Function integrals corresponding to a~solution of the Cauchy--Dirichlet problem for the heat equation in a~domain of a~Riemannian manifold
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 3-15
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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.
@article{FPM_2006_12_6_a0,
author = {Ya. A. Butko},
title = {Function integrals corresponding to a~solution of the {Cauchy--Dirichlet} problem for the heat equation in a~domain of {a~Riemannian} manifold},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {3--15},
publisher = {mathdoc},
volume = {12},
number = {6},
year = {2006},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a0/}
}
TY - JOUR AU - Ya. A. Butko TI - Function integrals corresponding to a~solution of the Cauchy--Dirichlet problem for the heat equation in a~domain of a~Riemannian manifold JO - Fundamentalʹnaâ i prikladnaâ matematika PY - 2006 SP - 3 EP - 15 VL - 12 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a0/ LA - ru ID - FPM_2006_12_6_a0 ER -
%0 Journal Article %A Ya. A. Butko %T Function integrals corresponding to a~solution of the Cauchy--Dirichlet problem for the heat equation in a~domain of a~Riemannian manifold %J Fundamentalʹnaâ i prikladnaâ matematika %D 2006 %P 3-15 %V 12 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a0/ %G ru %F FPM_2006_12_6_a0
Ya. A. Butko. Function integrals corresponding to a~solution of the Cauchy--Dirichlet problem for the heat equation in a~domain of a~Riemannian manifold. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 6, pp. 3-15. http://geodesic.mathdoc.fr/item/FPM_2006_12_6_a0/