Non-Homogeneous Boundary Value Problems and Applications : Vol. 1

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.

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.