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

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
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