Problems of diffraction type for elliptic pseudo-differential operators with variable symbols.Abstract In this paper we consider problems of diffraction type for elliptic el·lip·tic or el·lip·ti·cal adj. 1. Of, relating to, or having the shape of an ellipse. 2. Containing or characterized by ellipsis. 3. a. pseudo-differential operators In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory. with variable symbols depending on parameters. We compare the regularizators of a diffraction and a Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. , 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 differential operator In mathematics, any combination of derivatives applied to a function. It takes the form of a polynomial of derivatives, such as D2xx − D2xy ∙ D was considered, for example, by A.N.Tichonov and A.A. Samarsky ([6]). In the statement of this problem, the homogeneity Homogeneity The degree to which items are similar. of a medium is broken by a bounded domain provided that the solution satisfies the conditions of a maximal max·i·mal adj. 1. Of, relating to, or consisting of a maximum. 2. Being the greatest or highest possible. smoothness on the boundary of this domain. In [3] the analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development. a·nal·o·gous adj. 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 infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. . 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 A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] (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 boundary layer In fluid mechanics, a thin layer of flowing gas or liquid in contact with a surface (e.g., of an airplane wing or the inside of a pipe). The fluid in the boundary layer is subjected to shear forces. 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 (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. 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 right-hand side n → derecha right-hand side right n → rechte Seite f right-hand side n → lato destro 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 In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp norms of the function itself as well as its derivatives up to a given order. [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 Theta A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option. ].sup.+] (x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the extension of the function [f.sub.[+ or -]] on the whole Euclidean space Euclidean space In geometry, a two- or three-dimensional space in which the axioms and postulates of Euclidean geometry apply; also, a space in any finite number of dimensions, in which points are designated by coordinates (one for each dimension) and the distance between [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 multiplication, fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. 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 Fourier transform In mathematical analysis, an integral transform useful in solving certain types of partial differential equations. A function's Fourier transform is derived by integrating the product of the function and a kernel function (an exponential function raised to . For arbitrary function See under Arbitrary. See also: 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 complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series 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 phi n. Symbol  The 21st letter of the Greek alphabet.PHI, n See health information, protected. ] [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 The same. Contrast with heterogeneous. homogeneous - (Or "homogenous") Of uniform nature, similar in kind. 1. In the context of distributed systems, middleware makes heterogeneous systems appear as a homogeneous entity. For example see: interoperable network. 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 dif·fer·en·tia·ble adj. 1. That can be differentiated: differentiable species. 2. Mathematics Possessing a derivative. 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 A symbol used in Windows, starting with Windows 95, that maintains a short version of a long file or directory name for compatibility with Windows 3.1 and DOS. For example, the short version of a file named "Letter to Joe" would be LETTER~1. Then "Letter to Pat" becomes LETTER~2. "~" 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 fac·tor·ize tr.v. fac·tor·ized, fac·tor·iz·ing, fac·tor·iz·es Mathematics To factor. fac : 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 Noun 1. subspace - a space that is contained within another space mathematical space, topological space - (mathematics) any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional" 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 de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. [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 inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. [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 estimation In mathematics, use of a function or formula to derive a solution or make a prediction. Unlike approximation, it has precise connotations. In statistics, for example, it connotes the careful selection and testing of a function called an estimator. (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 theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. is true, which is the generalization gen·er·al·i·za·tion n. 1. The act or an instance of generalizing. 2. A principle, a statement, or an idea having general application. 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 New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : 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 discontinuous /dis·con·tin·u·ous/ (dis?kon-tin´u-us) 1. interrupted; intermittent; marked by breaks. 2. discrete; separate. 3. lacking logical order or coherence. 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 gen·er·al·ized adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. funcitons (1968). New York, Academic Press. [5] J. J. KOHN, and L. NIRENBERG, An algebra algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as of pseudo-differential operators. Comm See comms. .Pure and Applied Math. 18 (1968), 269-305. [6] A. N. TICHONOV, and A. A. SAMARSKI, Equations of mathematical physics mathematical physics Branch of mathematical analysis that emphasizes tools and techniques of particular use to physicists and engineers. It focuses on vector spaces, matrix algebra, differential equations (especially for boundary value problems), integral equations, integral (1990). Dover, New York Dover is a town in Dutchess County, New York, United States. The population was 8,565 at the 2000 census. The town was named after Dover in England, the home town of an early settler. The Town of Dover is located on the eastern boundary of the county. . [7] M. I. VISHIK, and G. I. ESKIN, Convolution convolution /con·vo·lu·tion/ (-loo´shun) a tortuous irregularity or elevation caused by the infolding of a structure upon itself. 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 Not to be confused with generic function. In mathematics, generalized functions are objects generalizing the notion of functions. There is more than one recognised theory. (2002). London, Taylor, and Francis Inc, 6-109. Igor Fedotov * and Ying Gai ([dagger]) Department of Mathematical Technology, Tshwane University of Technology Tshwane University of Technology (TUT) is a higher education institution in South Africa that came into being through a merger of three technikons — Technikon Northern Gauteng, Technikon North-West and Technikon Pretoria. , South Africa South Africa, Afrikaans Suid-Afrika, officially Republic of South Africa, republic (2005 est. pop. 44,344,000), 471,442 sq mi (1,221,037 sq km), S 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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