Elliptic Functional Differential Equations and Applications

Abstract

I Boundary Value Problems for Functional Differential Equations in One Dimension.- 1 Ordinary Differential Equations with Nonlocal Boundary Conditions.- 2 Difference Operators in One Dimension.- 3 The Boundary Value Problem for the Differential-Difference Equation.- 4 Generalized and Classical Solutions.- 5 Applications to Control Systems with Delay.- 6 The Boundary Value Problem for the Differential-Difference Equation with Degeneration.- Notes.- II The First Boundary Value Problem for Strongly Elliptic Differential-Difference Equations.- 7 Some Geometrical Constructions.- 8 Difference Operators in the Multidimensional Case.- 9 Necessary and Sufficient Conditions for Strong Ellipticity.- 10 Solvability and Spectrum.- 11 Smoothness of Generalized Solutions in Subdomains.- 12 Smoothness of Solutions on a Boundary of Neighboring Subdomains.- 13 Elliptic Differential Equations with Nonlocal Conditions on Shifts of Boundary.- Notes.- III Applications to the Mechanics of a Deformable Body.- 14 The Elastic Model.- 15 Variational and Boundary Value Problems.- 16 Smoothness of Solutions.- 17 The One-Dimensional Case.- Notes.- IV Semi-Bounded Differential-Difference Operators with Degeneration.- 18 Self-Adjoint Extension of a Semi-Bounded Differential-Difference Operator.- 19 The Spectrum of Semi-Bounded Differential-Difference Operators.- 20 Smoothness of Solutions of Equations with Degeneration.- Notes.- V Nonlocal Elliptic Boundary Value Problems.- 21 Nonlocal Elliptic Problems with a Parameter.- 22 Elliptic Equations with Nonlocal Boundary Conditions in a Cylinder.- 23 Elliptic Differential-Difference Equations in a Cylinder.- 24 Applications to the Multidimensional Diffusion Processes.- 25 Elliptic Problems with Nonlocal Conditions near the Boundary and Feller Semigroups.- Notes.- A Linear Operators.- B Functional spaces.- C Elliptic Problems.- List of Symbols.