The Cauchy Problem for Solutions of Elliptic Equations

Title page
Preface
Introduction
List of main notations
1 Function spaces
1.1 An abstract theory
1.1.1 Semilocal spaces
1.1.2 Functions of positive smoothness
1.1.3 Function spaces on closed sets
1.1.4 Dual spaces
1.1.5 Functions of negative smoothness
1.2 Spaces of smooth functions
1.2.1 Spaces of continuous functions
1.2.2 Spaces of functions of finite smoothness
1.2.3 The space of infinitely differentiable functions
1.2.4 Standard regularization
1.2.5 Approximation by C^∞ functions
1.2.6 Spectral synthesis in spaces of smooth functions
1.2.7 Smooth functions on closed sets
1.2.8 Dual spaces
1.2.9 Functions of negative smoothness
1.3 Hölder spaces
1.3.1 Spaces of Hölder functions
1.3.2 Hölder functions of finite smoothness
1.3.3 Hölder continuous functions
1.3.4 Standard regularization of Hölder functions
1.3.5 Approximation by C^∞ functions
1.3.6 Spectral synthesis in spaces of Hölder functions
1.3.7 Hölder functions on closed sets
1.3.8 Dual spaces
1.3.9 The negative Hölder spaces
1.4 Sobolev spaces
1.4.1 Lebesgue spaces
1.4.2 The spaces W^{s,q}_{opt}(X)
1.4.3 Standard regularization of Sobolev functions
1.4.4 Approximation by C^∞ functions
1.4.5 Geomettical properties of domains
1.4.6 Embedding theorems
1.4.7 Spectral synthesis in Sobolev spaces
1.4.8 Sobolev functions on closed sets
1.4.9 The negative Sobolev spaces
1.4.10 Duality
1.4.11 Fractional order Sobolev spaces
1.4.12 Besov spaces
1.4.13 Traces of Sobolev functions
2 Pseudodifferential operators in the spaces of distributions on closed sets
2.1 Calderon-Zygmund operators
2.1.1 The Kernel Theorem of L. Schwartz
2.1.2 Calderon-Zygmund kernels
2.1.3 Singular integrals
2.1.4 Extension to L^q(R^n)
2.1.5 Maximal operator
2.1.6 Classical examples
2.1.7 Calderon-Zygmund operators
2.1.8 The maximal function
2.1.9 Bounded mean oscillation
2.1.10 The Calderon-Zygmund theory
2.2 Pseudodifferential operators
2.2.1 The Fourier integral representation of Calderon-Zygmund operators
2.2.2 The definition
2.2.3 Symbols
2.2.4 Schwartz kernels of pseudodifferential operators
2.2.5 C*-algebra of pseudodifferential operators
2.2.6 Pseudohomogeneous kernels
2.2.7 Seeley's theorem
2.2.8 Operators on manifolds
2.2.9 Elliptic operators and parametrices
2.2.10 Symbols with limited smoothness
2.3 Boundedness theorems for pseudodifferential operators in local spaces
2.3.1 Fundamental theorem of calculus
2.3.2 Behavior in local Hölder spaces
2.3.3 Behavior in local Zygmund spaces
2.3.4 Behavior in local spaces of Hölder continuous functions
2.3.5 Behavior in local Sobolev spaces
2.3.6 Behavior in local Besov spaces
2.3.7 Behavior in local BMO spaces
2.3.8 Potential spaces
2.4 Boundedness theorems for pseudodifferential operators in non-local spaces
2.4.1 Surface layer potentials
2.4.2 Surface values of layer potentials
2.4.3 Symbols with the transmission property
2.4.4 Operators with the transmission property
2.4.5 Pseudodifferential operators on manifolds with boundary
2.4.6 Potential operators
2.4.7 Continuity in Hölder spaces
2.4.8 Continuity in Sobolev spaces
3 Capacity
3.1 Generalized form of capacity associated with a seminormed space
3.1.1 More on the traces of distributions
3.1.2 Removable singularities
3.1.3 Solutions regular at infinity
3.1.4 The equivalence of two forms of capacity
3.1.5 Capacitary extremals
3.1.6 Approximation on nowhere dense compact sets
3.1.7 The unified capacity
3.2 Capacity in spaces of smooth functions
3.2.1 Fundamental solutions of homogeneous elliptic equations
3.2.2 Orthogonal decomposition in the space of polynomials
3.2.3 A Laurent expansion at infinity
3.2.4 Higher order capacities
3.2.5 Examples
3.2.6 Other expressions for the capacity
3.2.7 Behavior under affine transformations
3.2.8 The capacity of a point
3.2.9 More on outer capacity
3.2.10 Comparison with Hausdorff measure
3.3 Capacity in Hölder spaces
3.3.1 A definition
3.3.2 Behavior under affine transformations
3.3.3 A nondegeneracy property
3.3.4 A further look at outer capacity
3.3.5 Hausdorff measure
3.3.6 Commensurability with Hausdorff content
3.3.7 Semiadditivity of the capacity
3.4 Capacity in Sobolev spaces
3.4.1 Bessel capacity
3.4.2 Metric properties of Bessel capacity
3.4.3 Quasicontinuous representatives of Sobolev functions
3.4.4 An application to spectral synthesis in Sobolev spaces
3.4.5 A brief review of higher order capacities
3.4.6 Comparison with Bessel capacity
3.4.7 Nguyen's theorem
4 Systems of differential equations with injective (surjective) symbols
4.1 Elliptic complexes
4.1.1 (Over-) underdetermined systems i
4.1.2 Complexes of differential operators
4.1.3 Resolutions of overdetermined systems
4.1.4 Laplacians
4.1.5 Parametrices of elliptic complexes
4.2 A solvability criterion for a system with surjective symbol in terms of convexity of supports
4.2.1 P-convex sets
4.2.2 Statement of the theorem
4.2.3 Proof of the necessity
4.2.4 . Proof of the sufficiency
4.2.5 Solvability in the space of distributions
4.3 Uniqueness condition for the Cauchy problem in the small
4.3.1 The sheaf of solutions
4.3.2 The uniqueness condition
4.3.3 Topological conditions for solvability
4.4 Left (right) fundamental solutions for a system with injective (surjective) symbol
4.4.1 Fundamental solutions to differential complexes
4.4.2 An existence theorem
4.4.3 Some examples
5 Coarse results on approximation on compact sets by solutions of a system with surjective symbol
5.1 Runge theorem for solutions of a system with surjective symbol
5.1.1 Problem of approximation
5.1.2 Some examples
5.1.3 A brief survey
5.1.4 The annihilator of sol(K)
5.1.5 P-convex hull
5.1.6 Runge theorem
5.2 Approximation of finitely smooth solutions by infinitely differentiable solutions
5.2.1 More on the hypoellipticity of elliptic complexes
5.2.2 An auxiliary result
5.2.3 Proof of the theorem
5.2.4 A generalization of the Stone-Weierstrass Theorem
5. 3 Approximation by potentials
5.3.1 A digression
5.3.2 Analogy with rational approximation
5.3.3 Examples
5.4 Localization property under approximation on compacta by solutions of a system with surjective symbol
5.4.1 The validity range
5.4.2 Localization property
5.4.3 The necessity of condition (U)_s
6 Approximation in spaces of smooth functions
6.1 Approximation of high order
6.1.1 Further look at the approximation problem
6.1.2 The main theorem
6.1.3 Notes
6.2 Approximation on the closure of a domain with the strong cone property
6.2.1 Approximation of lower order
6.2.2 The role of the connectedness of the complement
6.2.3 Walsh theorem
6.2.4 Bernstein theorems for elliptic equations
6.3 Approximation on nowhere dense compact sets
6.3.1 Hartogs-Rosenthal theorem for systems with surjective symbol
6.3.2 A generalization of the Lavrent'ev Theorem
6.3.3 Further remarks on the Hartogs-Rosenthal theorem
6.3.4 Systems elliptic in the sense of Douglis-Nirenberg
6.3.5 The case of totally disconnected compact sets
6.3.6 More on the Weierstrass Theorem
6.3.7 The general case
6.3.8 Overdetermined systems of canonical type
6.3.9 Approximation by harmonic vector fields
6.4 Capacitary criteria of Vitushkin type for approximation in spaces of smooth functions
6.4.1 A capacitary criterion
6.4.2 Discussion of the theorem
6.4.3 Approximation on compacta whose complements have the cone property
7 Approximation in Hölder spaces
7.1 Approximation of high order in Hölder spaces
7.1.1 Description of the annihilator of the subspace of solutions
7.1.2 The range s >= p
7.2 Approximation of lower order in Hölder spaces
7.2.1 A counterexample
7.2.2 A brief review
7.2.3 Reduction
7.3 Approximation criteria in terms of Hausdorff content
7.3.1 Approximation on compacta of measure zero
7.3.2 Approximation on nowhere dense compacta
7.3.3 Further results
7.4 Capacitary criteria of Vitushkin type for approximation in spaces of Hölder functions
7.4.1 A capacitary criterion
7.4.2 Discussion of the theorem
7.4.3 Approximation on compact a whose complements have the cone property
8 Approximation in Sobolev spaces
8.1 Approximation of high order in Sobolev spaces
8.1.1 The annihilator of sol(K) in W^{s,q}(K)^k
8.1.2 The range s >= p
8.2 Approximation of lower order in Sobolev spaces
8.2.1 Reducing approximation of lower order to a problem of spectral synthesis
8.2.2 Approximation in Sobolev spaces on compact sets by potentials with densities supported on the boundary
8.2.3 Degenerate cases of approximation in Sobolev spaces on compact sets with empty interior
8.2.4 Degenerate cases of approximation in Sobolev spaces on arbitrary compact sets
8.2.5 Uniform approximation on compact sets by potentials with densities supported on the boundary
8.2.6 Degenerate cases of uniform approximation on nowhere dense compact sets
8.2.7 Distinguished case of uniform approximation on nowhere dense compact sets
8.2.8 Absence of degenerate cases of uniform approximation on compact sets with nonempty interior
8.3 Approximation criteria in terms of Bessel capacity
8.3.1 The case of nowhere dense compact sets
8.3.2 The problem for arbitrary compact sets
8.3.3 Approximation criteria in terms of special capacities
8.3.4 Bounded point evaluations
8.4 Capacitary criteria of Vitushkin type for approximation in spaces of Sobolev functions
8.4.1 Statement of the theorem
8.4.2 Comments
8.4.3 Proof of the direct part, 1) => 2)
8.4.4 Proof of the converse part, 3) => 1)
9 Generalized boundary values of solutions of a system with injective symbol
9.1 Golubev series for solutions of elliptic equations
9.1.1 Statement of the main results
9.1.2 The converse theorem
9.1.3 A basic special case
9.1.4 Inductive limit topology in the space of solutions on a compact set
9.1.5 Banach spaces zq' (r ) K . . . . . . . . . . . . . . . . . . . . . . $
9.1.6 Inductive limit of the spaces zq' (r)K . . . . . . . . . . . . . . $
9.1.7 Another topology in the space of solutions on a compact set
9.1.8 The role of local connectedness
9.1.9 Equivalence of two topologies on Sol(K,P')
9.1.10 Conclusion of proof
9.1.11 A variant of Laurent-series expansion
9.1.12 Separation of singularities into atomic singularities
9.1.13 Representation of solutions by boundary integrals
9.1.14 Solutions with poles
9.1.15 An example for harmonic functions
9.1.16 Further results
9.1.17 Hyperfunctions
9.2 The Dirichlet problem for the generalized Laplacian by means of generalized functions
9.2.1 Green operators
9.2.2 Dirichlet systems
9.2.3 Green's formula for the generalized Laplacian
9.2.4 The Dirichlet problem
9.2.5 Function spaces
9.2.6 The operator related to the Dirichlet problem in the complete scale of Sobolev spaces
9.2.7 Fredholm operators
9.2.8 Theorem on a Complete Set of Isomorphisms
9.3 Traces on the boundary of generalized solutions of the Dirichlet equation
9.3.1 Weak solutions of the Dirichlet problem
9.3.2 Traces on the boundary of weak solutions to the Dirichlet equation
9.3.3 Traces on the boundary of solutions in the domain
9.3.4 Remarks
9.3.5 Traces of generalized solutions on parallel hypersurfaces
9.3.6 Solutions of finite order of growth near the boundary
9.3.7 Local increase of smoothness
9.3.8 Green's function
9.3.9 Problems with power singularities
9.4 Weak limit values on the boundary of solutions of a system with injective symbol
9.4.1 Green's formula for solutions of finite order of growth
9.4.2 Weak limit values
9.4.3 Equivalence of strong and weak limit values
9.4.4 A characterization
9.4.5 Miscellaneous
10 The Cauchy problem for a system with injective symbol
10.1 Green-type integral
10.1.1 Definition and simple properties
10.1.2 The Sokhotskii-Plemelj formulas
10.1.3 An application to the Cauchy problem
10.2 Iterations of the Green-type integral
10.2.1 Prologue
10.2.2 A theorem on iterations
10.2.3 The inner product h(.,.)
10.2.4 Solvability conditions for Pu = f
10.2.5 A remark about the 8-problem .................. $
10.2.6 An application to the Dirichlet problem
10.3 Solvability of the Cauchy problem in the class of distributions of finite order
10.3.1 Further look at the Cauchy problem
10.3.2 Tangential equation
10.3.3 Reduction to the Cauchy problem for the generalized Laplacian
10.3.4 Solvability of the Cauchy problem with data on the whole boundary
10.3.5 Criterion of solvability of the Cauchy problem with data on a boundary subset
10.3.6 A concluding remark
10.4 Carleman function
10.4.1 Definition
10.4.2 Existence
10.4.3 Carleman formula
10.4.4 Conditional stability of the Cauchy problem
10.4.5 The system of elasticity theory
11 Method of Fischer-Riesz equations in the Cauchy problem for a system with injective symbol
11.1 Operator-theoretic foundations of the method of Fischer-Riesz equations
11.1.1 Abstract problem in Hilbert spaces
11.1.2 Special bases
11.1.3 Solvability
11.1.4 Approximate solution
11.2 Hardy spaces
11.2.1 A further look at generalized boundary values
11.2.2 Generalized Hardy spaces
11.2.3 Boundary kernel function
11.2.4 Bergman formula
11.2.5 Relation with Green's function
11.3 Analysis of the Cauchy problem
11.3.1 Special bases for the Cauchy problem
11.3.2 Examples of special bases
11.3.3 Solvability of the Cauchy problem
11:3.4 Approximate solutions of the Cauchy problem
11.3.5 Zin's theorems
11.3.6 Traces of holomorphic functions on subsets of Shilov's boundary
11.3.7 Another approach
11.4 Analysis of the Dirichlet problem . .'
11.4.1 Basic assumptions
11.4.2 Special bases in the Dirichlet problem
11.4.3 Examples of special bases
11.4.4 A criterion of solvability of the Dirichlet problem
11.4.5 Regularization of solutions of the Dirichlet problem
11.4.6 Some calculations for the classical Dirichlet problem
12 Bases with double orthogonality in the Cauchy problem for a system with injective symbol
12.1 An operator-theoretic approach
12.1.1 The abstract framework
12.1.2 Abstract. Bergman Theory
12.1.3 Further horizons
12.1.4 An alternative method
12.1.5 Extremal property
12.2 Analysis of the Cauchy problem in terms of surface bases with double orthogonality
12.2.1 The main step
12.2.2 Surface bases
12.2.3 Analysis of the Cauchy problem
12.2.4 Notes
12.3 Analysis of the Cauchy problem in terms of solid bases with double orthogonality
12.3.1 Formulation of the problem
12.3.2 Green-type integral
12.3.3 Main lemma
12.3.4 The Cartan-Kähler Theorem
12.3.5 Extension problem
12.3.6 Solid bases
12.3.7 Solvability of the Cauchy problem
12.3.8 Approximate solution
12.3.9 Example for harmonic functions
12.3.10 A stability set
12.4 Applications to matrix factorizations of the Laplace equation
12.4.1 The Cauchy problem
12.4.2 Green-type integral
12.4.3 A solid basis of harmonic polynomials
12.4.4 An expansion of the fundamental solution
12.4.5 A solvability criterion
12.4.6 Regularization
Bibliography
Index of names
Subject index
Index of notation