A boundary-value problem with a shift for two functions on a riemannian surface with the boundary in the class of generalized functions

1975 ◽  
Vol 27 (2) ◽  
pp. 228-231 ◽  
Author(s):  
S. A. Yatsenko
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Generalized Functions ◽  
Riemannian Surface

A quasi-periodic boundary value problem for the Laplacian and the continuation of its resolvent

1979 ◽  
Vol 82 (3-4) ◽  
pp. 251-272 ◽  
Author(s):  
H. D. Alber
Keyword(s):  
Boundary Value Problem ◽  
Periodic Boundary ◽  
Differential Operators ◽  
Periodic Structures ◽  
Boundary Value ◽  
Plane Waves ◽  
Periodic Boundary Value Problem ◽  
Riemannian Surface ◽  
Analytic Perturbation Theory ◽  
Families Of Operators

SynopsisA quasi-periodic boundary value problem for the Helmholtz equation in an unbounded domain is considered. This problem arises from scattering of plane waves by periodic structures.Existence and uniqueness theorems are proved, and the continuation of the resolvent of this problem to a Riemannian surface is constructed. This construction makes no use of the continuation of the resolvent kernel but runs along the following lines:First a family of differential operators is defined, which is holomorphic in a generalized sense. Then, using a result from analytic perturbation theory about families of operators with compact resolvent, it is shown that the family of inverses of these differential operators gives the desired continuation.

scholarly journals Correctness conditions for boundary value problems

2020 ◽  
pp. 128-137
Author(s):  
D. Levkin ◽  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Mathematical Models ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Generalized Functions ◽  
Technological Processes ◽  
Multilayer Medium ◽  
Space Of Generalized Functions ◽  
Physical Fields

The article deals with the issues of mathematical modeling of technological systems that contain physical fields’ sources. It is believed that in the case of a simple spatial form of the object under study, the boundary value problems will be correct. The interest lies in mathematical models for nonlinear, multilayer objects under the influence of load sources, for which, using the traditional theory of existence and unity, it is impossible to guarantee the correctness of boundary value problems. The author considers boundary value problems for systems of differential and pseudo differential equations in a multilayer medium which describe the state of the studied systems under the action of discrete load sources. The correctness of such problems is proven using the theory of distributions over the space of generalized functions. The object of research is boundary value problems for systems of differential and pseudo differential equations in a multilayer medium. The aim of the research is to build correct boundary value problems, which underlie the calculated mathematical models of the process of action of physical fields on multilayer objects. The necessary and sufficient conditions for the correctness of the parabolic boundary value problem in the space of generalized functions are obtained in the article. It is shown that its solution is infinitely differentiated by a spatial variable. The results of the research can be used to obtain the conditions for the correctness of the boundary value problem for differential equations with variable coefficients. Note that, in some cases, the correctness of the calculated mathematical models determines the correctness of applied optimization mathematical models. The application of the author's research is possible when proving the correctness of boundary value problems for a number of technological processes. The universality of the research allows to widely usage of the results obtained in this work to improve the quality of technological processes.

Sign in / Sign up

Export Citation Format

Share Document