Non-homogeneous boundary value problems and applications

Author(s)

• Lions, Jacques Louis
• Magenes, E
• Kenneth, P.

Bibliographic Information

Non-homogeneous boundary value problems and applications

J.L. Lions, E. Magenes ; translated from the French by P. Kenneth

（Die Grundlehren der mathematischen Wissenschaften, Bd. 181-183）

Springer-Verlag, 1972-1973

• v. 1 : gw
• v. 1 : us
• v. 1 : pbk
• v. 2 : gw
• v. 2 : us
• v. 3 : gw
• v. 3 : us
• v. 3 : pbk

Other Title

Problèmes aux limites non homogènes et applications

Note

Translation of Problèmes aux limites non homogènes et applications

Bibliography: v. 1, p. [309]-357; v. 2, p. [208]-242; v. 3, p. [290]-308

Description and Table of Contents

Volume

v. 1 : gw ISBN 9783540053637

Description

1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R , with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj' 0 ~ i ~ 'V. By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and G , F being a space" on m" and the G/ s spaces" on am"; j we seek u in a function space u/t "on m" satisfying (1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v"])). Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2. We take as "working hypothesis" that, for fEF and gjEG , j the problem (1), (2) admits a unique solution u E U/t, which depends 3 continuously on the data . But for alllinear probIems, there is a large number of choiees for the space s u/t and {F; G} (naturally linke d together). j Generally speaking, our aim is to determine families of spaces 'ft and {F; G}, associated in a "natural" way with problem (1), (2) and con- j venient for applications, and also all possible choiees for u/t and {F; G} j in these families.

Table of Contents

• 1 Hilbert Theory of Trace and Interpolation Spaces.- 1. Some Function Spaces.- 1.1 Sobolev Spaces.- 1.2 The Case of the Entire Space.- 1.3 The Half-Space Case.- 1.4 Orientation.- 2. Intermediate Derivatives Theorem.- 2.1 Intermediate Spaces.- 2.2 Density and Extension Theorems.- 2.3 Intermediate Derivatives Theorem.- 2.4 A Simple Example.- 2.5 Interpolation Inequality.- 3. Trace Theorem.- 3.1 Continuity Properties of the Elements of W(a,b).- 3.2 Trace Theorem.- 4. Trace Spaces and Non-Integer Order Derivatives.- 4.1 Orientation. Definitions.- 4.2 "Intermediate Derivatives" and Trace Theorems.- 5. Interpolation Theorem.- 5.1 Main Theorem.- 5.2 Interpolation of a Family of Operators.- 6. Reiteration Properties and Duality of the Spaces [X, Y]0.- 6.1 Reiteration.- 6.2 Duality.- 7. The Spaces Hs(Rn) and Hs(?).- 7.1 Hs (Rn)-Spaces.- 7.2 Traces on the Boundary of a Half-Space.- 7.3 Hs (?)-Spaces.- 8. Trace Theorem in Hm(?).- 8.1 Extension and Density Theorems.- 8.2 Trace Theorem.- 9. The Spaces Hs(?), Real s ? 0.- 9.1 Definition by Interpolation.- 9.2 Trace Theorem in Hs(?).- 9.3 Interpolation of Hs(?)-Spaces.- 9.4 Regularity Properties of Hs(?)-Functions.- 10. Some Further Properties of the Spaces [X, Y]0.- 10.1 Domains of Semi-Groups.- 10.2 Application to Hs (Rn).- 10.3 Application to Hs (0, ?).- 11. Subspaces of Hs(?). The Spaces H0s(?).- 11.1 H0s(?)-Spaces.- 11.2 A Property of Hs(?), 0 ? s < 1/2.- 11.3 The Extension by 0 outside ?.- 11.4 Characterization of H0s(?)-Spaces.- 11.5 Interpolation of H0s(?)-Spaces.- 12. The Spaces H?s(?), s > 0.- 12.1 Definition. First Properties.- 12.2 Interpolation between the Spaces H?s(?), s > 0.- 12.3 Interpolation between $$H\frac{<!-- -->{<!-- -->{s_1}}}{0}(\Gamma )$$ and $${H^{ - {s_2}}}(\Omega )$$, si > 0.- 12.4 Interpolation between $${H^{<!-- -->{s_1}}}(\Omega )$$ and $${H^{ - {s_2}}}(\Omega )$$, si > 0.- 12.5 Interpolation between $${H^{<!-- -->{s_1}}}(\Omega )$$ and $$({H^{<!-- -->{s_2}}}(\Omega ))'$$.- 12.6 Interpolation between $$H\frac{<!-- -->{<!-- -->{s_1}}}{0}(\Omega )$$ and $$({H^{<!-- -->{s_2}}}(\Omega ))'$$.- 12.7 A Lemma.- 12.8 Differential Operators on Hs(?).- 12.9 Invariance by Diffeomorphism of Hs(?)-Spaces.- 13. Intersection Interpolation.- 13.1 A General Result.- 13.2 Example of Application (I).- 13.3 Example of Application (II).- 13.4 Interpolation of Quotient Spaces.- 14. Holomorphic Interpolation.- 14.1 General Result.- 14.2 Interpolation of Spaces of Continuous Functions with Hilbert Range.- 14.3 A Result Pertaining to Interpolation of Subspaces.- 15. Another Intrinsic Definition of the Spaces [X, Y]0.- 16. Compactness Properties.- 17. Comments.- 18. Problems.- 2 Elliptic Operators. Hilbert Theory.- 1. Elliptic Operators and Regular Boundary Value Problems.- 1.1 Elliptic Operators.- 1.2 Properly and Strongly Elliptic Operators.- 1.3 Regularity Hypotheses on the Open Set ? and the Coefficients of the Operator A.- 1.4 The Boundary Operators.- 2. Green's Formula and Adjoint Boundary Value Problems.- 2.1 The Adjoint of A in the Sense of Distributions or Formal Adjoint.- 2.2 The Theorem on Green's Formula.- 2.3 Proof of the Theorem.- 2.4 A Variant of Green's Formula.- 2.5 Formal Adjoint Problems with Respect to Green's Formula.- 3. The Regularity of Solutions of Elliptic Equations in the Interior of ?.- 3.1 Two Lemmas.- 3.2 A priori Estimates in Rn.- 3.3 The Regularity in the Interior of Q and the Hypoellipticity of Elliptic Operators.- 4. A priori Estimates in the Half-Space.- 4.1 A new Formulation of the Covering Condition.- 4.2 A Lemma on Ordinary Differential Equations.- 4.3 First Application: Proof of Theorem 2.2.- 4.4 A priori Estimates in the Half-Space for the Case of Constant Coefficients.- 4.5 A priori Estimates in the Half-Space for the Case of Variable Coefficients.- 5. A priori Estimates in the Open Set ? and the Existence of Solutions in Hs(?)-Spaces, with Real s ? 2m.- 5.1 A priori Estimates in the Open Set ?.- 5.2 Existence of Solutions in Hs(?)-Spaces, with Integer s ? 2m.- 5.3 Precise Statement of the Compatibility Conditions for Existence.- 5.4 Existence of Solutions in Hs(?)-Spaces, with Real s ? 2m.- 6. Application of Transposition: Existence of Solutions in Hs(?)-Spaces, with Real s ? 0.- 6.1 The Transposition Method
• Generalities.- 6.2 Choice of the Form L.- 6.3 The Spaces ? (?) and DAs(?).- 6.4 Density Theorem.- 6.5 Trace Theorem, and Green's Formula for the Space DAs(?), s ? 0.- 6.6 Existence of Solutions in DAs(?)-Spaces, with Real s ? 0.- 7. Application of Interpolation: Existence of Solutions in Hs(?)-Spaces, with Real s, 0 2m.- 7.1 New Properties of ?s(?)-Spaces.- 7.2 Use of Interpolation
• First Results.- 7.3 The Final Results.- 8. Complements and Generalizations.- 8.1 Continuity of Traces on Surfaces Neighbouring ?.- 8.2 A Generalization
• Application to Dirichlet's Problem.- 8.3 Remarks on the Hypotheses on A and Bj.- 8.4 The Realization of A in L2(?).- 8.5 Some Remarks on the Index of ?.- 8.6 Uniqueness and Surjectivity Theorems.- 9. Variational Theory of Boundary Value Problems.- 9.1 Variational Problems.- 9.2 The Problem.- 9.3 A Counter-Example.- 9.4 Variational Formulation and Green's Formula.- 9.5 "Concrete" Variational Problems.- 9.6 Coercive Forms and Problems.- 9.7 Regularity of Solutions.- 9.8 Generalizations (I).- 9.9 Generalizations (II).- 10. Comments.- 11. Problems.- 3 Variational Evolution Equations.- 1. An Isomorphism Theorem.- 1.1 Notation.- 1.2 Isomorphism Theorem.- 1.3 The Adjoint ?*.- 1.4 Proof of Theorem 1.1.- 2. Transposition.- 2.1 Generalities.- 2.2 Adjoint Isomorphism Theorem.- 2.3 Transposition.- 3. Interpolation.- 3.1 General Application.- 3.2 Characterization of Interpolation Spaces.- 3.3 The Case "? = 1/2".- 4. Example: Abstract Parabolic Equations, Initial Condition Problem (I).- 4.1 Notation.- 4.2 The Operator M.- 4.3 The Operator ?.- 4.4 Application of the Isomorphism Theorems.- 4.5 Choice of L in (4.20).- 4.6 Interpretation of the Problem.- 4.7 Examples.- 5. Example: Abstract Parabolic Equations, Initial Condition Problem (II).- 5.1 Some Interpolation Results.- 5.2 Interpretation of the Spaces ?1/2, ?*1/2.- 6. Example: Abstract Parabolic Equations, Periodic Solutions.- 6.1 Notation. The Operator ?.- 6.2 Application of the Isomorphism Theorems.- 6.3 Choice of L.- 6.4 Interpretation of the Problem.- 6.5 The Isomorphism of ?1/2 onto its Dual.- 7. Elliptic Regularization.- 7.1 The Elliptic Problem.- 7.2 Passage to the Limit.- 8. Equations of the Second Order in t.- 8.1 Notation.- 8.2 Existence and Uniqueness Theorem.- 8.3 Remarks on the Application of the General Theory of Section 1.- 8.4 Additional Regularity Results.- 8.5 Parabolic Regularization
• Direct Method and Application.- 9. Equations of the Second Order in t
• Transposition.- 9.1 Adjoint Isomorphism.- 9.2 Transposition.- 9.3 Choice of L.- 9.4 Trace Theorem.- 9.5 Variant
• Direct Method.- 9.6 Examples.- 10. Schroedinger Type Equations.- 10.1 Notation.- 10.2 Existence and Uniqueness Theorem.- 11. Schroedinger Type Equations
• Transposition.- 11.1 Adjoint Isomorphism.- 11.2 Transposition of (11.5).- 11.3 Choice of L.- 12. Comments.- 13. Problems.

Volume

v. 2 : gw ISBN 9783540054443

Description

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.

Table of Contents

• 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 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 Transposition.- 9.3 Formal Choice of L.- 10. Choice of f, $$ \vec g $$, u0, u1.- 10.1 Choice of f.- 10.2 The Space $$ D_{A + D_t^2}^{ - \left( {2r - 1} \right)}\left( Q \right) $$.- 10.3 Choice of gj.- 10.4 Choice of u0, u1.- 10.5 Conclusion.- 11. Trace Theorems in the Space $$ D_{A + D_t^2}^{ - \left( {2r - 1} \right)}\left( Q \right) $$.- 11.1 Density Theorem.- 11.2 Traces on ?.- 11.3 Continuity of the Trace on Neighbouring Surfaces.- 11.4 Traces on ?0.- 11.5 Continuity of the Trace on Sections Neighbouring ?0.- 11.6 Remark.- 12. Schroedinger Type Equations.- 12.1 Notation.- 12.2 First Regularity Theorem. Parabolic Regularization.- 12.3 Second Regularity Theorem.- 12.4 r-Isomorphism Theorem.- 12.5 Choice of L.- 12.6 Trace Theorem.- 13. Comments.- 14. Problems.- 6 Applications to Optimal Control Problems.- 1. Statement of the Problems for the Linear Parabolic Case.- 1.1 Notation.- 1.2 Optimization Problems.- 2. Choice of the Norms in the Cost Function.- 2.1 Reminder. Condition on K1(Q).- 2.2 Space Described by $$ \vec S\,y $$. Conditions on K2(?).- 2.3 Space Described by y(x, T
• u). Condition on K3(?).- 3. Optimality Condition for Quadratic Cost Functions.- 3.1 Notation.- 3.2 Optimality Condition.- 4. Optimality Condition and Green's Formula.- 4.1 Optimality Condition. Application of Section 3.2.- 4.2 The Isomorphisms ?i.- 4.3 The "Adjoint" Problem.- 4.4 New Form of the Optimality Condition.- 5. The Particular Case $$ \mu \,\, = \,\,m\,\, + \,\,\frac{1}{2} $$, E3 ? 0.- 5.1 Properties of y.- 5.2 Choice of K1(Q).- 5.3 Choice of K2(?) and K3(?).- 5.4 Adjoint Problem and Optimality Condition.- 6. Consequences of the Optimality Condition (I).- 6.1 Generalities.- 6.2 Consequences of Theorem 6.1.- 7. Consequences of the Optimality Condition (II).- 7.1 Additional Hypotheses.- 7.2 Optimality Condition.- 8. Complements on the Choice of the Spaces Ki.- 8.1 Orientation.- 8.2 Choice of K1(Q).- 8.3 Choice of K2(?).- 8.4 Choice of K3(?).- 9. Examples.- 10. Non-Parabolic Cases. Statement of the Problems. Generalities.- 10.1 Notation.- 10.2 Cost Function.- 10.3 Optimality Condition (I).- 10.4 Adjoint Problem.- 10.5 Green's Formula.- 10.6 Optimality Condition (II).- 10.7 Consequences.- 11. Applications. Examples.- 11.1 Control in the Boundary Conditions.- 11.2 Choice of K1.- 11.3 Choice of K2.- 11.4 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.

Volume

v. 3 : gw ISBN 9783540058328

Table of Contents

• 7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{<!-- -->{M_k}}}\left( H \right)$$.- 1.3 The Spaces $${D_{<!-- -->{M_k}}}\left( H \right)$$ and $${\varepsilon _{<!-- -->{M_k}}}\left( H \right)$$.- 2. Scalar-Valued Ultra-Distributions of Class Mk
• Generalizations.- 2.1 The Space $$D{'_{<!-- -->{M_k}}}\left( \Omega \right)$$.- 2.2 Non-Symmetric Spaces of Class Mk.- 2.3 Scalar Ultra-Distributions of Beurling-Type.- 3. Spaces of Analytic Functions and of Analytic Functionals.- 3.1 The Spaces H(H) and H'(H).- 3.2 The Spaces H(?) and H(?).- 4. Vector-Valued Functions of Class Mk.- 4.1 The Space $${D_{<!-- -->{M_k}}}\left( {\phi
• F} \right)$$.- 4.2 The Spaces $${D_{<!-- -->{M_k}}}\left( {H,F} \right)$$ and $${E_{<!-- -->{M_k}}}\left( {\phi
• F} \right)$$.- 4.3 The Spaces $${D_{ \pm ,{M_k}}}\left( {\phi
• F} \right)$$.- 4.4 Remarks on the Topological Properties of the Spaces $${D_{<!-- -->{M_k}}}\left( {\phi
• F} \right),{E_{<!-- -->{M_k}}}\left( {\phi
• F} \right),{D_{ \pm ,{M_k}}}\left( {\phi
• F} \right)$$.- 5. Vector-Valued Ultra-Distributions of Class Mk
• Generalizations.- 5.1 Recapitulation on Vector-Valued Distributions.- 5.2 The Space $$D{'_{<!-- -->{M_k}}}\left( {\phi
• F} \right)$$.- 5.3 The Space $$D{'_{ \pm ,{M_k}}}\left( {\phi
• F} \right)$$.- 5.4 Vector-Valued Ultra-Distributions of Beurling-Type.- 5.5 The Particular Case: F = Banach Space.- 6. Comments.- 8 Elliptic Boundary Value Problems in Spaces of Distributions and Ultra-Distributions.- 1. Regularity of Solutions of Elliptic Boundary Value Problems in Spaces of Analytic Functions and of Class Mk
• Statement of the Problems and Results.- 1.1 Recapitulation on Elliptic Boundary Value Problems.- 1.2 Statement of the Mk-Regularity Results.- 1.3 Reduction of the Problem to the Case of the Half-Ball.- 2. The Theorem on "Elliptic Iterates": Proof.- 2.1 Some Lemmas.- 2.2 The Preliminary Estimate.- 2.3 Bounds for the Tangential Derivatives.- 2.4 Bounds for the Normal Derivatives.- 2.5 Proof of Theorem 1.3.- 2.6 Complements and Remarks.- 3. Application of Transposition
• Existence of Solutions in the Space D'(?) of Distributions.- 3.1 Generalities.- 3.2 Choice of the Form L
• the Space ?(?) and its Dual.- 3.3 Final Choice of the Form L
• the Space Y.- 3.4 Density Theorem.- 3.5 Trace Theorem and Green's Formula in Y.- 3.6 The Existence of Solutions in the Space Y.- 3.7 Continuity of Traces on Surfaces Neighbouring ?.- 4. Existence of Solutions in the Space $$D{'_{<!-- -->{M_k}}}\left( \Omega \right)$$ of Ultra-Distributions.- 4.1 Generalities.- 4.2 The Space $${\Xi _{<!-- -->{M_k}}}\left( \Omega \right)$$ and its Dual.- 4.3 The Space $${Y_{<!-- -->{M_k}}}$$ and the Existence of Solutions in $${Y_{<!-- -->{M_k}}}$$.- 4.4 Application to the Regularity in the Interior of Ultra-Distribution Solutions of the Equation Au = f.- 5. Comments.- 6. Problems.- 9 Evolution Equations in Spaces of Distributions and Ultra-Distributions.- 1. Regularity Results. Equations of the First Order in t.- 1.1 Orientation and Notation.- 1.2 Regularity in the Spaces D+.- 1.3 Regularity in the Spaces $${D_{ + ,{M_k}}}$$.- 1.4 Regularity in Beurling Spaces.- 1.5 First Applications.- 2. Equations of the Second Order in t.- 2.1 Statement of the Main Results.- 2.2 Proof of Theorem 2.1.- 2.3 Proof of Theorem 2.2.- 3. Singular Equations of the Second Order in t.- 3.1 Statement of the Main Results.- 3.2 Proof of Theorem 3.1.- 4. Schroedinger-Type Equations.- 4.1 Statement of the Main Results.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 5. Stability Results in Mk-Classes.- 5.1 Parabolic Regularization.- 5.2 Approximation by Systems of Cauchy-Kowaleska Type (I).- 5.3 Approximation by Systems of Cauchy-Kowaleska Type (II).- 6. Transposition.- 6.1 Orientation.- 6.2 The Parabolic Case.- 6.3 The Second Order in t Case and the Schroedinger Case.- 7. Semi-Groups.- 7.1 Orientation.- 7.2 The Space of Vectors of Class Mk.- 7.3 The Semi-Group G in the Spaces D(A?
• Mk). Applications.- 7.4 The Transposed Settings. Applications.- 7.5 Another Mk-Regularity Result.- 8. Mk -Classes and Laplace Transformation.- 8.1 Orientation-Hypotheses.- 8.2 Mk -Regularity Result.- 8.3 Transposition.- 9. General Operator Equations.- 9.1 General Results.- 9.2 Application. Periodic Problems.- 9.3 Transposition.- 10. The Case of a Finite Interval ]0, T[.- 10.1 Orientation. General Problems.- 10.2 Space Described by v(0) as v Describes X.- 10.3 The Space $${\Xi _{<!-- -->{M_k}}}$$.- 10.4 Choice of L.- 10.5 The Space Y and Trace Theorems.- 10.6 Non-Homogeneous Problems.- 11. Distribution and Ultra-Distribution Semi-Groups.- 11.1 Distribution Semi-Groups.- 11.2 Ultra-Distribution Semi-Groups.- 12. A General Local Existence Result.- 12.1 Statement of the Result.- 12.2 Examples.- 13. Comments.- 14. Problems.- 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions.- 1. Regularity in the Interior of Solutions of Parabolic Equations.- 1.1 The Hypoellipticity of Parabolic Equations.- 1.2 The Regularity in the Interior in Gevrey Spaces.- 2. The Regularity at the Boundary of Solutions of Parabolic Boundary Value Problems.- 2.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 2.2 The Regularity in Gevrey Spaces.- 3. Application of Transposition: The Finite Cylinder Case.- 3.1 The Existence of Solutions in the Space D'(Q): Generalities, the Spaces X and Y.- 3.2 Space Described by ?v as v Describes X.- 3.3 Trace and Existence Theorems in the Space Y.- 3.4 The Existence of Solutions in the Spaces D's,r(Q) of Gevrey Ultra-Distributions, with r > 1, s ? 2m.- 4. Application of Transposition: The Infinite Cylinder Case.- 4.1 The Existence of Solutions in the Space D' (R
• D'(?)): The Space X_.- 4.2 The Existence of Solutions in the Space D'+ (R
• D'(?)): The Space Y+ and the Trace and Existence Theorems.- 4.3 The Existence of Solutions in the Spaces D'+,s(R
• D'r(?)), with r > 1, s ? 2m.- 4.4 Remarks on the Existence of Solutions and the Trace Theorems in other Spaces of Ultra-Distributions.- 5. Comments.- 6. Problems.- 11 Evolution Equations of the Second Order in t and of Schroedinger Type.- 1. Equations of the Second Order in t
• Regularity of the Solutions of Boundary Value Problems.- 1.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 1.2 The Regularity in Gevrey Spaces.- 2. Equations of the Second Order in t
• Application of Transposition and Existence of Solutions in Spaces of Distributions.- 2.1 Generalities.- 2.2 The Space $${D_{ - ,\gamma }}\left( {\left[ {0,T} \right]
• {D_\gamma }\left( {\bar \Omega } \right)} \right)$$ and its Dual.- 2.3 The Spaces X and Y.- 2.4 Study of the Operator ?.- 2.5 Trace and Existence Theorems in the Space Y.- 2.6 Complements on the Trace Theorems.- 2.7 The Infinite Cylinder Case.- 3. Equations of the Second Order in t
• Application of Transposition and Existence of Solutions in Spaces of Ultra-Distributions.- 3.1 The Difficulties in the Finite Cylinder Case.- 3.2 The Infinite Cylinder Case for m > 1.- 4. Schroedinger Equations
• Complements for Parabolic Equations.- 4.1 Regularity Results for the Schroedinger Equation.- 4.2 The Non-Homogeneous Boundary Value Problems for the Schroedinger Equation.- 4.3 Remarks on Parabolic Equations.- 5. Comments.- 6. Problems.- Appendix. Calculus of Variations in Gevrey-Type Spaces.

Volume

v. 1 : pbk ISBN 9783642651632

Description

1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R , with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj' 0 ~ i ~ 'V. By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and G , F being a space" on m" and the G/ s spaces" on am" ; j we seek u in a function space u/t "on m" satisfying (1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v"])). Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2. We take as "working hypothesis" that, for fEF and gjEG , j the problem (1), (2) admits a unique solution u E U/t, which depends 3 continuously on the data . But for alllinear probIems, there is a large number of choiees for the space s u/t and {F; G} (naturally linke d together). j Generally speaking, our aim is to determine families of spaces 'ft and {F; G}, associated in a "natural" way with problem (1), (2) and con j venient for applications, and also all possible choiees for u/t and {F; G} j in these families.

Table of Contents

• 1 Hilbert Theory of Trace and Interpolation Spaces.- 1. Some Function Spaces.- 1.1 Sobolev Spaces.- 1.2 The Case of the Entire Space.- 1.3 The Half-Space Case.- 1.4 Orientation.- 2. Intermediate Derivatives Theorem.- 2.1 Intermediate Spaces.- 2.2 Density and Extension Theorems.- 2.3 Intermediate Derivatives Theorem.- 2.4 A Simple Example.- 2.5 Interpolation Inequality.- 3. Trace Theorem.- 3.1 Continuity Properties of the Elements of W(a,b).- 3.2 Trace Theorem.- 4. Trace Spaces and Non-Integer Order Derivatives.- 4.1 Orientation. Definitions.- 4.2 "Intermediate Derivatives" and Trace Theorems.- 5. Interpolation Theorem.- 5.1 Main Theorem.- 5.2 Interpolation of a Family of Operators.- 6. Reiteration Properties and Duality of the Spaces [X, Y]0.- 6.1 Reiteration.- 6.2 Duality.- 7. The Spaces Hs(Rn) and Hs(?).- 7.1 Hs (Rn)-Spaces.- 7.2 Traces on the Boundary of a Half-Space.- 7.3 Hs (?)-Spaces.- 8. Trace Theorem in Hm(?).- 8.1 Extension and Density Theorems.- 8.2 Trace Theorem.- 9. The Spaces Hs(?), Real s ? 0.- 9.1 Definition by Interpolation.- 9.2 Trace Theorem in Hs(?).- 9.3 Interpolation of Hs(?)-Spaces.- 9.4 Regularity Properties of Hs(?)-Functions.- 10. Some Further Properties of the Spaces [X, Y]0.- 10.1 Domains of Semi-Groups.- 10.2 Application to Hs (Rn).- 10.3 Application to Hs (0, ?).- 11. Subspaces of Hs(?). The Spaces H0s(?).- 11.1 H0s(?)-Spaces.- 11.2 A Property of Hs(?), 0 ? s < 1/2.- 11.3 The Extension by 0 outside ?.- 11.4 Characterization of H0s(?)-Spaces.- 11.5 Interpolation of H0s(?)-Spaces.- 12. The Spaces H?s(?), s > 0.- 12.1 Definition. First Properties.- 12.2 Interpolation between the Spaces H?s(?), s > 0.- 12.3 Interpolation between $$H\frac{<!-- -->{<!-- -->{s_1}}}{0}(\Gamma )$$ and $${H^{ - {s_2}}}(\Omega )$$, si > 0.- 12.4 Interpolation between $${H^{<!-- -->{s_1}}}(\Omega )$$ and $${H^{ - {s_2}}}(\Omega )$$, si > 0.- 12.5 Interpolation between $${H^{<!-- -->{s_1}}}(\Omega )$$ and $$({H^{<!-- -->{s_2}}}(\Omega ))'$$.- 12.6 Interpolation between $$H\frac{<!-- -->{<!-- -->{s_1}}}{0}(\Omega )$$ and $$({H^{<!-- -->{s_2}}}(\Omega ))'$$.- 12.7 A Lemma.- 12.8 Differential Operators on Hs(?).- 12.9 Invariance by Diffeomorphism of Hs(?)-Spaces.- 13. Intersection Interpolation.- 13.1 A General Result.- 13.2 Example of Application (I).- 13.3 Example of Application (II).- 13.4 Interpolation of Quotient Spaces.- 14. Holomorphic Interpolation.- 14.1 General Result.- 14.2 Interpolation of Spaces of Continuous Functions with Hilbert Range.- 14.3 A Result Pertaining to Interpolation of Subspaces.- 15. Another Intrinsic Definition of the Spaces [X, Y]0.- 16. Compactness Properties.- 17. Comments.- 18. Problems.- 2 Elliptic Operators. Hilbert Theory.- 1. Elliptic Operators and Regular Boundary Value Problems.- 1.1 Elliptic Operators.- 1.2 Properly and Strongly Elliptic Operators.- 1.3 Regularity Hypotheses on the Open Set ? and the Coefficients of the Operator A.- 1.4 The Boundary Operators.- 2. Green's Formula and Adjoint Boundary Value Problems.- 2.1 The Adjoint of A in the Sense of Distributions or Formal Adjoint.- 2.2 The Theorem on Green's Formula.- 2.3 Proof of the Theorem.- 2.4 A Variant of Green's Formula.- 2.5 Formal Adjoint Problems with Respect to Green's Formula.- 3. The Regularity of Solutions of Elliptic Equations in the Interior of ?.- 3.1 Two Lemmas.- 3.2 A priori Estimates in Rn.- 3.3 The Regularity in the Interior of Q and the Hypoellipticity of Elliptic Operators.- 4. A priori Estimates in the Half-Space.- 4.1 A new Formulation of the Covering Condition.- 4.2 A Lemma on Ordinary Differential Equations.- 4.3 First Application: Proof of Theorem 2.2.- 4.4 A priori Estimates in the Half-Space for the Case of Constant Coefficients.- 4.5 A priori Estimates in the Half-Space for the Case of Variable Coefficients.- 5. A priori Estimates in the Open Set ? and the Existence of Solutions in Hs(?)-Spaces, with Real s ? 2m.- 5.1 A priori Estimates in the Open Set ?.- 5.2 Existence of Solutions in Hs(?)-Spaces, with Integer s ? 2m.- 5.3 Precise Statement of the Compatibility Conditions for Existence.- 5.4 Existence of Solutions in Hs(?)-Spaces, with Real s ? 2m.- 6. Application of Transposition: Existence of Solutions in Hs(?)-Spaces, with Real s ? 0.- 6.1 The Transposition Method
• Generalities.- 6.2 Choice of the Form L.- 6.3 The Spaces ? (?) and DAs(?).- 6.4 Density Theorem.- 6.5 Trace Theorem, and Green's Formula for the Space DAs(?), s ? 0.- 6.6 Existence of Solutions in DAs(?)-Spaces, with Real s ? 0.- 7. Application of Interpolation: Existence of Solutions in Hs(?)-Spaces, with Real s, 0 < s < 2m.- 7.1 New Properties of ?s(?)-Spaces.- 7.2 Use of Interpolation
• First Results.- 7.3 The Final Results.- 8. Complements and Generalizations.- 8.1 Continuity of Traces on Surfaces Neighbouring ?.- 8.2 A Generalization
• Application to Dirichlet's Problem.- 8.3 Remarks on the Hypotheses on A and Bj.- 8.4 The Realization of A in L2(?).- 8.5 Some Remarks on the Index of ?.- 8.6 Uniqueness and Surjectivity Theorems.- 9. Variational Theory of Boundary Value Problems.- 9.1 Variational Problems.- 9.2 The Problem.- 9.3 A Counter-Example.- 9.4 Variational Formulation and Green's Formula.- 9.5 "Concrete" Variational Problems.- 9.6 Coercive Forms and Problems.- 9.7 Regularity of Solutions.- 9.8 Generalizations (I).- 9.9 Generalizations (II).- 10. Comments.- 11. Problems.- 3 Variational Evolution Equations.- 1. An Isomorphism Theorem.- 1.1 Notation.- 1.2 Isomorphism Theorem.- 1.3 The Adjoint ?*.- 1.4 Proof of Theorem 1.1.- 2. Transposition.- 2.1 Generalities.- 2.2 Adjoint Isomorphism Theorem.- 2.3 Transposition.- 3. Interpolation.- 3.1 General Application.- 3.2 Characterization of Interpolation Spaces.- 3.3 The Case "? = 1/2".- 4. Example: Abstract Parabolic Equations, Initial Condition Problem (I).- 4.1 Notation.- 4.2 The Operator M.- 4.3 The Operator ?.- 4.4 Application of the Isomorphism Theorems.- 4.5 Choice of L in (4.20).- 4.6 Interpretation of the Problem.- 4.7 Examples.- 5. Example: Abstract Parabolic Equations, Initial Condition Problem (II).- 5.1 Some Interpolation Results.- 5.2 Interpretation of the Spaces ?1/2, ?*1/2.- 6. Example: Abstract Parabolic Equations, Periodic Solutions.- 6.1 Notation. The Operator ?.- 6.2 Application of the Isomorphism Theorems.- 6.3 Choice of L.- 6.4 Interpretation of the Problem.- 6.5 The Isomorphism of ?1/2 onto its Dual.- 7. Elliptic Regularization.- 7.1 The Elliptic Problem.- 7.2 Passage to the Limit.- 8. Equations of the Second Order in t.- 8.1 Notation.- 8.2 Existence and Uniqueness Theorem.- 8.3 Remarks on the Application of the General Theory of Section 1.- 8.4 Additional Regularity Results.- 8.5 Parabolic Regularization
• Direct Method and Application.- 9. Equations of the Second Order in t
• Transposition.- 9.1 Adjoint Isomorphism.- 9.2 Transposition.- 9.3 Choice of L.- 9.4 Trace Theorem.- 9.5 Variant
• Direct Method.- 9.6 Examples.- 10. Schroedinger Type Equations.- 10.1 Notation.- 10.2 Existence and Uniqueness Theorem.- 11. Schroedinger Type Equations
• Transposition.- 11.1 Adjoint Isomorphism.- 11.2 Transposition of (11.5).- 11.3 Choice of L.- 12. Comments.- 13. Problems.

Volume

v. 3 : pbk ISBN 9783642653957

Description

1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2) fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J am 1 v, where Pis a linear differential operator in m and where the Q/s are linear differential operators on am. In Volumes 1 and 2, we studied, for particular c1asses of systems {P, Qj}, problem (1), (2) in c1asses of Sobolev spaces (in general constructed starting from P) of positive integer or (by interpolation) non-integer order; then, by transposition, in c1asses of Sobolev spaces of negative order, until, by passage to the limit on the order, we reached the spaces of distributions of finite order. In this volume, we study the analogous problems in spaces of inlinitely dilferentiable or analytic Itlnctions or of Gevrey-type I~mctions and by duality, in spaces 01 distribtltions, of analytic Itlnctionals or of Gevrey- type ultra-distributions. In this manner, we obtain a c1ear vision (at least we hope so) of the various possible formulations of the boundary value problems (1), (2) for the systems {P, Qj} considered here.

Table of Contents

• 7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{<!-- -->{M_k}}}\left( H \right)$$.- 1.3 The Spaces $${D_{<!-- -->{M_k}}}\left( H \right)$$ and $${\varepsilon _{<!-- -->{M_k}}}\left( H \right)$$.- 2. Scalar-Valued Ultra-Distributions of Class Mk
• Generalizations.- 2.1 The Space $$D{'_{<!-- -->{M_k}}}\left( \Omega \right)$$.- 2.2 Non-Symmetric Spaces of Class Mk.- 2.3 Scalar Ultra-Distributions of Beurling-Type.- 3. Spaces of Analytic Functions and of Analytic Functionals.- 3.1 The Spaces H(H) and H'(H).- 3.2 The Spaces H(?) and H(?).- 4. Vector-Valued Functions of Class Mk.- 4.1 The Space $${D_{<!-- -->{M_k}}}\left( {\phi
• F} \right)$$.- 4.2 The Spaces $${D_{<!-- -->{M_k}}}\left( {H,F} \right)$$ and $${E_{<!-- -->{M_k}}}\left( {\phi
• F} \right)$$.- 4.3 The Spaces $${D_{ \pm ,{M_k}}}\left( {\phi
• F} \right)$$.- 4.4 Remarks on the Topological Properties of the Spaces $${D_{<!-- -->{M_k}}}\left( {\phi
• F} \right),{E_{<!-- -->{M_k}}}\left( {\phi
• F} \right),{D_{ \pm ,{M_k}}}\left( {\phi
• F} \right)$$.- 5. Vector-Valued Ultra-Distributions of Class Mk
• Generalizations.- 5.1 Recapitulation on Vector-Valued Distributions.- 5.2 The Space $$D{'_{<!-- -->{M_k}}}\left( {\phi
• F} \right)$$.- 5.3 The Space $$D{'_{ \pm ,{M_k}}}\left( {\phi
• F} \right)$$.- 5.4 Vector-Valued Ultra-Distributions of Beurling-Type.- 5.5 The Particular Case: F = Banach Space.- 6. Comments.- 8 Elliptic Boundary Value Problems in Spaces of Distributions and Ultra-Distributions.- 1. Regularity of Solutions of Elliptic Boundary Value Problems in Spaces of Analytic Functions and of Class Mk
• Statement of the Problems and Results.- 1.1 Recapitulation on Elliptic Boundary Value Problems.- 1.2 Statement of the Mk-Regularity Results.- 1.3 Reduction of the Problem to the Case of the Half-Ball.- 2. The Theorem on "Elliptic Iterates": Proof.- 2.1 Some Lemmas.- 2.2 The Preliminary Estimate.- 2.3 Bounds for the Tangential Derivatives.- 2.4 Bounds for the Normal Derivatives.- 2.5 Proof of Theorem 1.3.- 2.6 Complements and Remarks.- 3. Application of Transposition
• Existence of Solutions in the Space D'(?) of Distributions.- 3.1 Generalities.- 3.2 Choice of the Form L
• the Space ?(?) and its Dual.- 3.3 Final Choice of the Form L
• the Space Y.- 3.4 Density Theorem.- 3.5 Trace Theorem and Green's Formula in Y.- 3.6 The Existence of Solutions in the Space Y.- 3.7 Continuity of Traces on Surfaces Neighbouring ?.- 4. Existence of Solutions in the Space $$D{'_{<!-- -->{M_k}}}\left( \Omega \right)$$ of Ultra-Distributions.- 4.1 Generalities.- 4.2 The Space $${\Xi _{<!-- -->{M_k}}}\left( \Omega \right)$$ and its Dual.- 4.3 The Space $${Y_{<!-- -->{M_k}}}$$ and the Existence of Solutions in $${Y_{<!-- -->{M_k}}}$$.- 4.4 Application to the Regularity in the Interior of Ultra-Distribution Solutions of the Equation Au = f.- 5. Comments.- 6. Problems.- 9 Evolution Equations in Spaces of Distributions and Ultra-Distributions.- 1. Regularity Results. Equations of the First Order in t.- 1.1 Orientation and Notation.- 1.2 Regularity in the Spaces D+.- 1.3 Regularity in the Spaces $${D_{ + ,{M_k}}}$$.- 1.4 Regularity in Beurling Spaces.- 1.5 First Applications.- 2. Equations of the Second Order in t.- 2.1 Statement of the Main Results.- 2.2 Proof of Theorem 2.1.- 2.3 Proof of Theorem 2.2.- 3. Singular Equations of the Second Order in t.- 3.1 Statement of the Main Results.- 3.2 Proof of Theorem 3.1.- 4. Schroedinger-Type Equations.- 4.1 Statement of the Main Results.- 4.2 Proof of Theorem 4.1.- 4.3 Proof of Theorem 4.2.- 5. Stability Results in Mk-Classes.- 5.1 Parabolic Regularization.- 5.2 Approximation by Systems of Cauchy-Kowaleska Type (I).- 5.3 Approximation by Systems of Cauchy-Kowaleska Type (II).- 6. Transposition.- 6.1 Orientation.- 6.2 The Parabolic Case.- 6.3 The Second Order in t Case and the Schroedinger Case.- 7. Semi-Groups.- 7.1 Orientation.- 7.2 The Space of Vectors of Class Mk.- 7.3 The Semi-Group G in the Spaces D(A?
• Mk). Applications.- 7.4 The Transposed Settings. Applications.- 7.5 Another Mk-Regularity Result.- 8. Mk -Classes and Laplace Transformation.- 8.1 Orientation-Hypotheses.- 8.2 Mk -Regularity Result.- 8.3 Transposition.- 9. General Operator Equations.- 9.1 General Results.- 9.2 Application. Periodic Problems.- 9.3 Transposition.- 10. The Case of a Finite Interval ]0, T[.- 10.1 Orientation. General Problems.- 10.2 Space Described by v(0) as v Describes X.- 10.3 The Space $${\Xi _{<!-- -->{M_k}}}$$.- 10.4 Choice of L.- 10.5 The Space Y and Trace Theorems.- 10.6 Non-Homogeneous Problems.- 11. Distribution and Ultra-Distribution Semi-Groups.- 11.1 Distribution Semi-Groups.- 11.2 Ultra-Distribution Semi-Groups.- 12. A General Local Existence Result.- 12.1 Statement of the Result.- 12.2 Examples.- 13. Comments.- 14. Problems.- 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions.- 1. Regularity in the Interior of Solutions of Parabolic Equations.- 1.1 The Hypoellipticity of Parabolic Equations.- 1.2 The Regularity in the Interior in Gevrey Spaces.- 2. The Regularity at the Boundary of Solutions of Parabolic Boundary Value Problems.- 2.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 2.2 The Regularity in Gevrey Spaces.- 3. Application of Transposition: The Finite Cylinder Case.- 3.1 The Existence of Solutions in the Space D'(Q): Generalities, the Spaces X and Y.- 3.2 Space Described by ?v as v Describes X.- 3.3 Trace and Existence Theorems in the Space Y.- 3.4 The Existence of Solutions in the Spaces D's,r(Q) of Gevrey Ultra-Distributions, with r > 1, s ? 2m.- 4. Application of Transposition: The Infinite Cylinder Case.- 4.1 The Existence of Solutions in the Space D' (R
• D'(?)): The Space X_.- 4.2 The Existence of Solutions in the Space D'+ (R
• D'(?)): The Space Y+ and the Trace and Existence Theorems.- 4.3 The Existence of Solutions in the Spaces D'+,s(R
• D'r(?)), with r > 1, s ? 2m.- 4.4 Remarks on the Existence of Solutions and the Trace Theorems in other Spaces of Ultra-Distributions.- 5. Comments.- 6. Problems.- 11 Evolution Equations of the Second Order in t and of Schroedinger Type.- 1. Equations of the Second Order in t
• Regularity of the Solutions of Boundary Value Problems.- 1.1 The Regularity in the Space $$D\left( {\bar Q} \right)$$.- 1.2 The Regularity in Gevrey Spaces.- 2. Equations of the Second Order in t
• Application of Transposition and Existence of Solutions in Spaces of Distributions.- 2.1 Generalities.- 2.2 The Space $${D_{ - ,\gamma }}\left( {\left[ {0,T} \right]
• {D_\gamma }\left( {\bar \Omega } \right)} \right)$$ and its Dual.- 2.3 The Spaces X and Y.- 2.4 Study of the Operator ?.- 2.5 Trace and Existence Theorems in the Space Y.- 2.6 Complements on the Trace Theorems.- 2.7 The Infinite Cylinder Case.- 3. Equations of the Second Order in t
• Application of Transposition and Existence of Solutions in Spaces of Ultra-Distributions.- 3.1 The Difficulties in the Finite Cylinder Case.- 3.2 The Infinite Cylinder Case for m > 1.- 4. Schroedinger Equations
• Complements for Parabolic Equations.- 4.1 Regularity Results for the Schroedinger Equation.- 4.2 The Non-Homogeneous Boundary Value Problems for the Schroedinger Equation.- 4.3 Remarks on Parabolic Equations.- 5. Comments.- 6. Problems.- Appendix. Calculus of Variations in Gevrey-Type Spaces.

by "Nielsen BookData"

Details

• NCID: BA0043759X
• ISBN:
  • 3540053638
  • 0387053638
  • 9783642651632
  • 3540054448
  • 0387054448
  • 354005832X
  • 038705832X
  • 9783642653957
• LCCN: 71151407
• Country Code: gw
• Title Language Code: eng
• Text Language Code: eng
• Original Language Code: fre
• Place of Publication: Berlin ; New York
• Pages/Volumes: 3 v.
• Size: 24 cm
• Classification:
  • LCC : QA377
  • DC : 515/.35
• Subject Headings:
  • LCSH : Differential equations, Partial -- Numerical solutions
  • LCSH : Boundary value problems -- Numerical solutions
• Parent Bibliography ID:
  • BA00007422