Schauder estimates for higher order linear elliptic operator on manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:07:57Z http://mathoverflow.net/feeds/question/118284 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118284/schauder-estimates-for-higher-order-linear-elliptic-operator-on-manifold Schauder estimates for higher order linear elliptic operator on manifold Italo 2013-01-07T14:57:54Z 2013-01-07T16:25:21Z <p>Hi!</p> <p>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</p> <p><code>$$\left\|u\right\|_{C^{2k,\alpha}\left(M\right)}\leq C\left(L\right)\left\|f\right\|_{C^{0,\alpha}\left( M \right)}$$</code></p> <p>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 $\ker\left(L \right)$.</p> <p>If yes is there a reference for this kind of result?</p> <p>Thank you in advance.</p> http://mathoverflow.net/questions/118284/schauder-estimates-for-higher-order-linear-elliptic-operator-on-manifold/118292#118292 Answer by András Bátkai for Schauder estimates for higher order linear elliptic operator on manifold András Bátkai 2013-01-07T16:09:00Z 2013-01-07T16:09:00Z <p>This is probably more of a comment: Section 3.2 in <a href="http://books.google.hu/books?id=mWojiHzg9bEC&amp;printsec=frontcover" rel="nofollow">Lunardi's book</a> contains a broad overview on higher order parabolic problems with lots of references, including the most important Hölder estimates. </p> <p>On manifolds, you should be able to extend these results using a finite number of coordinate charts (by compactness) as in <a href="http://www.math.utsc.utoronto.ca/gpde/notes/schmfld.pdf" rel="nofollow">these notes</a>. But, this is not a reference...</p> http://mathoverflow.net/questions/118284/schauder-estimates-for-higher-order-linear-elliptic-operator-on-manifold/118295#118295 Answer by Liviu Nicolaescu for Schauder estimates for higher order linear elliptic operator on manifold Liviu Nicolaescu 2013-01-07T16:25:21Z 2013-01-07T16:25:21Z <p>The result is true with some caveats. Under your assumptions we have the following results.</p> <p><strong>1.</strong> If $u\in W^{2k,2}(M)$ and $Lu\in C^{j,\alpha}(M)$, then $u\in C^{2k+j,\alpha}(M)$.</p> <p><strong>2.</strong> 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</p> <p>$$\Vert u\Vert_{C^{2k+j,\alpha}} \leq C\Bigl(\; \Vert Lu\Vert_{C^{j,\alpha}}+ \Vert u\Vert_{C^{0,\alpha}}\;\Bigr)$$</p> <p><strong>3.</strong> 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</p> <p>$$\Vert u\Vert_{C^{2k+j,\alpha}} \leq C \Vert Lu\Vert_{C^{j,\alpha}}$$</p> <p>For proofs and more details see Chapter 10 of <a href="http://www3.nd.edu/~lnicolae/Lectures.pdf" rel="nofollow">these notes</a> and the references therein.</p>