Non-Homogeneous Boundary Value Problems and Applications

1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2) fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J am 1 f v, where Pis a linear differential operator in m and where the Q/s are linear differential operators on am. In Volumes 1 and 2, we studied, for particular c1asses of systems {P, Qj}, problem (1), (2) in c1asses of Sobolev spaces (in general constructed starting from P) of positive integer or (by interpolation) non-integer order; then, by transposition, in c1asses 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 inlinitely dilferentiable or analytic Itlnctions or of Gevrey-type I~mctions and by duality, in spaces 01 distribtltions, of analytic Itlnctionals or of Gevrey type ultra-distributions. In this manner, we obtain a c1ear 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.

Inhalt
7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 2. Scalar-Valued Ultra-Distributions of Class Mk; Generalizations.- 3. Spaces of Analytic Functions and of Analytic Functionals.- 4. Vector-Valued Functions of Class Mk.- 5. Vector-Valued Ultra-Distributions of Class Mk; Generalizations.- 6. Comments.- 8 Elliptic Boundary Value Problems in Spaces of Distributions and Ultra-Distributions.- 1. Regularity of Solutions of Elliptic Boundary Value Problems in Spaces of Analytic Functions and of Class Mk; Statement of the Problems and Results.- 2. The Theorem on Elliptic Iterates: Proof.- 3. Application of Transposition; Existence of Solutions in the Space D'(?) of Distributions.- 4. Existence of Solutions in the Space $$D{'_{{M_k}}}\left( \Omega \right)$$ of Ultra-Distributions.- 5. Comments.- 6. Problems.- 9 Evolution Equations in Spaces of Distributions and Ultra-Distributions.- 1. Regularity Results. Equations of the First Order in t.- 2. Equations of the Second Order in t.- 3. Singular Equations of the Second Order in t.- 4. Schroedinger-Type Equations.- 5. Stability Results in Mk-Classes.- 6. Transposition.- 7. Semi-Groups.- 8. Mk -Classes and Laplace Transformation.- 9. General Operator Equations.- 10. The Case of a Finite Interval ]0, T[.- 11. Distribution and Ultra-Distribution Semi-Groups.- 12. A General Local Existence Result.- 13. Comments.- 14. Problems.- 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions.- 1. Regularity in the Interior of Solutions of Parabolic Equations.- 2. The Regularity at the Boundary of Solutions of Parabolic Boundary Value Problems.- 3. Application of Transposition: The Finite Cylinder Case.- 4. Application of Transposition: The Infinite Cylinder Case.- 5. Comments.- 6.Problems.- 11 Evolution Equations of the Second Order in t and of Schroedinger Type.- 1. Equations of the Second Order in t; Regularity of the Solutions of Boundary Value Problems.- 2. Equations of the Second Order in t; Application of Transposition and Existence of Solutions in Spaces of Distributions.- 3. Equations of the Second Order in t; Application of Transposition and Existence of Solutions in Spaces of Ultra-Distributions.- 4. Schroedinger Equations; Complements for Parabolic Equations.- 5. Comments.- 6. Problems.- Appendix. Calculus of Variations in Gevrey-Type Spaces.