Non-Homogeneous Boundary Value Problems and Applications: Volume II (Paperback, Softcover Repri)

'4 Parabolic Evolution Operators. Hilbert Theory.- 1. Notation and Hypotheses. First Regularity Theorem.- 1.1 Notation.- 1.2 Statement of the Problems.- 1.3 (Formal) Green's Formulas.- 1.4 First Existence and Uniqueness Theorem (Statement).- 1.5 Orientation.- 2. The Spaces Hr, s(Q). Trace Theorems. Compatibility Relations.- 2.1 Hr, s-Spaces.- 2.2 First Trace Theorem.- 2.3 Local Compatibility Relations.- 2.4 Global Compatibility Relations for a Particular Case.- 2.5 General Compatibility Relations.- 3. Evolution Equations and the Laplace Transform.- 3.1 Vector Distribution Solutions.- 3.2 L2-Solutions.- 4. The Case of Operators Independent of t.- 4.1 Hypotheses.- 4.2 Basic Inequalities.- 4.3 Solution of the Problem.- 5. Regularity.- 5.1 Preliminaries.- 5.2 Basic Inequalities.- 5.3 An Abstract Result.- 5.4 Solution of the Boundary Value Problem.- 6. Case of Time-Dependent Operators. Existence of Solutions in the Spaces H2r m, m(Q), Real r ? 1.- 6.1 Hypotheses. Statement of the Result.- 6.2 Local Result in t.- 6.3 Proof of Theorem 6.1.- 6.4 Regular Non-Homogeneous Problems.- Adjoint Isomorphism of Order r.- 7.1 The Adjoint Problem.- 7.2 Adjoint Isomorphism of Order r.- 8. Transposition of the Adjoint Isomorphism of Order r. (I): Generalities.- 8.1 Transposition.- 8.2 Orientation.- 8.3 The Spaces H??, ??(Q), H??, ??(?), ?, ? ? 0.- 8.4 (Formal) Choice of L.- 9. Choice of f. The Spaces ?2rm,r(Q).- 9.1 The Space ?2rm,r(Q).- 9.2 The Space ??2rm,?r(Q).- 9.3 Choice of f. The Space D?(r?1)(P)(Q).- 10. Trace Theorems for the Spaces D?(r?1)(P)(Q), r ? 1.- 10.1 Density Theorem.- 10.2 Trace Theorem on ?.- 10.3 Continuity of the Trace on Surfaces Neighbouring ?.- 10.4 Trace Theorem on ?0.- 10.5 Continuity of the Trace on Sections Neighbouring ?.- 11. Choice of gj and uo. The Spaces H2?m ??(?).- 11.1 The Spaces H2?m ??(?).- 11.2 Choice of gj.- 11.3 Choice of uo.- 12. Transposition of the Adjoint Isomorphism of Order ?. (II): Results; Existence of Solutions in H2mr,r(Q)-Spaces, Real r ? 0.- 12.1 Final Choice of L.- 12.2 Results.- 12.3 Complements.- 13. State of the Problem. Complements on the Transposition of the Adjoint Isomorphism of Order 1.- 13.1 State of the Problem.- 13.2 Complements on the Transposition of the Adjoint Isomorphism of Order 1.- 13.3 Orientation.- 14. Some Interpolation Theorems.- 14.1 Notation. Statement of the Main Result.- 14.2 Outline of the Proof.- 14.3 First Auxiliary Interpolation Theorem.- 14.4 Second Auxiliary Interpolation Theorem.- 14.5 Third Auxiliary Interpolation Theorem.- 14.6 Proof of Theorem 14.1.- 15. Final Results; Existence of Solutions in the Spaces H2mr,r(Q), 0 r 1. Applications.- 15.1 Application of the Results of Section 14.- 15.2 Examples; Generalities.- 15.3 Examples (I).- 15.4 Examples (II).- 15.5 Some Complements on the Dirichlet Problem.- 16. Comments.- 17. Problems.- 5 Hyperbolic Evolution Operators, of Petrowski and of Schroedinger. Hilbert Theory.- 1. Application of the Results of Chapter 3 and General Remarks.- 1.1 Notation. Hypotheses.- 1.2 Application of the Results of Chapter 3.- 1.3 A Counter-Example.- 2. A Regularity Theorem (I).- 3. Regular Non-Homogeneous Problems.- 3.1 Statement of the Problem.- 3.2 The Compatibility Relations.- 3.3 The Case of the Dirichlet Problem.- 4. Transposition.- 4.1 Adjoint Isomorphism.- 4.2 Transposition.- 4.3 Choice of L.- 4.4 Conclusion.- 5. Interpolation.- 5.1 Statement of the Problem.- 5.2 Some Interpolation Results.- 5.3 Consequences.- 5.4 The Case of the Dirichlet Problem.- 6. Applications and Examples.- 6.1 General Results.- 6.2 Examples.- 7. Regularity Theorem (II).- 7.1 Statement.- 7.2 Proof of Theorem 7.1.- 8. Non-Integer Order Regularity Theorem.- 8.1 Orientation.- 8.2 Interpolation in r.- 8.3 Interpretation of the Space V(2r?1)m,2r(Q), r ? 1.- 9. Adjoint Isomorphism of Order r and Transposition.- 9.1 Adjoint Isomorphism of Order r.- 9.2 Transposit