Timeline for Variational formulation of an elliptic pde
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2020 at 21:13 | comment | added | inoc | I don’t understand your hints. | |
| Dec 14, 2020 at 20:37 | comment | added | Mateusz Kwaśnicki | I would start by extending $u$ to all of $\mathbb R^{n+1}$ so that, say, $u(x,y) = u(0,y)$ whenever $y < 0$. Sorry for not making this clear. | |
| Dec 14, 2020 at 11:52 | comment | added | inoc | I have to consider cutoff function and mollifier on what domain to get compact support in $\mathbb{R}^n\times[0,\infty)$? in $\mathbb{R}^n\times (0,\infty)$? | |
| Dec 14, 2020 at 11:11 | comment | added | Mateusz Kwaśnicki | Multiply $u$ by your favourite cutoff and convolve with your favourite mollifier to get a smooth, compactly supported approximation to $u$. This works in the usual Sobolev spaces, so it should work for $\cal W$, too. Get the "Hitchhiker's guide to the fractional Sobolev spaces" if you need references or details (this may be way too general for your needs, but off the top of my head I do not know a better source). | |
| Dec 14, 2020 at 11:02 | comment | added | inoc | What does you mean for standard mollify-and-cutoff argument? Can you give a little bit of details please? | |
| Dec 14, 2020 at 11:00 | comment | added | Mateusz Kwaśnicki | Finally, I doubt that $C_c^\infty(\mathbb R^n \times (0,\infty))$ is dense in $\mathcal W$: all functions in the former class have zero trace on the boundary. On the other hand, $C_c^\infty(\mathbb R^n \times [0, \infty))$ seems to be dense in $\mathcal W$ by the standard mollify-and-cutoff argument. | |
| Dec 14, 2020 at 10:58 | comment | added | Mateusz Kwaśnicki | Next, $u$ need not be in $W^{1,2}$ on the half-space, as someone pointed out to you in one of your other questions. | |
| Dec 14, 2020 at 10:57 | comment | added | Mateusz Kwaśnicki | @inoc: Write $w(s) = \int_0^S \partial_s u(s) ds - u(s)$. Then $w$ is weakly differentiable and $\partial_s w = 0$ almost everywhere, and therefore $w(s)$ is constant. Now just define $u_0 = w(s)$ (as this does not depend on $s$). | |
| Dec 14, 2020 at 8:31 | comment | added | inoc | I have another question: is true that $C^\infty_c(\mathbb{R}^n\times(0,\infty)$ is a dense subset of the class $\mathcal{W}$ wrt the norm $[u]_a+||u(\cdot,0)||_2$? If yes, how i can prove this fact? | |
| Dec 14, 2020 at 8:05 | comment | added | inoc | In section 4.1 there is write is "easily sees that the Bochner integral $\int_0^S\partial_su(s,\cdot)\,ds=u_0+u(S,0)$ for constant function $u_0\in L^2(\mathbb{R}^n)$ ", i don't understand the existence of such $u_0$. Maybe i can use fundamental theorem of calculus for Bochner integral ? But i can't prove $u\in W^{1,2}((0,\infty),L^2(\mathbb{R}^n))$. Can you give me more details please? | |
| Dec 14, 2020 at 8:01 | comment | added | Mateusz Kwaśnicki | This is written in Section 4.1 in the paper cited above. | |
| Dec 14, 2020 at 7:41 | comment | added | inoc | How i can prove that $u(\cdot,y)$ is continuous form $[0,\infty)$ to $L^2(\mathbb{R}^n)$? | |
| Dec 13, 2020 at 18:17 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |