Uniform Hopf Inequality

Question

There is a Uniform Hopf Inequality as follow:

Let $\Omega \subset \mathbb{R}^n$, $n \geq 1$ denote a smoothly bounded domain. Also let $\rho(x)=\mathrm{dist}(x,\partial \Omega)$, the distance function from $\partial \Omega$. Assume that $f ≥ 0$ belongs to $L^∞(Ω)$ and let $u$ denote the solution of $$ \begin{cases} -\Delta u = f & \Omega \\ u=0 & \partial \Omega \end{cases} $$ There exists $C>0$, independent of $f$, such that: $$ u(x) \geq C \rho(x) \int_{\Omega} f(y) \rho(y) \, dy. $$

I want to know that is there a similar type of result for heat equation or fractional one? I will be thanked if someone can give a reference.

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Edit: For the proof of this elliptic result see for example the Proposition A.4.2 in the Appendix of following book: Stable solutions of elliptic partial differential equations by Louis Dupaigne.

Answer 1

It seems that you are looking for so-called lower heat kernel bounds.

• For the heat equation the inequality you are looking for can, for instance, be found in Theorem 1.1 of "Zhang: The Boundary Behavior of Heat Kernels of Dirichlet Laplacians (Journal of Differential Equations, 2002)".

• For the fractional Laplacian the paper "Chen, Kim, Song: Heat kernel estimates for the Dirichlet fractional Laplacian (Journal of the European Mathematical Society, 2010)" seems to be relevant for your question.