Non Homogeneous Boundary Value Problems and Applications : Volume II 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(r1)(P)(Q) r  1 11. Choice of gj and uo. The Spaces H2m () 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 Condition for 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. I. In this second volume we continue at first the study of non­ homogeneous boundary value problems for particular classes of evolu­ tion equations. 1 In Chapter 4 we study parabolic operators by the method of Agranovitch Vishik [lJ; this is step (i) (Introduction to Volume I Section 4) i.e. the study of regularity. The next steps: (ii) transposition (iii) interpolation are similar in principle to those of Chapter 2 but involve rather considerable additional technical difficulties. In Chapter 5 we study hyperbolic operators or operators well­ defined in thesense of Petrowski or Schroedinger. Our regularity results (step (i)) seem to be new. Steps (ii) and (iii) are all3.logous to those of the parabolic case except for certain technical differences. In Chapter 6 the results of Chapter> 4 and 5 are applied to the study of optimal control problems for systems governed by evolution equations when the control appears in the boundary conditions (so that non homogeneous boundary value problems are the basic tool of this theory). Another type of application to the characterization of all well posed problems for the operators in question is given in the Ap­ pendix. Still other applications for example to numerical analysis will be given in Volume 3.