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Integral representations of nonnegative solutions for parabolic equations and elliptic Martin boundaries. (English) Zbl 1116.58026

The very interesting paper under review deals with the nonnegative solutions of the parabolic equation \((\partial_t+L)u=0\) in a cylinder \(D\times(0,T),\) where \(D\) is a noncompact domain of a Riemannian manifold.
Supposing that the associated heat kernel is intrinsically ultracontractive, the author establishes an integral representation theorem. Precisely, any nonnegative solution is represented uniquely by an integral over \((D\times\{0\})\cup (\partial_MD\times[0,t)),\) where \(\partial_MD\) is the Martin boundary of \(D\) for the associated elliptic operator \(L.\)
A unified application of that formula is also proposed to several concrete examples. It is shown moreover, that the intrinsic ultracontractivity of the heat kernel implies that the constant function \(1\) is a small perturbation of the elliptic operator \(L\) on \(D.\)

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
35C15 Integral representations of solutions to PDEs
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