section 4, but before that we will describe the exact influence of the conditions (10) on the behavior of the generalized solution.

First, we consider the case when the right-hand side function of the wave equation (1) has the representation

with . In particular, notice that in the case when the function is a harmonic polynomial in of order , whose coefficients are functions of (see the properties of in Section 3). For convenience further by “harmonic polynomial of order ” we will mean a function from that has the more general form (11). The coefficients are

and must have some special properties at (see, e.g., Lemma 17).

According to the results from [10] we know that the generalized solution of Problem may have a power type singularity at the origin , . In the present paper we study more accurately the exact behavior of the solution of Problem at . It is governed by the parameters

where ; and . We find the asymptotic formula for the generalized solution of Problem .

Theorem 4. Suppose that the right-hand side function has the form (11). Then the unique generalized solution of Problem belongs to and has the following asymptotic expansion at the singular point :

where

(i)the function and satisfies the a priori estimate

with constant independent of and ;(ii)the functions , , satisfy the equalities

with functions bounded and independent of ;(iii)if at least one of the constants in (16) is different from zero, then for the corresponding function there exists a direction with for , such that

After the case of the harmonic polynomials, here we deal with the more general situation when the right-hand side function is smooth, but it cannot be expanded simply as a sum (11). Now, Lemma 1 shows that the Problem is not Fredholm solvable.

Remark 5. Consider the operator

where is the unique classical solution to Protter Problem for the right-hand side function . According to Lemma 1 we have . This means that is not Fredholm operator for example in . On the other hand, the uniqueness result Theorem 18 shows that and could be a semi-Fredholm operator. A semi-Fredholm operator is a bounded operator that has a finite dimensional kernel or cokernel and closed range (see, e.g., [11]). Accordingly we need to find the range of .

The next result suggests that is a semi-Fredholm operator.

Theorem 6. Let the function belong to . Then the necessary and sufficient conditions for existence of bounded generalized solution of the Protter Problem are

for all , , and .
Moreover, this generalized solution and satisfies the a priori estimates

where the constant is independent of the function .

Obviously, the set of all functions from that satisfy the orthogonality conditions (19) is closed. Therefore, Theorem 6 shows that the operator defined in Remark 5 with a domain has a closed range in , and we get the following result.

Corollary 7. The operator is a semi-Fredholm operator from to .

We have briefly announced some of the results from this section in [12] with the assumption .

The main results in this work are discussed in Section 2 and the proofs are in Sections 3–7. In more detail the paper is organized as follows: estimates for the spherical functions involved in the representation of the solution are proven in Section 3. In Section 4 the necessary conditions for bounded solution Theorem 3 are proved. In Section 5 we consider some two-dimensional boundary value problems connected to Problem , the Problems and . Exact formulas for the solution of the Problem are presented in Lemma 20. In Section 6 the proofs of the main Theorems 4 and 6 are given based on the results from the previous sections and an asymptotic expansion formula for the generalized solution of the Problem (Theorem 22). The long and technical proof of Theorem 22 is postponed to Section 7.

2. Historical Remarks on the Main Results

Let us point out several related recent works on Protter problems. Necessary and sufficient conditions for the existence of solutions with fixed order of singularity were obtained in [10]. Similarly, for the -analogues of Protter problems, some results are presented in [13, 14]. For the problem with Dirichlet type boundary condition on , a formula for the asymptotic expansion of the singular solution can be found in [15], and the semi-Fredholm solvability is discussed in [16] for . A comparison of various recent results for Protter problems is made in [13].

Various authors adopted a variety of approaches to Protter problems over the last sixty years, for example, Wiener-Hopf method, special Legendre functions, a priori estimates, nonlocal regularization, and so forth (see [5] and references therein; see also [4, 10, 14, 17–19]). Alternatively, different multidimensional analogues of the classical Darboux problem for the wave equation are considered in [20–22], while for some related semilinear equations and systems see [23]. The existence of bounded or unbounded solutions for the wave equation in and , as well as for the Euler-Poisson-Darboux equation, has been studied in [4, 10, 17, 19, 24–26].

Regarding the Protter problems with lower order terms see [27] and references therein. Problems with more general boundary condition on are studied in [26, 27]. Some possible regularization methods involving integrodifferential or nonlocal terms can be found in [18].

For the Protter problems for equations of mixed hyperbolic-elliptic type proposed in [1], Aziz and Schneider [28] proved an uniqueness result in the linear case (see also [21]). Concerning nonexistence principle for nontrivial solution of semilinear mixed-type equations in multidimensional case, we refer to [29].

In 1960 Garabedian [30] proved the uniqueness of a classical solution of Proter problem. However, generally, Problem is not classically solvable and a necessary condition for the existence of a classical solution is the orthogonality of the right-hand side function to all solutions of the corresponding homogeneous adjoint Problem . Here, in Lemma 1, the solutions were constructed with the help of the functions defined by (6). The alternate representation in terms of the Gauss hypergeometric function

can be found in Khe [4]. In [3] there are some solutions for the three-dimensional analogue of the homogeneous Problem .

Let us look back at Theorem 3 and the necessary orthogonality conditions (10) for the existence of bounded solutions of Problem . Naturally, these conditions include the functions from Lemma 1. However, notice that there are also some others.

Remark 8. It is interesting that conditions (10) include the case of even . Notice that the functions are not classical solutions of the homogenous adjoint Problem . Actually, they satisfy the homogenous wave equation in and vanish on , but is not zero on . In addition, the functions have a singularity at the origin like ; however, this singularity is integrable in the domain .

Instead of imposing an infinite number of orthogonality conditions on , Popivanov and Schneider [3, 5] introduced the concept of generalized solution that allows the solution to have singularity on the inner cone . Here, Theorem 4 describes the effect of the parameters on the behavior of the generalized solution of Problem . The constants are defined by (13) and are obviously related to the orthogonality conditions (10). When the right-hand function is a harmonic polynomial (11), the asymptotic expansion in Theorem 4 shows that the generalized solution could be bounded only if all involved are zero.

Corollary 9. Suppose that the right-hand side function has the form (11) and satisfies the orthogonality conditions

for all ; and . Then the unique generalized solution of Problem belongs to , is bounded, and satisfies the a priori estimate

On the other hand, without any orthogonality conditions on , the following result is obtained.

Corollary 10. The generalized solution of Problem with a right-hand side function in the form (11) satisfies the a priori estimate

The influence of the orthogonality conditions (22) on the exact behavior of the generalized solution is clarified by case (iii) of Theorem 4. It shows that, for fixed indexes , the corresponding condition (22) “controls” one power-type singularity.

In Corollaries 9 and 10 the emphasis is on the extreme cases: when all orthogonality conditions (22) are fulfilled or, alternatively, when none of them are satisfied. In both cases the exact behavior of the solution is given. The estimate (24), presented here, is analogous to known estimates for Protter problems in [5] and in [24]. It is interesting that singularities of the generalized solutions are isolated at the origin and do not propagate in the direction of the bicharacteristics on the characteristic cone . Traditionally, it is assumed that the wave equation, with sufficiently smooth right-hand side, cannot have a solution with an isolated singular point as in Hörmander [31, Chapter 24.5]. The case here is different since the point of singularity lies on the noncharacteristic part of the boundary , as well as on the characteristic part .

Remark 11. The Problem in with harmonic polynomial on the right-hand side is also studied in [10]. However, the explicit asymptotic expansion here has no analogue in [10], where only the behavior of the singularities is given. Additionally, if the orthogonality conditions (22) are fulfilled, Corollary 9 states that the generalized solution is in fact bounded, while the estimates in [10, Theorem 1.1] still allow the solution to have some logarithmic singularities.

Remark 12. Let us compare Protter problems in (as treated in [13, 14]) and here (see also [10, 15]). In both cases the study of these BVPs is based on the properties of the special Legendre functions. Instead of Legendre polynomials here, in the three-dimensional case, the Legendre functions with noninteger indexes are used (for their properties see [32]). One can easily modify both these techniques to obtain similar results for the -dimensional problems, for even (analogous to case) or for odd (the present case ). Some related results for Protter problems in are presented in [17, 24].

In the general case when the right-hand side function is smooth enough, Theorem 6 implies that the necessary conditions (10) for existence of bounded solutions from Theorem 3 are also sufficient. Further, this means that there are no other nontrivial classical solutions of the homogenous adjoint Problem except those listed in Lemma 1.

Remark 13. We point out the differences between Theorem 6 and the results from [16] for Protter Problem with Dirichlet type boundary condition on . First, notice that in the case when right-hand side function is harmonic polynomial of order the solution of Problem may have worse singularity (like ; see Theorem 4) than the solution of Problem (like ; see [15]). For the general case we are able to reduce the assumptions on ; in Theorem 6 we assume , while in [16, Theorem 1.1] smoother is required. In order to achieve this, we rely on the more accurate estimates for the special functions proven in Section 3.

3. Estimates for the Special Functions

For the proof of the main results we will need some properties of the spherical functions in . They are naturally expressed on the unit sphere in spherical polar coordinates. Let us introduce polar coordinates in :

where , , . Then the spherical functions, expressed in terms of and as in the traditional definition on (see [9]), are given by

and . Here are the Legendre polynomials defined by the Rodrigues’ formula

while are the associated Legendre polynomials that can be defined as

The functions satisfy the differential equation

and form a complete orthonormal system in (see [9]).

Using Cartesian coordinates as in Section 1, one can define the spherical functions as for , or by

where are constants. In the present paper, we keep the same notation for the radial extension of the spherical function to ; that is, for . According to the properties of , the function is a homogenous harmonic polynomial of order in the variables , , .

We will need some estimates for and the special functions involved in the representations of the solutions of the Protter problems. Let us start with the Legendre polynomials .

Lemma 14. The following estimates hold for :

Proof. The estimates (31) are proved, for example, in [10]. Here we will show that (32) holds. Using Bonnet’s recursion formula

we get and

Thus

From here and the equalities and we get the estimate (32) by induction.

In order to study the first derivatives of the generalized solution of the Protter problem, we will need to estimate also the first derivatives with respect to of , the radial extension of the spherical function to :

Using the so-called addition theorem for Legendre polynomials we get the following result.

Lemma 15. For the functions satisfy the equalities

Proof. From the definition (26) of it follows that

According to the addition theorem (see [33])

With and one derives the equality

This means that

Using (39) again, we get the required property of the derivatives of . Directly from (36) for the squares of the derivatives with respect to we have

and from the definition (26) of we find

Put , and differentiation of (39) with respect to and gives

With we derive

Analogously, after differentiating (39) twice with respect to ,

Then substituting and we have

Since we conclude that

As a direct consequence, we find the next estimates for the radial extension of the spherical functions to .

Corollary 16. For the functions satisfy the inequalities

Proof. Obviously

The estimates (50) and (51) follow directly from the Cauchy-Schwarz inequality and Lemma 15:

Finally (52) follows from the equality (42):

Generally, if the function is smooth enough, it can be represented as a harmonic series. In order to estimate the coefficients of the series we will use the next result. It is based on the fact that the spherical functions are eigenfunctions for the Laplace operator on the sphere .

Lemma 17. Let .

(1)(See [16]). Let . Then it has the representation

and for

(2)Suppose ; then for

Proof. The estimate in case follows directly from the fact that the spherical functions are eigenfunctions for the spherical Laplacean :

That is, (29) shows that . In fact is bounded by the Fourier coefficient of :

For detail proof of (1) see [16, 34].
Next we will prove case with . Let ; then

Integrating by parts gives

where

Therefore using (52) we have

To prove case it remains to substitute for in this estimate.

4. Necessary Conditions for Bounded Solution

Here we will prove of the necessity of the orthogonality conditions (10) for the existence of a bounded solution.

Proof of Theorem 3. Let be a bounded generalized solution of Problem . Let us fix a function such that for , and for .
For fixed indexes , , , consider the functions

for . Obviously, vanishes on a neighborhood of and on . Therefore according to Definition 2 we have

and thus

We want to prove that for all , , . Notice that and then the integral is convergent. The function and thus

Therefore it is sufficient to prove that when , the right-hand side of the equality (67) tends to zero for , , .
Using the fact that the functions are solutions of the homogenous wave equation in , straightforward computations show that

where , while and stand for and , respectively. Expressed with spherical polar coordinates in ,

For simplicity, we consider only one of the terms from the definition of the function (see (6)). Let us denote

defined in , and consider the subsets and .
Since the functions , , , and are bounded, the function in , and is zero in , it is sufficient to prove that the integrals

tend to zero as , for , , and .
We get from the estimate in and thus

For let us compute first

because could be negative only when . Therefore, we have

and thus .
Finally, to evaluate , we use the estimate

and find

that shows that .

5. Previous Results

In this section we quote some results from [10] that will be essentially the starting point for the proofs of Theorems 4 and 6. We start with the following uniqueness result.

Theorem 18 (see [10]). The Problem has at most one generalized solution.

In [10] the right-hand side function of the wave equation (1) is fixed as a harmonic polynomial (11). Then the following existence result for the generalized solution is valid.

Theorem 19 (see [10]). Suppose that the right-hand side has the form (11) where . Then, the unique generalized solution of the Problem in exists and has the form

In fact, the function from (78) is the solution of a two-dimensional boundary value problem that involves only the corresponding coefficient from (11). In order to formulate this BVP, it is natural to introduce polar coordinates in : ; and are such that , , and

In the special case when has the form

according to Theorem 19, we may look for a solution of the same form

Then we can reduce the (3+1)-D Protter problem to some BVPs in . From the properties of the spherical functions it follows that the function is a solution of the equation

with right-hand side , in the domain , bounded by

Thus, we arrive at the next two-dimensional problems.

Problem P21. Find a solution of (82) in the domain which satisfies the boundary conditions

Finally, we substitute

and get the following problem.

Problem P22. Find a solution of the equation

in the domain with boundary conditions

In [10] the solution of Problem is constructed with the help of the Riemann’s function

for equation (86) found by Copson [35]. The problem is reduced to an integral equation of Volterra type. Then this integral equation is solved using some formulas from the book by Samko et al. [36] and the properties of the Mellin transform. According to formulas , , and in the proof of Theorem 3.1 from [10] we can write down the following result.

Lemma 20. The solution of Problem is given by

where

Finally, we will need the relation between the functions , with , defined in (6) and the Legendre polynomials .

Lemma 21 (see [10]). Define the functions

Then the equality

holds for with some nonzero constants .

6. Proofs of the Main Results

In this section we will prove Theorems 4 and 6. The proofs are based on an asymptotic expansion formula for the solution of the two-dimensional Problem . In order to formulate it, let us concentrate first on Problem with right-hand side functions of the form

with fixed and . We will use the results stated in Section 5. The unique generalized solution of Problem also has the form

The function is the solution of Problem with the function as a right-hand side in (82). We are interested in the exact behavior of at . One expects it to depend on the constants

that correspond to defined in (13) for the Problem . For simplicity, denote further

Theorem 22. Let and . Then the generalized solution of Problem belongs to and has the following asymptotic expansion at :

where

(1)the functions are independent of , bounded, and satisfy ;(2)the function satisfies the following estimates:

and for

If also and , then

and if additionally and , then

where in all inequalities the constant is independent of and .

The proof of this result is quite long and technical and we leave it for Section 7. Here we will use Theorem 22 to prove Theorems 4 and 6.

Proof of Theorem 4. Assume that the right-hand side function is a harmonic polynomial (11). Then the unique generalized solution also is a harmonic polynomial (78), according to Theorem 19. Furthermore, the functions are solutions of Problem with right-hand side that can be represented as

When we have and obviously

The definition of functions from Lemma 1 and (103) gives the identity

which shows that , according to their definitions (96) and (13).
Now we can apply Theorem 22 for the functions and . Using (104) and (105) we get the expansion

where , , and . Summing up over and one gets the desired expansion.
Finally, to prove property (iii), let us fix a direction with , , and . Then for the functions from (16) we have

And, thus, there are some nonzero constants such that

Therefore property (iii) follows from the fact that the spherical functions are linearly independent.

Besides Theorem 22, the proof of Theorem 6 relies on and the estimates from Section 3 for the special functions.

Proof of Theorem 6. The function can be represented as

The generalized solution of Problem could be formally written in the form

where is the solution of Problem with right-hand side . We will prove that the series (110) and its derivatives are uniformly convergent in for and that is bounded.
According to Lemma 17 the series for and its first derivatives uniformly converge. For the derivatives with respect to and with respect to , there holds the relation

and therefore . A similar argument shows that and and we can apply Theorem 22.
First, using Lemma 17 case with and , we see that for

On the other hand, Lemma 17 case with and gives

Next we study the series (110). Using the notations of Theorem 22, we know that when for , the function satisfies

and then for from (100), (101), and (102) we have the following estimates:

or alternatively

and for the derivatives

Then applying the estimate (49) for the spherical functions from Corollary 16 and substituting (113) and (114) in (117) we find

On the other hand, first using (118) with (112) and then (52) for the sum of derivatives of , we get

Combining (119) for the derivatives of with (113), (114), and (115) gives

For the case we have and representation (103) shows that . Therefore Theorem 22 gives

After this preparation, we are ready to estimate the Fourier series (110) and its first derivatives:

Therefore we have since for each fixed the series (110) uniformly converges in the set and the same holds for its first derivatives.
Finally, we will prove that the function defined as the series (110) is the generalized solution of Problem . First, notice that the function is the generalized solution of Protter Problem with right-hand side function . Thus satisfies the boundary conditions and just like all the terms . The proof of case from Definition 2 is straightforward; for a test function and we have

and the uniform convergence in of the series (110) and its first derivatives allow us to take the limit in this equality. Therefore

and we see that is the generalized solution of Problem with right-hand side .

7. Proof of the Asymptotic Expansion in the Two-Dimensional Case

The proof of Theorem 22 is based on the results stated in Section 5. In particular, according to Lemma 20, the solution of Problem can be constructed with help of the substitutions , as

where is defined by the formulas (89), (90), (91), and

One can see that generally the integral in (90) blows up when approaches , and thus has singularity at even for smooth functions .

Proof of Theorem 22. The proof is as follows.
(A) Proof of the Asymptotic Formula. We will study the behavior of the function

given by the integral representation (89) from Lemma 20. The smoothness of in (89) away from the point follows directly from the smoothness of the function . Next we will derive the asymptotic expansion of at .
First we will find the relation between the constants and the function defined by (91). Let us compute the integral

Now we can apply Lemma 21 to get

and we conclude that

Next we consider the case . The simpler case will be discussed separately later in the proof.
Let us expand the Legendre polynomial in formula (90) using (27):

where . For we get

Applying (132) we find

where

Now, we want to estimate the function . First, let us look more carefully to the representation (91) of the function with :

Since all the arguments of here are in the interval and for , it is obvious that

and therefore

where the constant is independent of and .
We will need also a more accurate estimate for with higher power of . The first two integrals in (137) are bounded by . For the last term

more computations are required. First, using the estimate (32) from Lemma 14, we have

with some and the constant is independent of . Therefore

Now, notice that the value of will not change if we add to . This is based on the equality

that holds, because when is odd number , while for even indices , , the polynomial is an even function and by the definition of the Legendre polynomials (see (27)) it follows that

Thus

and we can apply (142) to conclude that for there is and a constant independent of and , such that

Now we apply the expansion (135) of in the definition (89) of and find that

where the function is smooth (see (89)) and . Consider the term

According to (146) the last integral is bounded:

Thus, we find the expansion

where

and . Let us point out that , while the fact that the functions are bounded follows from the estimate

The representation (150) holds for , while when we have simply and can substitute it in formulas (89), (90), and (91) for the solution of Problem . Straightforward computations lead to

where

Therefore, the representation (150) obviously stays true in the case .
Finally, let us return to the generalized solution of Problem and to the coordinates and :

Here, the function is given by

and therefore , while functions , defined by

are independent of and are obviously bounded. To complete the proof of case , notice that this definition gives

(B) Proof of the Estimates of . Next, for , we will estimate the function and its first derivatives.
First, we will study the behavior of at . The function is given by (156), where is defined in (150) as

Since the argument of the Legendre polynomial in (160) varies in the interval ,

where the constant is independent of and . Thus applying the estimate (139) for we get

Therefore

where the constant is independent of and ; that is, (100) holds.
Now we consider the derivatives of . Thus we need to evaluate the derivatives of ; integrating (159) by parts we get

For defined by (136), using (138), we find

where the constant is independent of and . In fact, we can remove the coefficient here for smoother functions . In order to do this, let us rewrite as

For and we will need also :

Then estimates for the derivatives of are required:

Notice that for the argument of in the last term

since the function is homogenous. Therefore we can replace there the derivative with respect to with . Integrating by parts we find

Recall that , and thus and in the last integral we have

Hence, using (31) from Lemma 14, we get that

Analogously for , differentiating one more time the expression for , applying again (169) in the last integral, and integrating by parts the last three terms, we find the estimate

Applying (138), (172), and (173) to and in (166) and (167) we find

To evaluate the derivatives of we need also to study the derivatives of defined by (160):

Notice that in these integrals and therefore

Then it follows from Lemma 14 that

Now we are ready to estimate the derivatives of from (164). For after integration by part we have

Then, applying (174), it follows that

Similarly, for , we find

and thus we get

Finally, to prove (102), notice that

and therefore

where the constant is independent of and .
The estimate (101) is a straightforward consequence of formulas (172) for and (164) for , while the case follows directly from the representation (153).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Nedyu Popivanov and Todor Popov was partially supported by the Bulgarian NSF under Grant DCVP 02/1/2009 and by Sofia University Grant 94/2014. The research of Allen Tesdall was partially supported by Research Foundation of CUNY Grant 66237-00 44. The authors would like to thank the anonymous referees for the constructive comments and suggestions to improve the quality of the paper.

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