The Functional Laplace Operator and Classical Diffusion Equations. Boundary Value Problems for Uniform Domains. Harmonic Controlled Systems

Abstract

We deal here with boundary value problems for the equations

$$ \Delta \textup{F}=\textup{O},\ \Delta \textup{U}=\textup{F} $$

(1)

, in which the operator Δ is defined as the iterated variational derivative, i.e. the strong Laplacian (see 4.1):

$$ \Delta F = \int\limits_a^b {{{F''}_{{x^2}\left( t \right)}}dt} $$

(2)

.