Author | Lions, J. L. author |
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Title | Non-Homogeneous Boundary Value Problems and Applications [electronic resource] : Volume II / by J. L. Lions, E. Magenes |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1972 |
Connect to | http://dx.doi.org/10.1007/978-3-642-65217-2 |
Descript | XII, 244 p. online resource |
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 $$ D_{A + D_t̂2}̂{ - \left( {2r - 1} \right)}\left( Q \right) $$ -- 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 Greenโs 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