Non-Homogeneous Boundary Value Problems and Applications

4 Parabolic Evolution Operators. Hilbert Theory.- 1. Notation and Hypotheses. First Regularity Theorem.- 2. The Spaces Hr, s(Q). Trace Theorems. Compatibility Relations.- 3. Evolution Equations and the Laplace Transform.- 4. The Case of Operators Independent of t.- 5. Regularity.- 6. Case of Time-Dependent Operators. Existence of Solutions in the Spaces H2r m, m(Q), Real r ? 1.- Adjoint Isomorphism of Order r.- 8. Transposition of the Adjoint Isomorphism of Order r. (I): Generalities.- 9. Choice of f. The Spaces ?2rm,r(Q).- 10. Trace Theorems for the Spaces D?(r?1)(P)(Q), r ? 1.- 11. Choice of gj and uo. The Spaces H2?m ??(?).- 12. Transposition of the Adjoint Isomorphism of Order ?. (II): Results; Existence of Solutions in H2mr,r(Q)-Spaces, Real r ? 0.- 13. State of the Problem. Complements on the Transposition of the Adjoint Isomorphism of Order 1.- 14. Some Interpolation Theorems.- 15. Final Results; Existence of Solutions in the Spaces H2mr,r(Q), 0 < r < 1. Applications.- 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.- 2. A Regularity Theorem (I).- 3. Regular Non-Homogeneous Problems.- 4. Transposition.- 5. Interpolation.- 6. Applications and Examples.- 7. Regularity Theorem (II).- 8. Non-Integer Order Regularity Theorem.- 9. Adjoint Isomorphism of Order r and Transposition.- 10. Choice of f, $$ \vec g $$, u0, u1.- 11. Trace Theorems in the Space $$ D_{A + D_t^2}^{ - \left( {2r - 1} \right)}\left( Q \right) $$.- 12. Schroedinger Type Equations.- 6 Applications to Optimal Control Problems.- 1. Statement of the Problems for the Linear Parabolic Case.- 2. Choice of the Norms in the Cost Function.- 3. Optimality Conditionfor Quadratic Cost Functions.- 4. Optimality Condition and Green's Formula.- 5. The Particular Case $$ \mu \,\, = \,\,m\,\, + \,\,\frac{1}{2} $$, E3 ? 0.- 6. Consequences of the Optimality Condition (I).- 7. Consequences of the Optimality Condition (II).- 8. Complements on the Choice of the Spaces Ki.- 9. Examples.- 10. Non-Parabolic Cases. Statement of the Problems. Generalities.- 11. Applications. Examples.- 12. Comments.- 13. Problems.- Boundary Value Problems and Operator Extensions.- 1. Statement of the Problem. Well-Posed Spaces.- 1.1 Notation.- 2. Abstract Boundary Conditions.- 2.1 Boundary Spaces and Operators.- 2.2 Characterization of Well-Posed Spaces.- 3. Example 1. Elliptic Operators.- 3.1 Notation.- 3.2 The Boundary Operators and Spaces.- 3.3 Consequences.- 3.4 Various Remarks.- 4. Example 2. Parabolic Operators.- 4.1 Notation.- 4.2 The Boundary Operators and Spaces.- 4.3 Consequences.- 5.1 Notation.- 5.2 Formal Results.- 6. Comments and Problems.