FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2006, VOLUME 12, NUMBER 6, PAGES 3-15

Function integrals corresponding to a solution of the Cauchy--Dirichlet problem for the heat equation in a domain of a Riemannian manifold

Ya. A. Butko

Abstract

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A solution of the Cauchy--Dirichlet problem is represented as the limit of a sequence of integrals over finite Cartesian powers of the domain of a manifold considered. It is shown that these limits coincide with the integrals with respect to surface measures of the Gauss type on the set of trajectories in the manifold. Moreover, each of the integrands is a combination of elementary functions of the coefficients of the equation considered and geometric characteristics of the manifold. Also, a solution of the Cauchy--Dirichlet problem in the domain of the manifold is represented as the limit of a solution of the Cauchy problem for the heat equation on the whole manifold under infinite growth of the absolute value of the potential outside the domain. The proof uses some asymptotic estimates for Gaussian integrals over Riemannian manifolds and the Chernoff theorem.

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Last modified: February 26, 2007