Non-Homogeneous Boundary Value Problems and Applications Volume II

Non-Homogeneous Boundary Value Problems and Applications Volume II / by Jacques Louis Lions, Enrico Magenes.

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. th...

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Bibliographic Details
Main Authors: Lions, Jacques Louis. (Author, http://id.loc.gov/vocabulary/relators/aut), Magenes, Enrico. (http://id.loc.gov/vocabulary/relators/aut)
Corporate Author: SpringerLink (Online service)
Format: Electronic eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1972.
Edition:1st ed. 1972.
Series:Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 182
Springer eBook Collection.
Subjects:
Numerical analysis.
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Table of Contents:
  • 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 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.

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