Non-Homogeneous Boundary Value Problems and Applications: Vol. 1 (Paperback, Softcover Repri)

'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 0.- 12.1 Definition. First Properties.- 12.2 Interpolation between the Spaces H?s(?), s 0.- 12.3 Interpolation between $$Hrac{{{s_1}}}{0}(\Gamma )$$ 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 $$Hrac{{{s_1}}}{0}(\Omega )$$ 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