## Continual means and boundary value problems in functional spaces (Google eBook)The fates of important mathematical ideas are varied. Sometimes they are instantly appreciated by the specialists and constitute the foundation of the development of theories or methods. It also happens, however, that even ideas uttered by distinguished mathematicians are surrounded with respectful indifference for a long time, and every effort of inter preters and successors has to be made in order to gain for them the merit deserved. It is the second case that is encountered in the present book, the author of which, the Leningrad mathematician E.M. Polishchuk, reconstructs and develops one of the dir.ctions in functional analysis that originated from Hadamard and Gateaux and was newly thought over and taken as the basis of a prospective theory by Paul Levy. Paul Levy, Member of the French Academy of Sciences, whose centenary of his birthday was celebrated in 1986, was one of the most original mathe matiCians of the second half of the 20th century. He could not complain about a lack of attention to his ideas and results. Together with A.N. Kolmogorov, A.Ya. Khinchin and William Feller, he is indeed one of the acknowledged founders of the theory of random processes. In the proba bility theory and, to a lesser degree, in functional analysis his work is well-known for its conceptualization and scope of the problems posed. |

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### Contents

INTRODUCTION | 9 |

Function domains | 15 |

The functional Laplace operator | 33 |

THE LAPLACE AND POISSON EQUATIONS FOR A NORMAL DOMAIN | 41 |

solutions of classical boundary value problems Applications | 70 |

THE FUNCTIONAL LAPLACE OPERATOR AND CLASSICAL DIFFUSION | 85 |

Boundary value problems for uniform domains | 99 |

Harmonic control systems | 114 |

GENERAL ELLIPTIC FUNCTIONAL OPERATORS ON FUNCTIONAL RINGS | 131 |

The generalized functional Poisson equation | 149 |

REFERENCES | 156 |

### Common terms and phrases

addition analogous arguments assume boundary conditions boundary value problems bounded compact extension considered const constructed continual means continuous function converges uniformly definition denote derivative differential equations Dirac measure domain G ellipsoid elliptic operators English transl exists finite func function domain function space functional defined functional Dirichlet problem functional F functional Laplace operator functional T[x Furthermore Gateaux formula Gateaux ring given GSteaux functional harmonic functional heat equation Hilbert space holds implies integral interval Laplace operator Laplacian Lemma Let F LEVY Levy's linear Math mean value measurable function Moreover norm normal domain notation obtain paper POLISHCHUK parameters Poisson equation polynomials proof proved relation respect satisfying the conditions Section semigroup set G set of functions solution sphere subset summable suppose surface of type theory tion tional topology uniform domain uniformly continuous valid variables vector Volterra x e G