Abstract
In this paper, we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. There are no assumptions on the regularity of the boundary which is allowed to have arbitrary singularities. We give applications of the developed analysis to obtain a priori estimates for solutions of boundary value problems that are elliptic within the constructed calculus.
Highlights
We are interested in questions devoted to the global solvability and further properties of boundary value problems in Rn
Given a problem for some pseudodifferential operator A with fixed boundary conditions in a domain Ω ⊂ Rn, the main idea for our analysis is to develop a suitable pseudo-differential calculus in which the given boundary value problem can be solved and its solution can be efficiently estimated
Section 3: we introduce elements of the global theory of distributions DL (Ω) in Ω adapted to the boundary value problem LΩ
Summary
We are interested in questions devoted to the global solvability and further properties of boundary value problems in Rn. A global analysis of pseudo-differential operators on the torus based on the Fourier series representations of functions with further applications to the spectral theory was originated by Agranovich [1], with further developments of its different aspects by Agranovich [2], Amosov [3], Elschner [22], McLean [42], Melo [43], Prossdorf and Schneider [50], Saranen and Wendland [58], Turunen and Vainikko [70], Vainikko and Lifanov [71], and others Most of these papers deal with one-dimensional cases or with classes of operators rather than with classes of symbols. Section 10: the notion of difference operators is used to define Hormandertype classes induced by the boundary value problem LΩ and to develop elements of its symbolic calculus. An application is given to obtain a priori estimates for solutions to boundary value problems to elliptic operators
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
