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Hheepp
Hheepp
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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.

===============

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.

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.

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.

===============

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.

Changed the integration variable in the second displayed formula, as to not conflict with the rest of the formula.
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Jochen Glueck
Jochen Glueck
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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 $$ \begin{cases} -\Delta u = f & \Omega \\ u=0 & \partial \Omega \end{cases} $$ Then thereThere exists $C>0$, independent of $f$, such that: $$ u(x) \geq C \rho(x) \int_{\Omega} f(x) \rho(x) \, dx. $$$$ u(x) \geq C \rho(x) \int_{\Omega} f(y) \rho(y) \, dy. $$

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.

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 $$ \begin{cases} -\Delta u = f & \Omega \\ u=0 & \partial \Omega \end{cases} $$ Then there exists $C>0$ such that: $$ u(x) \geq C \rho(x) \int_{\Omega} f(x) \rho(x) \, dx. $$

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.

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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Hheepp
Hheepp
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  • 12

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 $$ \begin{cases} -\Delta u = f & \Omega \\ u=0 & \partial \Omega \end{cases} $$ Then there exists $C>0$ such that: $$ u(x) \geq C \rho(x) \int_{\Omega} f(x) \rho(x) \, dx. $$

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.