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Function integrals corresponding to a solution of the Cauchy-Dirichlet problem for the heat equation in a domain of a Riemannian manifold. (English) Zbl 1151.35375

J. Math. Sci., New York 151, No. 1, 2629-2638 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 3-15 (2006).
Summary: 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 the manifold considered. It is shown that these limits coincide with the integrals with respect to surface measures of Gauss type on the set of trajectories in the manifold. Moreover, the integrands are 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 an 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.

MSC:

35K05 Heat equation
47D06 One-parameter semigroups and linear evolution equations
47N20 Applications of operator theory to differential and integral equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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