General boundary value problems for pseudo-differential equations and related difference equations
© Vasilyev; licensee Springer. 2013
Received: 1 March 2013
Accepted: 11 September 2013
Published: 7 November 2013
The author develops the theory of pseudo-differential equations and boundary value problems in non-smooth domains. A model pseudo-differential equation in a canonical flat domain is reduced to a system of linear difference equations.
It is well known that the term ‘elliptic boundary value problem’ means not only satisfying certain equation in inner points of a manifold, but satisfying some boundary conditions as well. But it is not enough. These boundary conditions have been assiciated with inner equation, and these conditions of concordance are called Shapiro-Lopatinskii conditions. Indeed, this condition is a non-vanishing determinant of a certain system of linear algebraic equations. But it is true for smooth boundary only, because it may be locally straightened by diffeomorphism transforming to a hyper-plane. So, locally, one considers a model problem with frozen coefficients in a half-space. Originating boundary conditions are determined by an infinite number of solutions, which are linear combinations of some arbitrary functions. For unique defining of these functions, one needs certain additional conditions. Such conditions are more usable as traces of certain pseudo-differential operators on a boundary. Of course, there is a possibility for other kinds of boundary conditions. This phenomenon was explained mathematically in papers by Vishik and Eskin (see ) using the concept of factorization for an elliptic symbol of a pseudo-differential operator. The author tried to apply this idea to a multidimensional situation and non-smooth boundary , and for some cases, typical analogues of the mentioned results were obtained. However, for general boundary conditions, this problem is very hard, and maximum that we can obtain is the equivalent system of linear difference equations for defining arbitrary functions. This is the main conclusion of this paper.
2 General boundary value problem
where , are positive constants.
is defined, generally speaking, on the set only;
- (2)admits an analytical continuation into the radial tube domain over the cone , which satisfies the following estimate:
Similar properties must have the factor with change instead of and instead of æ.
The number æ is called an index of wave factorization of with respect to the cone .
The space consists of functions from with support in (closure of ).
in the space if the right-hand side is taken from the space of distributions which admit a continuation lu on the whole , and this continuation belongs to . The norm in is defined by formula . We suppose the symbol admits the wave factorization.
The following theorem is valid .
where , are arbitrary functions from , , respectively, , , , Q is arbitrary polynomial satisfying the estimate (∗) for , and formula (2) describes all possible solutions of equation (1).
and, analogously, for .
3 Originating difference equations
and then denotes the restriction , , on ; is the restriction . Analogously, we define , , , , , .
One can also use analogous arguments for the second integral.
can be applied.
Theorem 2 Boundary value problem (1), (3) is uniquely solvable if and only if the system of difference equations (8) is uniquely solvable.
The simple difference system of first order was found and described by the author recently . This example shows that even for the simplest cases, we obtain the system of difference equations of first order with variable coefficients. There are no methods for solving such systems, and we need to develop new approaches for this purpose.
The author is very grateful to referees for their remarks and useful suggestions.
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