Non-homogeneous boundary value problems and applications

TLDR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.

Abstract: 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.