Is there a smooth function $u$ such that $u = 1$ in $B_r(0)$, $u=0$ in $\mathbb R^N \setminus B_{2r}(0)$, and $$ |\nabla u| \le Cr^{-1}, \quad |\Delta u |\le Cr^{-2}, \quad |(-\Delta)^s| u \le Cr^{-2s} $$ holds?
5
-
$\begingroup$ Sure there is. Take any cut-off $u$ satisfying your conditions for $r = 1$, and set $u_r(x) = u(r^{-1}x)$. Then $(-\Delta)^s u_r(x) = r^{-2s} (-\Delta)^s u(r^{-1} x)$, and thus $u_r$ satisfies your conditions for the given $r$. $\endgroup$– Mateusz KwaśnickiCommented Mar 27, 2021 at 22:05
-
$\begingroup$ @MateuszKwaśnicki Thank you! Does the answer remain the same if you consider the problem replacing $\mathbb R^N$ with a bounded domain $\Omega \subset \mathbb R^N$ and considering the spectral fractional Laplacian? $\endgroup$– user173196Commented Mar 27, 2021 at 22:15
-
$\begingroup$ As long as you stay away from the boundary — I suppose the answer is "yes": all one needs to know is the asymptotic behaviour of the kernel of $(-\Delta)^s$ near the diagonal, and this is well known. I can look up a reference if you need. If $0$ is at the boundary — I am not sure, I never worked with the spectral fractional Laplacian (the Neumann one; for the Dirichlet one this makes no sense when $0$ is on the boundary). $\endgroup$– Mateusz KwaśnickiCommented Mar 27, 2021 at 22:19
-
$\begingroup$ @MateuszKwaśnicki Yes, please, let me know if you find the reference: I'd be very interested to know the case of the Neumann fractional Laplacian and $u$ compactly supported in the interior of $\Omega$ $\endgroup$– user173196Commented Mar 27, 2021 at 22:30
-
1$\begingroup$ This seems to go back at least to [Farkas, Jacob, Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions, Math. Nachr. 224 (2001): 75–104] and [Jacob, Schilling, Some Dirichlet spaces obtained by subordinate reflected diffusions. Rev. Mat. Iberoamericana 15 (1999): 59–91]. Generally, the kernel of $(-\Delta_\Omega)^s$ is given by $c_s \int_0^\infty k_t^\Omega(x,y) t^{-1-s} dt$, where $k_t^\Omega(x,y)$ is the heat kernel for $-\Delta_\Omega$. This gives appropriate scaling of the kernel, and consequently the desired estimate. $\endgroup$– Mateusz KwaśnickiCommented Mar 27, 2021 at 23:24
Add a comment
|