Non-Homogeneous Boundary Value Problems and Applications : Volume III

1. Our essential objective is the study of the linear, non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2) fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J am 1 < f < v, where Pis a linear differential operator in m and where the Q/s are linear differential operators on am. In Volumes 1 and 2, we studied, for particular c1asses of systems {P, Qj}, problem (1), (2) in c1asses of Sobolev spaces (in general constructed starting from P) of positive integer or (by interpolation) non-integer order; then, by transposition, in c1asses of Sobolev spaces of negative order, until, by passage to the limit on the order, we reached the spaces of distributions of finite order. In this volume, we study the analogous problems in spaces of inlinitely dilferentiable or analytic Itlnctions or of Gevrey-type I~mctions and by duality, in spaces 01 distribtltions, of analytic Itlnctionals or of Gevrey­ type ultra-distributions. In this manner, we obtain a c1ear vision (at least we hope so) of the various possible formulations of the boundary value problems (1), (2) for the systems {P, Qj} considered here.

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.