On some pseudo-differential equations and transmutation operators

Vladimir Vasilyev

doi:10.1088/1742-6596/1479/1/012009

Abstract

We describe some operators for solving model elliptic pseudo-differential equations in special canonical domains. It helps us to write a general solution of corresponding pseudo-differential equation in an explicit form. Moreover, knowing a general solution we can choose additional (possibly boundary) conditions to determine uniquely the solution. All considerations we give in Sobolev–Slobodetskii spaces.

Highlights

For studying pseudo-differential equations on manifolds the main difficulty is to obtain invertibility conditions for a model pseudo-differential equation in a so-called canonical domain
We describe some operators for solving model elliptic pseudo-differential equations in special canonical domains
It helps us to write a general solution of corresponding pseudodifferential equation in an explicit form

Summary

Introduction

For studying pseudo-differential equations on manifolds the main difficulty is to obtain invertibility conditions for a model pseudo-differential equation in a so-called canonical domain. 2. Elliptic symbols and wave factorization We will consider the operators in the Sobolev – Slobodetskii space Hs(Rm) with norm. We are interested in studying invertibility of the operators in corresponding Sobolev – Slobodetskii spaces. Definition Wave factorization of symbol A(ξ) with respect to the cone C is called its representation in the form. For concrete cones it is possible to calculate such operators, but before we will give the main theorem. To formulate this theorem we will introduce a special integral operator [11]. 4. Examples We will give some calculations for the operator Vφ for two concrete cones

A flat case

A spatial case