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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.

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  • $\begingroup$ As the question is currently worded, $C$ is allowed to depend on $f$ - but then I don't understand the purpose of the integral term in the displayed formula. Do you mean that $C$ is indepedent of $f$? $\endgroup$ – Jochen Glueck Sep 8 '19 at 19:21
  • $\begingroup$ @Jochen Glueck: Yes $C$ is independent of $f$. $\endgroup$ – Hheepp Sep 9 '19 at 6:11
  • $\begingroup$ Thanks for your response; I included this in the question. May I ask whether you could provide a reference for this (elliptic) result? $\endgroup$ – Jochen Glueck Sep 9 '19 at 7:19
  • $\begingroup$ @Jochen Glueck: for example see the Proposition A.4.2 in the Appendix of following book: Stable solutions of elliptic partial differential equations by Louis Dupaigne. $\endgroup$ – Hheepp Sep 9 '19 at 7:33
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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.

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