Timeline for Schauder estimates for higher order linear elliptic operator on manifold
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 5, 2014 at 18:19 | comment | added | Liviu Nicolaescu | Dear David, follow the proof in Lemma 10.4.9 where you replace the $L^{k,2}$-norms with the $C^{k,\alpha}$-norms and the $L^2$ estimates with Holder estimates. Observe that $C^\alpha$ convergence implies $L^2$-convergence. | |
| Feb 5, 2014 at 14:26 | comment | added | David P | Dear Liviu, I tried to look up in your notes the inequality you wrote in item 3. The closest thing is Lemma 10.4.9 but it speaks about $L^1$ norms. Do you really mean $C^{k, \alpha}$ norms in your post above? | |
| Jan 10, 2013 at 15:05 | vote | accept | Italo | ||
| Jan 9, 2013 at 15:57 | comment | added | Liviu Nicolaescu | The comment space is not enough for my answer. Write directly to my e-mail address nicolaescu.1 at nd.edu and I will reply to your e-mail address. | |
| Jan 9, 2013 at 15:54 | comment | added | Italo | 2) i've no idea on how to get the last inequality, for the $L^{p}$ case i have the generalized poincare inequality, but for the $C^{2k,\alpha}(M)$ i don't know how to go on. Thank you in advance! | |
| Jan 9, 2013 at 15:50 | comment | added | Italo | So i get elliptic estimates of type ` $$\left\|u_n\right\|_{C^{2k,\alpha}(M)}\leq C(L,M,g)\left(\left\|L(u_n)\right\|_{C^{0,\alpha}(M)} +\left\|u_n \right\|_{C^{0,\alpha}(M)} \right)$$ ` but now i can't get the $C^{0,\alpha}(M)$-convergence for $u_n$. | |
| Jan 9, 2013 at 15:46 | comment | added | Italo | Dear Liviu i have some questions: 1) how do you get the local $C^{2k,\alpha}$-regularity of $u$ if you only know that $u\in W^{2k,2}(M)$? I had this idea but it brought me nowhere: i can't use the elliptic estimates (thm 10.3.1 part b) of your notes as they are because you assume the $C^{k+j}$-regularity for $u$ so i mollify $u$ with a smoothing kernel $\rho_{\frac{1}{n}}$ obtaining a sequence of smooth functions $u_n=u\star \rho_{\frac{1}{n}}$ and i use cutoffs to come to the euclidean case. | |
| Jan 8, 2013 at 16:13 | comment | added | Liviu Nicolaescu | Sure, you can ask me any questions. | |
| Jan 8, 2013 at 15:33 | comment | added | Italo | Thank you very much Liviu! I'm reading your notes, once i work out the details i'll accept the answer! If i need, can i ask you some details? | |
| Jan 7, 2013 at 16:25 | history | answered | Liviu Nicolaescu | CC BY-SA 3.0 |