Schauder estimates for higher order linear elliptic operator on manifold

Hi!

Let $(M,g)$ be a smooth compact riemannian manifold without boundary. Let $L$ be a linear elliptic operator on $M$ of order $2k$ with smooth coefficients. Suppose i have $u\in W^{2k,2}(M)$ and $f\in C^{0,\alpha}(M)$ such that $$L(u)=f$$ Do i have Schauder estimates of type

$$\left\|u\right\|_{C^{2k,\alpha}\left(M\right)}\leq C\left(L\right)\left\|f\right\|_{C^{0,\alpha}\left( M \right)}$$

I can assume also that $L$ (it would be better for $L$ of general type) is self adjoint and $u$ is $L^2$-orthogonal to $\ker\left(L \right)$.

If yes is there a reference for this kind of result?

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Dear Italo I have tried to edit your question, it is ok for you? – Giuseppe Jan 7 '13 at 15:23
Perfect! What was the problem? – Italo Jan 7 '13 at 15:24
When MathJax seems having problems, the basic solution is to enclose between backtips any TeX code containing underscores or asterisks. – Giuseppe Jan 7 '13 at 15:38
Thank you very much! – Italo Jan 7 '13 at 15:42

The result is true with some caveats. Under your assumptions we have the following results.

1. If $u\in W^{2k,2}(M)$ and $Lu\in C^{j,\alpha}(M)$, then $u\in C^{2k+j,\alpha}(M)$.

2. There exists $C>0$ depending only on $M$, $L$, $j$, and $\alpha$ such that, for any $u\in C^{2k+j,\alpha}(M)$ we have

$$\Vert u\Vert_{C^{2k+j,\alpha}} \leq C\Bigl(\; \Vert Lu\Vert_{C^{j,\alpha}}+ \Vert u\Vert\_{C^{0,\alpha}}\;\Bigr)$$

3. There exists $C>0$ depending only on $M$, $L$, $j$, and $\alpha$ such that, for any $u\in C^{2k+j,\alpha}(M)\cap (\ker L)^\perp$ (the $\perp$ refers to the $L^2$-inner product) we have

$$\Vert u\Vert_{C^{2k+j,\alpha}} \leq C \Vert Lu\Vert_{C^{j,\alpha}}$$

For proofs and more details see Chapter 10 of these notes and the references therein.

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+1 Great answer! – András Bátkai Jan 7 '13 at 16:29
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? – Italo Jan 8 '13 at 15:33
Sure, you can ask me any questions. – Liviu Nicolaescu Jan 8 '13 at 16:13
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. – Italo Jan 9 '13 at 15:46
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 $u_n$. – Italo Jan 9 '13 at 15:50

This is probably more of a comment: Section 3.2 in Lunardi's book contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates.

On manifolds, you should be able to extend these results using a finite number of coordinate charts (by compactness) as in these notes. But, this is not a reference...

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