Timeline for Applications of finite speed of propagation property
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 16, 2023 at 6:17 | comment | added | pxchg1200 | okay, I see. Are there exists some reference that provided the strict proof of conclusion (*)? | |
| Aug 16, 2023 at 4:32 | comment | added | Terry Tao | $w - w_\Omega$ only solves the wave equation in the interior and exterior of $\Omega$; on the boundary $\partial \Omega$ there will be a distributional forcing term. | |
| Aug 16, 2023 at 2:03 | comment | added | pxchg1200 | @TerryTao However, according to the arguments above, it seems that we can choose $r_0$ as large as possible, such that $\Omega\times [0, m\cdot d(y,\partial\Omega)]\subset K(x_0,r_0)$ as $m\to+\infty$. This seems very strange to me. | |
| Aug 16, 2023 at 2:01 | comment | added | pxchg1200 | @TerryTao Thank you for your comments! But I am still confuse about the restriction $t\leq d(y,\partial\Omega)$. Here is my understanding: After extended $w_{\Omega}$ to the entire Euclidean space, for any fixed $y\in \Omega$, $u=w-w_{\Omega}$ satisfies $\partial_{t}^{2}u-\Delta u=0$ in $\mathbb{R}^n\times (0,+\infty)$, and for any $x_0\in \Omega$, we can find $B(x_0,r_0)$ ($r_0$ large enough) such that $\Omega\times [0,d(y,\partial\Omega)] \subset K(x_0,r_0)$. Then applying the finite speed of propagation property to $u$ in the set $B(x_0,r_0)\times\{t=0\}$. Am I misunderstanding something? | |
| Aug 15, 2023 at 14:54 | comment | added | Terry Tao | (Strictly speaking, one needs a version of the finite speed of propagation property that also allows for an inhomogeneous term, but this can be deduced from the version you stated via Duhamel's formula.) | |
| Aug 15, 2023 at 14:51 | comment | added | Terry Tao | Apply the finite speed of propagation property to $u = w - w_\Omega$ (where $w_\Omega$ is extended arbitrarily to the entire Euclidean space), and note that $\Omega$ contains $B(y,t)$ whenever $y \in \Omega$ and $t \leq d(y,\partial \Omega)$. (If one is worried about the discontinuities of the extension of $w_\Omega$, one can convolve $u$ by some mollifier (approximation to the identity) if desired, though with the theory of distributions this is not really necessary.) | |
| Aug 15, 2023 at 14:20 | history | asked | pxchg1200 | CC BY-SA 4.0 |