The author presents the theory of integration in infinite dimensional spaces, called continual means, and its applications to boundary value problems in function spaces. This theory goes back to earlier work of R. Gâteaux [Bull. Soc. Math. Fr. 47, 47–70, 70–96 (1919; JFM 47.0382.01)] and P. Lévy [Leçons d’analyse fonctionnelle. Paris: Gauthier-Villars (1922; JFM 48.0453.01) and Les problèmes concrets d’analyse fonctionnelle. ibid. (1951; Zbl 0043.32302)] and was almost forgotten for a long time. The material covered in the present book is to be found in various papers of the author. The contents of the four chapters is as follows.
Functional classes and function domains, mean values, harmonicity and the Laplace operator in function spaces. The Laplace and Poisson equations for a normal domain. The functional Laplace operator and classical diffusion equations, boundary value problems for uniform domains, harmonic controlled systems. General elliptic functional operators on functional rings.