Previous |  Up |  Next


Lévy laplacian; maximum principle; Dirichlet and Poisson problem
We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.
[1] Averbuch V.I., Smoljanov O.G., Fomin S.V.: Generalization of functions and differential equations in linear spaces II., Differential operators and their Fourier transform (in Russian). Trudy Moskov. Mat. Obshch. 27 (1975), 247-262.
[2] Daleckij J.L., Fomin S.V.: Measures and Differential Equations in Infinite Dimensional Spaces (in Russian). Nauka, Moscow, 1983. MR 0720545
[3] Mingarelli A.B., Wang S.: A maximum principle and related problems for a Laplacian in Hilbert space. Differential Equations and Dynamical Systems 1 (1993), 1 23-34. MR 1385791 | Zbl 0885.35142
[4] Gochberg I.C., Krejn M.G.: Introduction to the Theory of Linear Operators in Hilbert space (in Russian). Nauka, Moscow, 1965.
[5] Lévy P.: Problèmes Concrets d'analyse Fonctionnelle. Paris, Gauthier-Villars, 1951. MR 0041346 | Zbl 0155.18201
[6] Šilov G.E.: On some questions of analysis in Hilbert space I. (in Russian). Functional Anal. Appl. 1 (1967), 2 81-90. MR 0213916
[7] Nemirovskij A.S., Šilov G.E.: On the axiomatic description of Laplace's operator for functions on Hilbert space (in Russian). Functional Anal. Appl. 3 (1969), 79-85. MR 0253088
[8] Sikirjavyj V.Ja.: The invariant Laplace operator as an operator of pseudospherical differentiation (in Russian). Moscow Univ. Math. Bull. 3 (1972), 66-73. MR 0305143
Partner of
EuDML logo