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.

Location: http://mech.math.msu.su/~fpm/eng/k06/k066/k06601h.htm

Last modified: February 26, 2007