# Problems of diffraction type for elliptic pseudo-differential operators with variable symbols.

Abstract

In this paper we consider problems of diffraction type for elliptic pseudo-differential operators with variable symbols depending on parameters. We compare the regularizators of a diffraction and a Dirichlet problem, and we prove that the regularizator of a diffraction problem tends to the regularizator of a Dirichlet problem as the parameter of the external domain tends to zero.

Keywords and Phrases: Pseudo-differential, Diffraction, Elliptic, Regularizator.

1. Introduction

In this paper we consider problems of diffraction type for elliptic pseudo-differential operators. In more details, we consider simultaneously two pseudo-differential equations elliptic with parameters in different domains with a common boundary. A classical diffraction problem for differential operators was considered, for example, by A.N.Tichonov and A.A. Samarsky ([6]). In the statement of this problem, the homogeneity of a medium is broken by a bounded domain provided that the solution satisfies the conditions of a maximal smoothness on the boundary of this domain. In [3] the analogous problem for pseudo-differential equations was studied, but the main result was obtained only for the case of pseudo-differential operators with constant symbols. In this article we consider the same problem for pseudo-differential equations with variable symbols depending on two parameters, under the condition that one of the parameters tends to infinity. For example, we consider a diffraction problem in [R.sup.n.sub.+] = {x [member of] [R.sup.n], [x.sub.n] [greater than or equal to] 0} and in [R.sup.n.sub.-] (where [R.sup.n.sub.-] = R - [R.sup.n.sub.+]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where A and B are pseudo-differential operators of order [m.sub.1] and [m.sub.2] elliptic with parameter q and p, respectively. If p is big, then the solution in the half space [R.sup.n.sub.-] has the form of a boundary layer with respect to [x.sub.n]. For instance, the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([x.sub.n] < 0) is boundary layer function. If [epsilon] = 1/p tends to zero, this function approaches zero for [x.sub.n] < 0.

It is possible to prove that if the symbols of operators A and B don't depend on x, then we can find an exact solution of problem (1.1) (see [3])which is defined by the inverse operator. That is, if we write the problem (1.1) in the form Au=f where A={[P.sup.+] A, P-B}, then u=[A.sup.-1]f In the case when A and B depend on x, the inverse operator can not be defined explicitly but if we can find an operator R such that Rf=f+Tf where the operator T has the small norm, we say that the operator R is the regularizator (1) of problem (1.1). We are going to evaluate the difference between the regularizators for problem (1.1) and the Dirichlet problem (1.2) below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

We prove that the regularizator of the Dirichlet problem (1.2) can be obtained as a limit case in the diffraction problem (1.1) as p = (1/[epsilon]) tends to infinity ([epsilon] [right arrow] 0). We shall use the technique of the theory of pseudo-differential operators developed in [5], [7] and the notations of [2].

2. Notations and Properties

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) be a space of distributions u(x), x=(x', [x.sub.n]) = ([x.sub.1], [x.sub.2],..., [x.sub.n-1], [x.sub.n]) [member of] [R.sup.n] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where p, q are real non-negative parameters,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

The norm on the right-hand side of (2.1) is the usual norm in [L.sub.2] ([R.sup.n.sub.[xi]]).

If p = q = 1, then the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) coincides with the ordinary Sobolev space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]). Since [L.sub.2] and [H.sub.0] are the notations of the same space we shall write further [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We introduce also the spaces [H.sub.s] ([R.sup.n.sub.+]) and [H.sub.s] ([R.sup.n.sub.-]) of functions [f.sub.+] and [f.sub.-] defined in [R.sup.n.sub.+] = {x [member of] [R.sup.n]: [x.sub.n] > 0}, ([R.sup.n.sub.-]) = {x [member of] [R.sup.n]: [x.sub.n] < 0}, ([R.sup.n.sub.-]) = [R.sup.n]\[R.sup.n.sub.+], respectively, with the norms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

and [[theta].sup.+] (x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the extension of the function [f.sub.[+ or -]] on the whole Euclidean space [R.sup.n] such that the extension belongs to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]).

We state some properties of the operator [[PI].sup.[+ or -]]:

* 1. The operator [[PI].sup.[+ or -]] is defined on smooth decreasing functions by the formula (2.3). Since the operator of multiplication of the Heaviside function [theta].sup.[+ or -]] (x) is bounded in [H.sub.0] ([R.sup.n.sub.x]), the operator [[PI].sup.[+ or -]] is bounded in the space [H.sub.0] ([R.sup.n.[xi]]) being the dual of [H.sub.0] ([R.sup.n.sub.x]) with respect to the Fourier transform. For arbitrary function [??]([xi]) [member of] H0 ([R.sup.n.sub.-]) the formula (2.3) is understood as the closure of the opeator [[PI].sup.[+ or -]].

2. If [??] ([xi]) [member of] [H.sub.0] (([R.sup.n.sub.[xi]]), then this function can be represented as the sum [??] ([xi]) = [??] ([xi]) + [??] ([xi]), where [??][+ or -] ([xi]) = [[PI].sup.[+ or -]] [??] ([xi]).

3. Since [theta].sup.+] (x) = 0 for [x.sub.n] < 0 ([[theta].sup.-] (x) = 0 for [x.sub.n] > 0), the function [[PI].sup.+] ([xi]) ([[PI].sup.-] [??] ([xi])) admits an analytic continuation in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0).

4. If a function [??] ([xi]) ([??] ([xi])) [member of] [H.sub.0] ([R.sup.n.sub.[xi]]) and may be extended in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0), then [[PI].sup.[+ or -]] [??] = 0.

5. If the functions [[PI].sup.[+ or -]] [??] ([xi]) and [[PI].sup.[+ or -]] [[??].sub.[+ or -]] ([xi]) [??] ([xi])]make sense, where [??] ([xi]) ([??] ([xi])) admit an analytic continuation in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0)), then [[PI].sup.[+ or -]] [[??].sub.[+ or -]] ([xi])] [??] [xi] [[PI].sup.[+ or -]] [??] ([xi])]

Let f ={[f.sub.+], [f.sub.-]} [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]) [x] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (R.sup.n.sub.-]. On this product space we can introduce a natural operation of addition and multiplication by a function [PHI] [member of] [C.sup.[infinity]] ([R.sup.n]) by the following rule: If f ={[f.sub.+], [f.sub.-]} and g ={[g.sub.+], [g.sub.-]}, then f + g ={[f.sub.+] + [g.sub.+], [f.sub.-] + [g.sub.-]} and [phi]f ={[phi][f.sub.+], [phi][f.sub.-]}. We can also introduce a natural norm on this set.

Let A and B be two pseudo-differential operators whose symbols are [sigma] (A) = a(x, [xi], q) and [signma] (B) = b(x, [sigma], p), respectively. Recall that a pseudo-differential operator corresponding to the symbol a(x, [xi]) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

We suppose that the symbols a and b depend on parameters q and p (where q [less than or equal to] p), respectively, and satisfy the following conditions:

* 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. The functions a(x, [xi], q) and b(x, [xi], p) are homogeneous of order [m.sub.1] and [m.sub.2], ([m.sub.1] and [m.sub.2] are positive) with respect to [xi], q and [xi], p, respectively.

3. The operators A and B are elliptic with parameter, i.e. a(x, [xi], q) [not equal to] 0 for real [xi] and for q + |[xi]| [not equal to] = 0, and b(x, [xi], p) 6= 0 for real [xi] and for p + |[xi]| [not equal to] = 0.

4. For every value of multi-indexes [alpha] = ([[alpha].sub.1], [[alpha].sub.2],..., [[alpha].sub.n]), [beta] = ([[beta].sub.1], [[beta].sub.2],..., [[beta].sub.n]), the following estimations hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

|alpha]| = [[alpha].sub.1] + [[alpha].sub.2] + ... + [[alpha].sub.n], |[beta]| = [[beta].sub.1] + [[beta].sub.2] + ... + [[beta].sub.n]

5. The symbols a and b can be represented in the form

a(x, [xi], q) = a(1, [xi], q) + a'(x, [xi], q),

b(x, [xi], p) = b(1, [xi], p) + b'(x, [xi], p)

where a'(x, [xi], q) and b'(x, [xi], p) are infinitely differentiable functions with respect to x, with compact supportt, i.e. they belong to [C.sup.[[infinity].sub.0] ([R.sup.n.sub.x]).

We remark that a pseudo-differential operator with a symbol satisfying the condition 5. can be defined by the following formula, which is equivalent to the formula (2.4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

where the tilde "~" dentoes the Fourier transform with respect to the first argument. In [7] M.Vishik and G. Eskin have proved that symbols satisfying the conditions 1. - 5. admit the following factorization:

a(x, [xi], q) = [a.sub.+](x, [xi]', [[xi].sub.n], q)[a.sub.-](x, [xi]', [[xi].sub.n], q), (2.7)

and

b(x, [xi], p) = [b.sub.+](x, [xi]', [[xi].sub.n], p)b-(x, [xi]', [[xi].sub.n], p) (2.8)

where [a.sub.+](x, [xi]', [[xi].sub.n], q), b+(x, [xi]', [[xi].sub.n], p) ([a.sub.-](x, [xi]', [[xi].sub.n], q), [b.sub.-](x, [xi]', [[xi].sub.n], p)) are functions admitting an analytic continuation in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0) and they remain homogeneous with respect to [xi], q ([xi], p). Suppose that ord [a.sub.+](x, [xi]', [[xi].sub.n], q) = [k.sub.1], ord [b.sub.-](x, [xi]', [[xi].sub.n], p) = [k.sub.2] [greater than or equal to] 0, (k = [k.sub.1] + [k.sub.2] > 0) and the orders do not depend on x.

3. Evaluation of The Di_erence Between The Regularizators of Di_raction and of Dirichlet Problems

Consider a function f[member of] [H.sub.k-m] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We also introduce the couple operator ([3]) Au={[P.sup.+] Au, [P.sup.-] Bu}where [P.sup.+] ([P.sup.-]) is the restriction operator of distributions on [R.sub.+] ([R.sub.-]) (it is clear that for ordinary functions it coincides with Heaviside function [[theta].sup.+] ([[theta].sup.-])) and the operator A (B) has the symbol a(x, [xi], q) (b(x, [xi], p)).

We consider the following diffraction problem

Au = f [member of] [H.sub.k-m], u [member of] [H.sub.k] ([R.sup.n]) (3.1)

It follows from [3] that problem (3.1) has a unique solution for su_ciently large values of parameters p and q. The proof is based on construction of the regularizator of this problem which has the following form:

[??]f = [R.sub.1] [[[theta].sup.+] [R.sub.-]E[f.sub.+] + [[theta].sup.-][R.sub.+]E[f.sub.-]] Or equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where [R.sub.1], [R.sub.-] and [R.sub.+] are pseudo-differential operators with symbols

[[a.sub.+](x, [xi]', [[xi].sub.n], q)[b.sub.-][(x, [xi]', [[xi].sub.n], p)].sup.-1], [b.sub.-](x, [xi]', [[xi].sub.n], p) [[a.sub.-][(x, [xi]', [[xi].sub.n], q)].sup.-1]

and

[a.sub.+](x, [xi]', [[xi].sub.n], q) [[b.sub.+][(x, [xi]', [[xi].sub.n], p)].sup.-1]

respectively.

Consider at the same time with problem (3.1) the following Dirichlet problem

[P.sup.+] A(D, x, q)[u.sup.(0).sub.+] = [f.sub.+](x), [u.sup.(0).sub.+] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]) (3.3)

here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (R.sup.n.sub.+]) is the subspace of the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) ([k.sub.1] [greater than or equal to] 0) of functions, which vanish on [R.sup.n.sub.-]. The regularizator of this equation was constructed by M. Vishik and G. Eskin in [7] and it has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

We shall prove that if p [right arrow] [infinity],then [??]f [right arrow] [R.sub.+][f.sub.+]. It means that the regularizator of Dirichlet problem (3.3) may be obtained as a limit case of the problem (3.1) when p approaches infinity. We represent the difference of these two operators (3.2) and (3.4) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Since the smoothness of this difference is [k.sub.1], we estimate the norm of I f in the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Using 1 = [[theta].sup.+] + [[theta].sup.-], we transform the term 1/[B.sub.-](x,D,p) [[theta].sup.+] [B.sub.](x,D,p)/[A.sub.-](x,D,q) E[f.sub.+](x) as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Substituting (3.7) into (3.6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

We consider separately the operator 1/[B.sub.-](x,D,p). Let us set p = 1/[epsilon] and transform this operator as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

Moreover we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, for [N.sub.2] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

Substituting (3.10) into (3.9), for [N.sub.1] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

where we denote

[T.sub.-] = [B.sub.-](x, [epsilon]D, 1) - [B.sub.-](x, 0, 1)/[B.sub.-](x, 0, 1)[B.sub.-](x, [epsilon]D, 1) (3.13)

with the symbol

[sigma]([T.sub.-]) = [b.sub.-](x, [epsilon][xi], 1) - [b.sub.-](x, 0, 1)/[b.sub.-](x, 0, 1)[b.sub.-](x, [epsilon][xi], 1) (3.14)

We expand this symbol [sigma]([T.sub.-]) as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.15)

By virtue of assumption (4) for homogeneous symbols given in section 2, we have the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.16)

Denoting

[??]([xi], [epsilon]) = F [[B.sub.-](x, [epsilon]D, 1)/[A.sub.-] (x,D, q) E[f.sub.+](x)] = F [h(x, [epsilon])] (3.17)

and applying the estimation (3.16) to (3.12), by the extension theory we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.19)

Considering (3.10) and (3.17) it is easy to verify that the norm N4 admits the estimation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.20)

So it remains to evaluate N3. We remark that [F.sup.-1] [-I[epsilon]/([epsilon][[xi].sub.n]-i)]= [[theta].sup.-] e xn/E is the so called function in the type of boundary layer. It follows (3.19) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.21)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.22)

Here "prime" denotes the norm over the boundary. Using the formula [||h(x', 0, [epsilon])||'.sub.0] [less than or equal to] c [||h(x, [epsilon])||.sup.+].sub.[delta] + 1/2] where 0 < [delta] < 1/2 "+" denotes the norm over the upper half-space. Taking into account the norm of boundary layer function [||[[theta].sup.-]e [[x.sub.n]/E||.sub.0] [??] it follows (3.16) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.23)

Substituting (3.17) into (3.23) we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.24)

Using the evaluation (3.18), (3.20) and (3.24) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or more roughly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.25)

Considering the inequality (3.11) for [N.sub.2] and the inequality (3.25) for [N.sub.1], it follows (3.8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is to say

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.26)

Thus the following theorem is true, which is the generalization of the result in [3]:

Theorem 1. Let

f [member of]{[f.sub.+], [f.sub.-]} [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]) [x] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.-] [equivalent to] H (3.27)

and [??] be the regularizator of problem (3.1) provided the condition f [member of] [H.sub.k-m] is replaced by (3.27). Further, let [R.sub.+] be the regularizator of problem (3.3) with [f.sub.+] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]), then for the operator I If=[??]f-[R.sub.+][f.sub.+]defined by (3.5), the estimation (3.26) is true.

Received March 16, 2005, Accepted December 17, 2005.

References

[1] R. A. ADAMS, Sobolev Space (1975). New York: Academic Press.

[2] I. A. FEDOTOV, Di_raction problems and the extension of Hs-functions. Math. Nachr 241 (2002), 56-64.

[3] I. A. FEDOTOV, Discontinuous pseudo-differential operators. Uspekhi Mat. Nauk 24 6 (1970), 193-194. (Math Review 50, (6017)).

[4] I. M. GEL'FAND, and G. E. SILOV, Generalized funcitons (1968). New York, Academic Press.

[5] J. J. KOHN, and L. NIRENBERG, An algebra of pseudo-differential operators. Comm.Pure and Applied Math. 18 (1968), 269-305.

[6] A. N. TICHONOV, and A. A. SAMARSKI, Equations of mathematical physics (1990). Dover, New York.

[7] M. I. VISHIK, and G. I. ESKIN, Convolution Equations in a Bounded Domain (Russian). Uspekhi Mat. Nauk 20 3 (1965) 89-152. English transl. in, Russian Math. Surv. 20 (1964), 85-151.

[8] V. S. VLADIMIROV, Methods of the Theory of Generalized Functions (2002). London, Taylor, and Francis Inc, 6-109.

Igor Fedotov * and Ying Gai ([dagger])

Department of Mathematical Technology,

Tshwane University of Technology, South Africa

Private Bag X680, Pretoria, 0001, Republic of South Africa

* E-mail:fedotovi@tut.ac.za

([dagger]) E-mail:yingandy@yahoo.com.cn

(1) More general definition of regularizator is given, for example, in [3] or [7].

In this paper we consider problems of diffraction type for elliptic pseudo-differential operators with variable symbols depending on parameters. We compare the regularizators of a diffraction and a Dirichlet problem, and we prove that the regularizator of a diffraction problem tends to the regularizator of a Dirichlet problem as the parameter of the external domain tends to zero.

Keywords and Phrases: Pseudo-differential, Diffraction, Elliptic, Regularizator.

1. Introduction

In this paper we consider problems of diffraction type for elliptic pseudo-differential operators. In more details, we consider simultaneously two pseudo-differential equations elliptic with parameters in different domains with a common boundary. A classical diffraction problem for differential operators was considered, for example, by A.N.Tichonov and A.A. Samarsky ([6]). In the statement of this problem, the homogeneity of a medium is broken by a bounded domain provided that the solution satisfies the conditions of a maximal smoothness on the boundary of this domain. In [3] the analogous problem for pseudo-differential equations was studied, but the main result was obtained only for the case of pseudo-differential operators with constant symbols. In this article we consider the same problem for pseudo-differential equations with variable symbols depending on two parameters, under the condition that one of the parameters tends to infinity. For example, we consider a diffraction problem in [R.sup.n.sub.+] = {x [member of] [R.sup.n], [x.sub.n] [greater than or equal to] 0} and in [R.sup.n.sub.-] (where [R.sup.n.sub.-] = R - [R.sup.n.sub.+]) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where A and B are pseudo-differential operators of order [m.sub.1] and [m.sub.2] elliptic with parameter q and p, respectively. If p is big, then the solution in the half space [R.sup.n.sub.-] has the form of a boundary layer with respect to [x.sub.n]. For instance, the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([x.sub.n] < 0) is boundary layer function. If [epsilon] = 1/p tends to zero, this function approaches zero for [x.sub.n] < 0.

It is possible to prove that if the symbols of operators A and B don't depend on x, then we can find an exact solution of problem (1.1) (see [3])which is defined by the inverse operator. That is, if we write the problem (1.1) in the form Au=f where A={[P.sup.+] A, P-B}, then u=[A.sup.-1]f In the case when A and B depend on x, the inverse operator can not be defined explicitly but if we can find an operator R such that Rf=f+Tf where the operator T has the small norm, we say that the operator R is the regularizator (1) of problem (1.1). We are going to evaluate the difference between the regularizators for problem (1.1) and the Dirichlet problem (1.2) below:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

We prove that the regularizator of the Dirichlet problem (1.2) can be obtained as a limit case in the diffraction problem (1.1) as p = (1/[epsilon]) tends to infinity ([epsilon] [right arrow] 0). We shall use the technique of the theory of pseudo-differential operators developed in [5], [7] and the notations of [2].

2. Notations and Properties

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) be a space of distributions u(x), x=(x', [x.sub.n]) = ([x.sub.1], [x.sub.2],..., [x.sub.n-1], [x.sub.n]) [member of] [R.sup.n] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where p, q are real non-negative parameters,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

The norm on the right-hand side of (2.1) is the usual norm in [L.sub.2] ([R.sup.n.sub.[xi]]).

If p = q = 1, then the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) coincides with the ordinary Sobolev space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]). Since [L.sub.2] and [H.sub.0] are the notations of the same space we shall write further [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] instead of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We introduce also the spaces [H.sub.s] ([R.sup.n.sub.+]) and [H.sub.s] ([R.sup.n.sub.-]) of functions [f.sub.+] and [f.sub.-] defined in [R.sup.n.sub.+] = {x [member of] [R.sup.n]: [x.sub.n] > 0}, ([R.sup.n.sub.-]) = {x [member of] [R.sup.n]: [x.sub.n] < 0}, ([R.sup.n.sub.-]) = [R.sup.n]\[R.sup.n.sub.+], respectively, with the norms

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

and [[theta].sup.+] (x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the extension of the function [f.sub.[+ or -]] on the whole Euclidean space [R.sup.n] such that the extension belongs to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]).

We state some properties of the operator [[PI].sup.[+ or -]]:

* 1. The operator [[PI].sup.[+ or -]] is defined on smooth decreasing functions by the formula (2.3). Since the operator of multiplication of the Heaviside function [theta].sup.[+ or -]] (x) is bounded in [H.sub.0] ([R.sup.n.sub.x]), the operator [[PI].sup.[+ or -]] is bounded in the space [H.sub.0] ([R.sup.n.[xi]]) being the dual of [H.sub.0] ([R.sup.n.sub.x]) with respect to the Fourier transform. For arbitrary function [??]([xi]) [member of] H0 ([R.sup.n.sub.-]) the formula (2.3) is understood as the closure of the opeator [[PI].sup.[+ or -]].

2. If [??] ([xi]) [member of] [H.sub.0] (([R.sup.n.sub.[xi]]), then this function can be represented as the sum [??] ([xi]) = [??] ([xi]) + [??] ([xi]), where [??][+ or -] ([xi]) = [[PI].sup.[+ or -]] [??] ([xi]).

3. Since [theta].sup.+] (x) = 0 for [x.sub.n] < 0 ([[theta].sup.-] (x) = 0 for [x.sub.n] > 0), the function [[PI].sup.+] ([xi]) ([[PI].sup.-] [??] ([xi])) admits an analytic continuation in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0).

4. If a function [??] ([xi]) ([??] ([xi])) [member of] [H.sub.0] ([R.sup.n.sub.[xi]]) and may be extended in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0), then [[PI].sup.[+ or -]] [??] = 0.

5. If the functions [[PI].sup.[+ or -]] [??] ([xi]) and [[PI].sup.[+ or -]] [[??].sub.[+ or -]] ([xi]) [??] ([xi])]make sense, where [??] ([xi]) ([??] ([xi])) admit an analytic continuation in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0)), then [[PI].sup.[+ or -]] [[??].sub.[+ or -]] ([xi])] [??] [xi] [[PI].sup.[+ or -]] [??] ([xi])]

Let f ={[f.sub.+], [f.sub.-]} [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]) [x] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (R.sup.n.sub.-]. On this product space we can introduce a natural operation of addition and multiplication by a function [PHI] [member of] [C.sup.[infinity]] ([R.sup.n]) by the following rule: If f ={[f.sub.+], [f.sub.-]} and g ={[g.sub.+], [g.sub.-]}, then f + g ={[f.sub.+] + [g.sub.+], [f.sub.-] + [g.sub.-]} and [phi]f ={[phi][f.sub.+], [phi][f.sub.-]}. We can also introduce a natural norm on this set.

Let A and B be two pseudo-differential operators whose symbols are [sigma] (A) = a(x, [xi], q) and [signma] (B) = b(x, [sigma], p), respectively. Recall that a pseudo-differential operator corresponding to the symbol a(x, [xi]) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

We suppose that the symbols a and b depend on parameters q and p (where q [less than or equal to] p), respectively, and satisfy the following conditions:

* 1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. The functions a(x, [xi], q) and b(x, [xi], p) are homogeneous of order [m.sub.1] and [m.sub.2], ([m.sub.1] and [m.sub.2] are positive) with respect to [xi], q and [xi], p, respectively.

3. The operators A and B are elliptic with parameter, i.e. a(x, [xi], q) [not equal to] 0 for real [xi] and for q + |[xi]| [not equal to] = 0, and b(x, [xi], p) 6= 0 for real [xi] and for p + |[xi]| [not equal to] = 0.

4. For every value of multi-indexes [alpha] = ([[alpha].sub.1], [[alpha].sub.2],..., [[alpha].sub.n]), [beta] = ([[beta].sub.1], [[beta].sub.2],..., [[beta].sub.n]), the following estimations hold:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

|alpha]| = [[alpha].sub.1] + [[alpha].sub.2] + ... + [[alpha].sub.n], |[beta]| = [[beta].sub.1] + [[beta].sub.2] + ... + [[beta].sub.n]

5. The symbols a and b can be represented in the form

a(x, [xi], q) = a(1, [xi], q) + a'(x, [xi], q),

b(x, [xi], p) = b(1, [xi], p) + b'(x, [xi], p)

where a'(x, [xi], q) and b'(x, [xi], p) are infinitely differentiable functions with respect to x, with compact supportt, i.e. they belong to [C.sup.[[infinity].sub.0] ([R.sup.n.sub.x]).

We remark that a pseudo-differential operator with a symbol satisfying the condition 5. can be defined by the following formula, which is equivalent to the formula (2.4):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

where the tilde "~" dentoes the Fourier transform with respect to the first argument. In [7] M.Vishik and G. Eskin have proved that symbols satisfying the conditions 1. - 5. admit the following factorization:

a(x, [xi], q) = [a.sub.+](x, [xi]', [[xi].sub.n], q)[a.sub.-](x, [xi]', [[xi].sub.n], q), (2.7)

and

b(x, [xi], p) = [b.sub.+](x, [xi]', [[xi].sub.n], p)b-(x, [xi]', [[xi].sub.n], p) (2.8)

where [a.sub.+](x, [xi]', [[xi].sub.n], q), b+(x, [xi]', [[xi].sub.n], p) ([a.sub.-](x, [xi]', [[xi].sub.n], q), [b.sub.-](x, [xi]', [[xi].sub.n], p)) are functions admitting an analytic continuation in the half-plane Im [[xi].sub.n] > 0 (Im [[xi].sub.n] < 0) and they remain homogeneous with respect to [xi], q ([xi], p). Suppose that ord [a.sub.+](x, [xi]', [[xi].sub.n], q) = [k.sub.1], ord [b.sub.-](x, [xi]', [[xi].sub.n], p) = [k.sub.2] [greater than or equal to] 0, (k = [k.sub.1] + [k.sub.2] > 0) and the orders do not depend on x.

3. Evaluation of The Di_erence Between The Regularizators of Di_raction and of Dirichlet Problems

Consider a function f[member of] [H.sub.k-m] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We also introduce the couple operator ([3]) Au={[P.sup.+] Au, [P.sup.-] Bu}where [P.sup.+] ([P.sup.-]) is the restriction operator of distributions on [R.sub.+] ([R.sub.-]) (it is clear that for ordinary functions it coincides with Heaviside function [[theta].sup.+] ([[theta].sup.-])) and the operator A (B) has the symbol a(x, [xi], q) (b(x, [xi], p)).

We consider the following diffraction problem

Au = f [member of] [H.sub.k-m], u [member of] [H.sub.k] ([R.sup.n]) (3.1)

It follows from [3] that problem (3.1) has a unique solution for su_ciently large values of parameters p and q. The proof is based on construction of the regularizator of this problem which has the following form:

[??]f = [R.sub.1] [[[theta].sup.+] [R.sub.-]E[f.sub.+] + [[theta].sup.-][R.sub.+]E[f.sub.-]] Or equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

where [R.sub.1], [R.sub.-] and [R.sub.+] are pseudo-differential operators with symbols

[[a.sub.+](x, [xi]', [[xi].sub.n], q)[b.sub.-][(x, [xi]', [[xi].sub.n], p)].sup.-1], [b.sub.-](x, [xi]', [[xi].sub.n], p) [[a.sub.-][(x, [xi]', [[xi].sub.n], q)].sup.-1]

and

[a.sub.+](x, [xi]', [[xi].sub.n], q) [[b.sub.+][(x, [xi]', [[xi].sub.n], p)].sup.-1]

respectively.

Consider at the same time with problem (3.1) the following Dirichlet problem

[P.sup.+] A(D, x, q)[u.sup.(0).sub.+] = [f.sub.+](x), [u.sup.(0).sub.+] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]) (3.3)

here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (R.sup.n.sub.+]) is the subspace of the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]) ([k.sub.1] [greater than or equal to] 0) of functions, which vanish on [R.sup.n.sub.-]. The regularizator of this equation was constructed by M. Vishik and G. Eskin in [7] and it has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

We shall prove that if p [right arrow] [infinity],then [??]f [right arrow] [R.sub.+][f.sub.+]. It means that the regularizator of Dirichlet problem (3.3) may be obtained as a limit case of the problem (3.1) when p approaches infinity. We represent the difference of these two operators (3.2) and (3.4) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

Since the smoothness of this difference is [k.sub.1], we estimate the norm of I f in the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n]). We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

Using 1 = [[theta].sup.+] + [[theta].sup.-], we transform the term 1/[B.sub.-](x,D,p) [[theta].sup.+] [B.sub.](x,D,p)/[A.sub.-](x,D,q) E[f.sub.+](x) as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Substituting (3.7) into (3.6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.8)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

We consider separately the operator 1/[B.sub.-](x,D,p). Let us set p = 1/[epsilon] and transform this operator as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

Moreover we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, for [N.sub.2] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

Substituting (3.10) into (3.9), for [N.sub.1] we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

where we denote

[T.sub.-] = [B.sub.-](x, [epsilon]D, 1) - [B.sub.-](x, 0, 1)/[B.sub.-](x, 0, 1)[B.sub.-](x, [epsilon]D, 1) (3.13)

with the symbol

[sigma]([T.sub.-]) = [b.sub.-](x, [epsilon][xi], 1) - [b.sub.-](x, 0, 1)/[b.sub.-](x, 0, 1)[b.sub.-](x, [epsilon][xi], 1) (3.14)

We expand this symbol [sigma]([T.sub.-]) as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.15)

By virtue of assumption (4) for homogeneous symbols given in section 2, we have the following inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.16)

Denoting

[??]([xi], [epsilon]) = F [[B.sub.-](x, [epsilon]D, 1)/[A.sub.-] (x,D, q) E[f.sub.+](x)] = F [h(x, [epsilon])] (3.17)

and applying the estimation (3.16) to (3.12), by the extension theory we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.18)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.19)

Considering (3.10) and (3.17) it is easy to verify that the norm N4 admits the estimation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.20)

So it remains to evaluate N3. We remark that [F.sup.-1] [-I[epsilon]/([epsilon][[xi].sub.n]-i)]= [[theta].sup.-] e xn/E is the so called function in the type of boundary layer. It follows (3.19) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.21)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.22)

Here "prime" denotes the norm over the boundary. Using the formula [||h(x', 0, [epsilon])||'.sub.0] [less than or equal to] c [||h(x, [epsilon])||.sup.+].sub.[delta] + 1/2] where 0 < [delta] < 1/2 "+" denotes the norm over the upper half-space. Taking into account the norm of boundary layer function [||[[theta].sup.-]e [[x.sub.n]/E||.sub.0] [??] it follows (3.16) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.23)

Substituting (3.17) into (3.23) we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.24)

Using the evaluation (3.18), (3.20) and (3.24) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or more roughly

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.25)

Considering the inequality (3.11) for [N.sub.2] and the inequality (3.25) for [N.sub.1], it follows (3.8) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

That is to say

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.26)

Thus the following theorem is true, which is the generalization of the result in [3]:

Theorem 1. Let

f [member of]{[f.sub.+], [f.sub.-]} [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]) [x] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.-] [equivalent to] H (3.27)

and [??] be the regularizator of problem (3.1) provided the condition f [member of] [H.sub.k-m] is replaced by (3.27). Further, let [R.sub.+] be the regularizator of problem (3.3) with [f.sub.+] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([R.sup.n.sub.+]), then for the operator I If=[??]f-[R.sub.+][f.sub.+]defined by (3.5), the estimation (3.26) is true.

Received March 16, 2005, Accepted December 17, 2005.

References

[1] R. A. ADAMS, Sobolev Space (1975). New York: Academic Press.

[2] I. A. FEDOTOV, Di_raction problems and the extension of Hs-functions. Math. Nachr 241 (2002), 56-64.

[3] I. A. FEDOTOV, Discontinuous pseudo-differential operators. Uspekhi Mat. Nauk 24 6 (1970), 193-194. (Math Review 50, (6017)).

[4] I. M. GEL'FAND, and G. E. SILOV, Generalized funcitons (1968). New York, Academic Press.

[5] J. J. KOHN, and L. NIRENBERG, An algebra of pseudo-differential operators. Comm.Pure and Applied Math. 18 (1968), 269-305.

[6] A. N. TICHONOV, and A. A. SAMARSKI, Equations of mathematical physics (1990). Dover, New York.

[7] M. I. VISHIK, and G. I. ESKIN, Convolution Equations in a Bounded Domain (Russian). Uspekhi Mat. Nauk 20 3 (1965) 89-152. English transl. in, Russian Math. Surv. 20 (1964), 85-151.

[8] V. S. VLADIMIROV, Methods of the Theory of Generalized Functions (2002). London, Taylor, and Francis Inc, 6-109.

Igor Fedotov * and Ying Gai ([dagger])

Department of Mathematical Technology,

Tshwane University of Technology, South Africa

Private Bag X680, Pretoria, 0001, Republic of South Africa

* E-mail:fedotovi@tut.ac.za

([dagger]) E-mail:yingandy@yahoo.com.cn

(1) More general definition of regularizator is given, for example, in [3] or [7].

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Author: | Gai, Ying |
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Publication: | Tamsui Oxford Journal of Mathematical Sciences |

Date: | May 1, 2006 |

Words: | 3113 |

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