Wave Factorization of Elliptic Symbols: Theory and Applications

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 "e;freezing of coefficients"e; principle. The first main difference in our exposition begins at the point when the "e;model problem"e; 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.