Timeline for Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2022 at 10:11 | comment | added | Tibeku | @GiorgioMetafune Thanks for your help so much! Is it still true for an elliptic operator $L$ in divergence form? | |
| Mar 16, 2022 at 22:01 | comment | added | Giorgio Metafune | For convex sets and a general pure second order operator $L$ the same argument as above applies, since $d$ is concave and then $Ld \leq 0$. | |
| Mar 16, 2022 at 8:54 | comment | added | Tibeku | @GiorgioMetafune Thanks for your answer so much! May I ask one more question? Does this conclusion still hold for domain diffeomorphic to a convex polyhedron (esp, in dim 2 diffeomorphic to a plane sector)? Or dose this conclustion hold in convex domain for variable coefficient elliptic operator rather than only the laplacian operator $\Delta$? | |
| Mar 11, 2022 at 14:56 | comment | added | Giorgio Metafune | This should be true. If $\Omega$ is convex, then $\Delta d$ is concave (see mathoverflow.net/questions/291308/…). If $\Omega$ is also smooth, this implies $\Delta d \leq 0$ and then $d_t-\Delta d \geq 0$. The maximum principle for the heat equation gives $|u(t,x)| \leq Cd(x)$. This estimate does not depend on smoothness if $\Omega$ is convex and should give the result by some approximation, I guess. | |
| Mar 11, 2022 at 13:37 | comment | added | Tibeku | @GiorgioMetafune Thank you so much for your help! May I ask one more question? Lemma 4.6.1 in book Heat kernels and spectral theory requires the domain to be $C^2$ and Lemma 4.6.1 does not hold in non-smooth domain. Do you think in convex polyhedron domain case there still holds: if $|u_0(x)|\leq C_0d(x)$, then $|u(t,x)|\leq Cd(x)$ where constant $C$ independent of $t$? | |
| Mar 11, 2022 at 13:06 | comment | added | Giorgio Metafune | Yes, sure. The first eigenfunction is positive in the interior, so any estimate near the boundary extends to a global estimate, enlarging the constant $C$. | |
| Mar 11, 2022 at 12:47 | comment | added | Tibeku | @GiorgioMetafune Thanks for your answer so much! It has been really helpful for me. Could you explain moreover, why my assumption is equivalent to $|u_0(x)|\leq C\phi(x)$, since here I only assume $|u_0(x)|\leq C{\rm dist}(x,\partial\Omega)$ when $x$ is sufficiently near the boundary. | |
| Mar 11, 2022 at 10:01 | comment | added | Giorgio Metafune | Your assumption is equivalent to saying that $|u_0(x)| \leq C\phi(x)$, where $\phi$ is the first eigenfuntion. Then $|u(t,x)| \leq Ce^{-\lambda t}\phi(x)$, with $\lambda>0$ the first eigenvalue. For the boudary nehaviour of $\phi$ see for example Lemma 4.6.1, Heat kernels and spectral theory by Davies. | |
| S Mar 11, 2022 at 9:05 | review | First questions | |||
| Mar 11, 2022 at 10:48 | |||||
| S Mar 11, 2022 at 9:05 | history | asked | Tibeku | CC BY-SA 4.0 |