Timeline for Elliptic regularity of second order pseudos
Current License: CC BY-SA 3.0
12 events
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| Jun 14, 2015 at 16:33 | comment | added | user74880 | Also, another point: since the construction of $\widetilde{P}$ depends on $\tau$, and hence $\varphi$, should one be concerned that the constants of equivalence in $(\widetilde{P} u, u) \cong \Vert u\Vert^2_{H^1}$ are dependent on $\varphi$? | |
| Jun 14, 2015 at 9:15 | history | edited | B K | CC BY-SA 3.0 |
Deleted incorrect thoughts about deducing an affirmative answer also to the first question from the calculations.
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| Jun 14, 2015 at 9:13 | comment | added | B K | Well, your ``nitpicking'' was absolutely right, since I made a mistake. I thought that the density of $C_c^\infty(\mathbb{R}^n)$ in $H^2(\mathbb{R}^n)$ would directly imply the density of $C_c^\infty(M\backslash\{q\})$ in $H^2(M)$, but that's not the case. And also it does not necessarily hold that $\varphi_n\to 1_M$ in $H^2(M)$ implies $\varphi_n u\to u$ in $H^1(M)$. So I think only the first part of my answer is correct, and I don't know anymore whether the answer to your first question is in fact affirmative. I edited the answer accordingly. Of course now my answer is not complete anymore. | |
| Jun 12, 2015 at 23:20 | comment | added | user74880 | Rather, it is better to take a sequence $u_n \in H^2(M) \cap C^\infty_c(M \setminus \{q\})$ such that $u_n \to u$ in $H^2$-norm. I hope you don't mind my nitpicking. | |
| Jun 12, 2015 at 22:47 | comment | added | user74880 | Also, it is not clear to me that $\varphi_n \to 1_M$ in $H^2(M)$-norm implies $\varphi_n u \to u$ in $H^1(M)$-norm. Is this obvious? | |
| Jun 12, 2015 at 22:18 | comment | added | user74880 | On second thoughts, in dimensions $n \geq 3$, I am convinced. But the case $n = 2$ seems a bit odd to me. | |
| Jun 12, 2015 at 21:03 | comment | added | user74880 | One last detail: could you please elaborate a little on why $C^\infty_c(M\setminus \{q\})$ is dense in $H^2(M)$? It seems to me (intuitively) that any sequence of approximating functions would have to go to $0$ sharply at $q$ resulting in a high gradient norm. What am I missing here? | |
| Jun 12, 2015 at 21:00 | vote | accept | user74880 | ||
| Jun 12, 2015 at 17:14 | history | edited | B K | CC BY-SA 3.0 |
added 534 characters in body
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| Jun 12, 2015 at 16:44 | comment | added | B K | Yes, that should work. I edited the answer accordingly. | |
| Jun 12, 2015 at 14:12 | comment | added | user74880 | Nice answer. Since the two norms are equivalent with equivalence constants independent of $\varphi$, is it possible to make a limiting argument to answer the first question affirmatively? | |
| Jun 12, 2015 at 8:09 | history | answered | B K | CC BY-SA 3.0 |