Non-Homogeneous Boundary Value Problems and Applications: Vol. 3

Description

1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu == f in (9, an open set in R N , (2) fQjU == gj on 8(9 (boundp,ry of (f)), lor on a subset of the boundary 8(9 1 < i < v, where P is a linear differential operator in (9 and where the Q/s are linear differen tial operators on 8(f). In Volumes 1 and 2, we studied, for particular classes of systems {P, Qj}, problem (1), (2) in classes of Sobolev spaces (in general constructed starting from L2) of positive integer or (by interpolation) non-integer order; then, by transposition, in classes of Sobolev spaces of negative order, until, by passage to the limit on the order, we reached the spaces of distributions of finite order. In this volume, we study the analogous problems in spaces of infinitely differentiable or analytic functions or of Gevrey-type functions and by duality, in spaces of distributions, of analytic functionals or of Gevrey- type ultra-distributions. In this manner, we obtain a clear vision (at least we hope so) of the various possible formulations of the boundary value problems (1), (2) for the systems {P, Qj} considered here.