Continual Means and Boundary Value Problems in Function Spaces

1. Functional Classes and Function Domains. Mean Values. Harmonicity and the Laplace Operator in Function Spaces.- 1. Functional classes.- 2. Function domains.- 2.1. Uniform domains.- 2.2. Normal domains.- 3. Continual means.- 3.1. The mean over a uniform domain.- 3.2. The mean value ma, RF over the Hilbert sphere and its main properties.- 3.3. The spherical mean of a Gateaux functional.- 3.4. Functionals as rarndom variables.- 3.5. The Dirac measure in a function space. The centre of a function domain. Harmonicity.- 4. The functional Laplace operator.- 4.1. Definitions and properties.- 4.2. Spherical means and the Laplace operator in the Hilbert co-ordinate space 12.- 2 Chapter 2. The Laplace and Poisson Equations For a Normal Domain.- 5. Boundary value problems for a normal domain with boundary values on the Gateaux ring.- 5.1. Functional Laplace and Poisson equations.- 5.2. The fundamental, functional of a surface S.- 5.3. Examples.- 5.4. The maximum principle and uniqueness of solutions.- 5.5. The exterior Dirichlet problem.- 5.6. The deviation H - F.- 6. Semigroups of continual means. Relations to the probability solutions of classical boundary value problems. Applications of the integral over a regular measure.- 6.1. Semigroups of means over Hilbert spheres.- 6.2. The operators ms, n and the probability solutions of classical boundary value problems in the space Em.- 6.3. Regular measures and the extension of the Gateaux ring.- 3. The Functional Laplace Operator and Classical Diffusion Equations. Boundary Value Problems for Uniform Domains. Harmonic Controlled Systems.- 7. Boundary value problems with strong Laplacian and their parallelism to classical parabolic equations.- 7.1. The functional Laplacian and the classical parabolic operator.- 7.2. Dual problems and an analogy table.- 8. Boundary value problems for uniform domains.- 8.1. Functional and classical Dirichlet problems.- 8.2. The Dirichlet problem for operators.- 8.3. The functional Neumann problem.- 8.4. Properties of the Poisson equation.- 9. Harmonic control systems.- 9.1. Normal control domain.- 9.2. Uniform control domain.- 4. General Elliptic Functional Operators on Functional Rings.- 10. The Dirichlet problem in the space of summable functions and related topics.- 10.1. Functional elliptic operators of general type.- 10.2. Compact extensions of function domains. Compact restrictions.- 10.3. Averaging M?,?F of a functional $$[F in overline R$$ with respect to a family of transition densities of diffusion processes.- 11. The generalized functional Poisson equation.- Comments.- References.