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Is there a regularity theory for elliptic equations "optimized" for coefficients in a Sobolev space $W^{k,r}$?

By this, I mean a result (and the according elliptic estimates) that gives you (for $k$ large enough) $u \in W^{k+l,p}$ if $Lu \in W^{k,q}$ and if the coefficients of $L$ are in $W^{k,r}$ (in divergence form, for some $1 < p,q,r < \infty$).

The results in the book of Gilbarg-Trudinger require the coefficients to be in $C^{k,1}$ (by embedding one loses too much to get somewhere). Is $l=1$ possible for a second order operator? Or even $l=2$? If yes, for which $k,p,q,r$?

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up vote 4 down vote accepted

I doubt there is such result (I may be wrong, somebody please correct me if I am wrong here). Say we consider the boundary value problem for $$ -\mathrm{div}(A \nabla u) = f, \tag{1} $$ where $A = (a_{ij})$ and elliptic (coercive and bounded). For certain bounded smooth domain $\Omega\subset \mathbb{R}^n$, consider $a_{ij}\in W^{k,r}(\Omega)$ that can be continuously embedded into Hölder spaces, let's say $r >n$: $$ W^{k,r}(\Omega) \hookrightarrow C^{k-1,1-n/r}(\Omega), $$ and this embedding makes $a_{ij}$ lose $1$ differentiability at least!

For general regularity result like you said, for (1) say $p=q=2$, even if we wanna get interior estimates on the $l=2$ differentiability lifting $$ \|u\|_{H^{k+2}_{loc}} \leq \|f\|_{H^k}, $$ $a_{ij}$ has to be in $C^{k,1}$. We can't get such luxury for $W^{k,r}$-coefficients.

For $l=1$, if the coefficient space has the continuous embedding, then 1 differentiability lifting is natural like $f\in L^2$, and $u\in H^1$. If not, then the coefficient may be unbounded and (1) is not an elliptic problem any more.

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