# Uniform Hopf Inequality

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.

• 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 $f$? – Jochen Glueck Sep 8 '19 at 19:21
• @Jochen Glueck: Yes $C$ is independent of $f$. – Hheepp Sep 9 '19 at 6:11
• Thanks for your response; I included this in the question. May I ask whether you could provide a reference for this (elliptic) result? – Jochen Glueck Sep 9 '19 at 7:19
• @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. – Hheepp Sep 9 '19 at 7:33