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To summarize briefly, this book is devoted to an exposition of the foundations of pseudo differential equations theory in non-smooth domains. The elements of such a theory already exist in the literature and can be found in such papers and monographs as [90,95,96,109,115,131,132,134,135,136,146, 163,165,169,170,182,184,214-218]. In this book, we will employ a theory that is based on quite different principles than those used previously. However, precisely one of the standard principles is left without change, the "freezing of coefficients" principle. The first main difference in our exposition begins at the point when the "model problem" appears. Such a model problem for differential equations and differential boundary conditions was first studied in a fundamental paper of V. A. Kondrat'ev [134]. Here also the second main difference appears, in that we consider an already given boundary value problem. In some transformations this boundary value problem was reduced to a boundary value problem with a parameter . -\ in a domain with smooth boundary, followed by application of the earlier results of M. S. Agranovich and M. I. Vishik. In this context some operator-function R('-\) appears, and its poles prevent invertibility; iffor differential operators the function is a polynomial on A, then for pseudo differential operators this dependence on . -\ cannot be defined. Ongoing investigations of different model problems are being carried out with approximately this plan, both for differential and pseudodifferential boundary value problems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Distributions and Their Fourier transforms

Abstract
Distributions are linear continuous functionals over the space of so-called basic functions. Space of basic functions we choose as the Schwartz space S(ℝ m ) (ℝ m is m — dimensional Euclidean space) of infinitely differentiable on ℝ m functions decreasing under |x| → ∞ more rapidly than any power of |x|−1, x = (x 1, ..., x m ), \(|x| = \sqrt {{x_1}^2 + \ldots + {x_m}^2} .\) We determine the counting number of norms in S(ℝ m ) by the formula
$$||\varphi |{|_p} = \sup {(1 + |x{|^2})^{p/1}}|{D^a}\varphi (x)|,\varphi \in S({\mathbb{R}^m}),p = 0,1 \ldots ,|a| \leqslant p$$
(1.1.1)
where \({D^a}\varphi = \frac{{{\partial ^{|\alpha |}}\varphi }}{{\partial {x_1}^{{\alpha _1}} \ldots \partial {x_m}^{{\alpha _m}}}}\), α is multi index, |α| = α 1 + ⋯ + α m ; with the help of these norms we define the convergence concept in S(ℝ m ). Namely we say the sequence φ 1, ..., φ k , ... of functions from S(ℝ m ) converges to function φS(ℝ m ) iff ∥φ k φ p → 0, k → ∞ for all ρ = 0, 1, ... The last statement, by virtue of (1.1.1) is equivalent to saying that x α D β φ k (x) uniformly tends to zero under k → ∞ for arbitrary multiindex \(\alpha ,\beta ,{x^\alpha } \equiv {x_1}^{{\alpha _1}} \ldots {x_m}^{{\alpha _m}}\).
Vladimir B. Vasil’ev

Chapter 2. Multidimensional complex analysis

Abstract
A cone C ⊂ ℝ m (with top at the origin) is point set which has properly: if yC then λyC for all λ > 0.
Vladimir B. Vasil’ev

Chapter 3. Sobolev-Slobodetskii spaces

Abstract
Let s be an arbitrary real number. By definition the Sobolev-Slobodetskii space H s (ℝ m ) consists of distributions u for which their Fourier transforms are locally integrable functions ũ(ξ)such that
$$ \left\| u \right\|_s^2 = {\int\limits_{{\mathbb{R}^m}} {\left| {\mu \left( \xi \right)} \right|} ^2}{\left( {1 + \left| \xi \right|} \right)^{2s}}d\xi < + \infty .$$
(3.1.1)
Vladimir B. Vasil’ev

Chapter 4. Pseudodifferential operators and equations in a half-space

Abstract
Let A(ξ) be a locally integrable function satisfying the inequality
$$ \left| {A\left( \xi \right)} \right| \leqslant C{\left( {1 + \left| \xi \right|} \right)^\alpha }. $$
(4.1.1)
Vladimir B. Vasil’ev

Chapter 5. Wave factorization

Abstract
It was clearly shown in the previous section that, in the theory of boundary value problems for elliptic pseudodifferential equations which was constructed by M.I.Vishik and G.I.Eskin, the key role was played the symbol factorization of elliptic pseudodifferential operators on one of variables with the other variables fixed. Depending on factorization index one needs to consider the different boundary value problems reflecting the singularities of appeared situation. When studying the pseudodifferential equation in cone (angle) it proves that analogous role will play the wave factorization related to exit into compex domain with respect to all variables at once. Such factorization permits one to use a multivariable variant of the Wiener-Hopf method and to obtain for a pseudodifferential operator in angle the same solvability picture which we had in the half-space case. This approach has been applied to some classes of differential operators too, but as of now it has not been useful. In this section we will introduce the concept of wave factorization for symbols of pseudodifferential operators and we will verify that the set of symbols which admit the wave factorization is large enough.
Vladimir B. Vasil’ev

Chapter 6. Diffraction on a quadrant

Abstract
Diffraction theory, one of the most important areas of mathematical physics, has recently been under intensive development. Integral or more generally pseudodifferential equations take an exceptional place in diffraction theory; as a rule diffraction problems reduce to one or another of them [69,123,57,118]. This reduction to boundary equations is achieved by different methods, depending on the particular equation type, and the investigations in many cases lead to existence and uniqueness theorems. One of the methods used for solving the boundary equations thus attained is the Wiener-Hopf method, or the factorization method, which has been successfully applied to some diffraction problems [166­168,175,240,256,257]. We suggested here a multidimensional generalization of this method, and with its help we will study pseudodifferential equations arising from a diffraction problem on a quadrant, obtained in [168]. The solution in the simplest case of this problem can be written in explicit form and is more appealing than the formula found in [168].
Vladimir B. Vasil’ev

Chapter 7. The problem of indentation of a wedge-shaped punch

Abstract
One important class of problems considered in elasticity theory is the class of so-called contact problems. Its investigation was begun at the end of 19th century; it significantly advanced [67,106,141,142,143,178,202,212] by application of different methods and results of differential and integral equations theory. In recent years methods of integral equations (potential theory) have been greatly extended because, as a rule, contact problems reduce to equations of this type [192,193]. But solution of these integral equations meets with serious mathematical difficulties. A large number of papers is devoted to the study of special types of such equations [106,178] when a punch has fixed form (for example, the punch is circular, elliptical, or wedge-shaped, etc.), and in these papers they develop asymptotic methods of solution.
Vladimir B. Vasil’ev

Chapter 8. Equations in an infinite plane angle

Abstract
In this section we will put these “semi-empirical” studies of Chapters 6,7 related to concrete applied problems, on a stable mathematical base and we will investigate the questions of solvability of a number of general classes of pseudodifferential equations in cones.
Vladimir B. Vasil’ev

Chapter 9. General boundary value problems

Abstract
Here we will consider the question of correct statement of a boundary value problem for pseudodifferential equation in an angle on a plane which we will give on the basis of Theorem 8.1.2, and we will begin with Dirichlet and Neumann problems. Everywhere below in Section 9.1 we assume:
$$m = 2, n = 2, f = 0, a = 1.$$
Vladimir B. Vasil’ev

Chapter 10. The Laplacian in a plane infinite angle

Abstract
This section is a “realization” of methods, which are developed in Chapter 9 applied to the Laplacian and contains concrete calculations. Similar [238,239] and more complicated problems [94,134,207,208] were considered earlier by other methods, and as a result, it gave the possibility to obtain the theorem of existence and uniqueness of solution under some restrictions on parameters of functional spaces, size of angle and so on. In this section we have chosen similar functional spaces in which our methods give the possibility to formulate the conditions under fulfilling of which the solution of the posed problem exists and is unique (including explicit construction for solution in terms of Fourier and Mellin transforms). Other approaches one can find in papers which are contained in the list of references from [135].
Vladimir B. Vasil’ev

Chapter 11. Problems with potentials

Abstract
Here we consider again the equation
$$\left( {A{u_ + }} \right)\left( x \right) = f\left( x \right),x \in C_ + ^a$$
(11.1)
, to show how one can use Theorem 8.1.3 and what kind of boundary value problems correspond to this case. We assume \(\wp - s = - 1 + \delta ,\left| \delta \right| < 1/2\), æ is index of wave factorization of symbol \(A\left( \xi \right) \in C_a^\infty \).
Vladimir B. Vasil’ev

Backmatter

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