I'm reading the book of T. Aubin Some nonlinear problems in Riemannian geometry. In chapter 3 he introduces elliptic operators on manifolds, but then he gives regularity results for elliptic operators on bounded open sets of $\mathbb{R}^n$. How these results extend to the manifold setting? More precisely, let $(M,g)$ be a $n$-dimensional smooth compact Riemannian manifold without boundary, $T$ be a linear elliptic differential operator of order $2m$ with (in local coordinates) smooth coefficients.
Let $f\in L^{p}\left(M\right)$ with $1< p<\infty $ and suppose there exist $u\in L^{q}\left(M\right)$ with $1< q<\infty$ s.t. $$\int_{M}u T^{*}\left(v\right)=\int_{M}fv$$ for all $v\in C^{\infty}\left(M\right)$ with $T^{*}$ the adjoint of $T$ w.r.t. the metric $g$. What can be said about $u$? Which kind of estimates does it satisfy? How can i use the theory on bounded open sets of euclidean spaces to get elliptic estimates on $u$?

EDIT: I'd like to prove the following proposition: Let $\left(M,g\right)$ compact Riemannian manifold without boundary, $T$ a linear elliptic operator of order $2m$ with smooth coefficients. Let $f\in L^{p}\left(M\right)$ with $1< p<\infty $ and suppose there exist $u\in L^{q}\left(M\right)$ with $1< q<\infty$ s.t. $$\int_{M}u T^{*}\left(v\right)=\int_{M}fv$$ for all $v\in C^{\infty}\left(M\right)$ with $T^{*}$ the adjoint of $T$ w.r.t. the metric $g$. Then $u\in W^{2m,p}\left(M\right)$ and satisfies the estimate

$$\left\|u\right\|_{W^{2m,p}\left(M\right)} \le \left(M,T,g,m,p\right)\left(\left\|f\right\|_{L^{p}\left(M\right)}+ \left\|u\right\|_{L^{p}\left(M\right)} \right)$$

Proof (tentative): I do the convolution $u_k = J_{\frac{1}{k}} \star u$ with $J_{\epsilon}$ the smoothing kernel. So i have

$$u_k\rightarrow u $$

in $L^{q}$ moreover for every $v\in C^{\infty}\left(M\right)$

$$\lim_{k\rightarrow +\infty}\int_{M}T\left(u_k\right)v=\lim_{k\rightarrow +\infty}\int_{M}u_k T^{*}\left(v\right)=\int_{M}u T^{*}\left(v\right)=\int_{M}f v $$

so $T\left(u_k\right)$ converges weakly in $L^p\left(M\right)$ to $f$ and because of this

$$\left\|T\left(u_k\right)\right\|_{L^{p}\left(M\right)}\leq C\left(f\right)$$

Now take an open covering $\lbrace B_{r,j}\rbrace_{j\in J}$ with $B_{r,j}$ open sets diffeomorphic to balls of radius $r$ sufficiently small and $\chi_j$ cutoff functions s.t. $\chi_{j}=1$ identically on $B_{2r,j}$ and $\chi_{j}=0$ identically outside $B_{3r,j}$. By Local theory i have that

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq C_j\left(\left\|T\left(\chi_j u_k\right)\right\|_{L^p\left(B_{3r,j}\right)}+ \left\|\chi_j u_k\right\|_{L^p\left(B_{3r,j}\right)}\right)\leq C_j\left(\left\|T\left(\chi_j u_k\right)\right\|_{L^p\left(B_{3r,j}\right)}+ \left\|u_k\right\|_{L^p\left(M\right)}\right)$$

Thanks to Deane i made some steps forward,

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq C_j\left(\left\|\chi_j T\left(u_k\right)\right\|_{L^p\left(B_{3r,j}\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} + C_j\left(T\right)\left\|\chi_j \right\|_{W^{2m,\infty}\left(B_{3r,j}\right)}\left\|u_k\right\|_{W^{2m-1,p}\left(B_{3r,j}\right)}\right)$$

that becomes

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq C_j\left(\left\|T\left(u_k\right)\right\|_{L^p\left(M\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} + C_j\left(T\right)\left\|\chi_j \right\|_{W^{2m,\infty}\left(B_{3r,j}\right)}\left\|u_k\right\|_{W^{2m-1,p}\left(B_{3r,j}\right)}\right)$$

By Ehrling's lemma (thm $7.30$ of Renardy-Rogers book Introduction to PDEs) i have that $\forall \epsilon_j>0$ exist $c\left(\epsilon_j\right)>0$ s.t.

$$\left\| u_k\right\|_{W^{2m-1,p}\left(B_{3r,j}\right)}\leq \epsilon_j\left\|u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}+ c\left(\epsilon_j\right)\left\|u_k\right\|_{L^p\left(B_{3r,j}\right)}$$

that implies

$$\left\|\chi_j u_k\right\|_{W^{2m,p}\left(B_{3r,j}\right)}\leq \tilde{C}_j\left(T,\epsilon_j,\chi_j\right)\left[ \left\|T\left(u_k\right)\right\|_{L^p\left(M\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} \right] + \epsilon_jC_j\left(T\right)\left\|\chi_j\right\|_{W^{2m,\infty}\left(B_{3r,j}\right)}\left\|u_k\right\|_{W^{2m,p}\left(M\right)}$$

Choosing $\epsilon_j$ small enough i have at last

$$\left\| u_k\right\|_{W^{2m,p}\left(M\right)}\leq \tilde{C}\left(T,\epsilon_j,\chi_j\right)\left[ \left\|T\left(u_k\right)\right\|_{L^p\left(M\right)}+ \left\|u_k\right\|_{L^p\left(M\right)} \right]$$

Now i suppose $p\geq q$ so

$$\left\| u_k\right\|_{W^{2m,p}\left(M\right)}\leq \tilde{C}\left(T,\epsilon_j,\chi_j\right)\left[ C\left(f\right)+ \left\|u\right\|_{L^q\left(M\right)} \right]$$

and by Sobolev embedding the best i can say is that $u\in W^{2m-1,p}\left(M\right)$ and it satisfies

$$\left\|u\right\|_{W^{2m-1,p}\left(M\right)}\leq C\left( C\left(f\right)+ \left\|u\right\|_{L^{q}\left(M\right)} \right)$$

How can i gain the $W^{2m,p}$ regularity and the estimates? How can i deal with the case $q$ is smaller than $p$?

  • 2
    $\begingroup$ I'm not sure if this will answer your question, but here's how it works for linear elliptic operators. The key is usually to to get Sobolev estimates (e.g. Garding's inequality), so one covers $M$ by coordinate neighborhoods, embeds each neighborhood in a torus (where the Sobolev norms are easy to write down), gets uniform estimates on each neighborhood, and patches the estimates together to get global estimates (assuming $M$ is compact). The patching part - which might really be what you're asking about - uses a partition of unity. $\endgroup$ – Paul Siegel Dec 28 '12 at 16:38
  • $\begingroup$ With these assumptions i start only with a $u$ that has a summability property and satisfies a PDE in weak form. I'd like to use theorem 3.57 of the book of Aubin locally with a partition of unity to get the same global result. I tried the following: i mollify u with smoothing kernels and i obtain a sequence of smooth functions $u_n$ tending to $u$ in $L^{q}$ s.t. $T(u_n)$ tends weakly to $f$ in $L^{p}$. Then i choose a partition of unity $\left\{(U_j,\chi_j)\right\}_{j\in J}$ and i'd like to use theorem 3.57 on $\chi_j u_n$ on each $U_j$ but i can't get the result i want. $\endgroup$ – student Dec 28 '12 at 17:14
  • 2
    $\begingroup$ Your assumptions imply that the function $u$ is in the Sobolev space $W^{2m,p}(M),$ and its $W^{2m,p}$ norm can be bounded in terms of its $L^q$ norm and the $L^p$ norm of $f$ (with a constant that depends on the geometry of $(M,g)$, on its dimension, on $p,q,m$). The proof of these facts is a simple application of the local results, using a partition of unity (as Paul Siegel says), and is something that you should really work out by yourself in detail at least once. $\endgroup$ – YangMills Dec 28 '12 at 22:05
  • 2
    $\begingroup$ Perhaps you could post in your question what exactly you've been able to do so far and indicate where your efforts fall shirt if what you expect to be true. $\endgroup$ – Deane Yang Dec 28 '12 at 22:14
  • $\begingroup$ I wrote what i've been able to do and what i expect to be true, i'll really appreciate a hint on how to go on with the proof. $\endgroup$ – student Dec 30 '12 at 17:46

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