A Gaussian lower bound for the Dirichlet heat kernel. (English) Zbl 0801.35035

Van den Berg, M.

Bull. Lond. Math. Soc. 24, No. 5, 475-477 (1992).

From the introduction: Let \(D\) be an open set in \(\mathbb{R}^ m\), \(m= 2,3,\dots\), with boundary \(\partial D\), and let \(p_ D(x,y;t)\) be the heat kernel associated to the parabolic operator \(-\Delta_ D+ \partial/\partial t\), where \(\Delta_ D\) is the Dirichlet Laplacian for \(D\). For a non-empty open set \(D\) in \(\mathbb{R}^ m\) with boundary \(\partial D\) and \(\varepsilon\geq 0\) we define \(d(z)= \inf_{x\in \partial D} | z-x|\), \(D_ \varepsilon= \{z\in D: d(z)\geq \varepsilon\}\). Furthermore, let \(d_ \varepsilon(x,y)\) be the infimum of lengths of arcs in \(D_ \varepsilon\) with endpoints \(x\) and \(y\). If there is no arc in \(D_ \varepsilon\) with endpoints \(x\) and \(y\), we define \(d_ \varepsilon(x,y)=+\infty\). The main result of this paper is the following
Theorem: Let \(D\) be a non-empty open set in \(\mathbb{R}^ m\), \(m= 2,3,\dots\). Let \(\varepsilon> 0\), \(x\in D\) and \(y\in D\). Then for all \(t>0\), \[ p_ D(x,y;t)\geq e^{-d^ 2_ \varepsilon(x,y)/(4t)} p_{B_ \varepsilon}(0,0;t), \] where \(B_ \varepsilon\) is the open ball in \(\mathbb{R}^ m\) with radius \(\varepsilon\) and centre 0.

MSC:

35K05	Heat equation
58J35	Heat and other parabolic equation methods for PDEs on manifolds

Keywords:

Gaussian lower bound; Dirichlet heat kernel