Non-Homogeneous Boundary Value Problems and Applications

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 < ½.- 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 "? = ½".- 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 ?½, ?*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 ?½ 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.