# Regularity of elliptic PDE with coefficients in some Sobolev space

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 $k,p,q,r$?

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.

I think the best answer is to look at the paper

Di Fazio, G., Hakim, D.I. & Sawano, Y. Elliptic equations with discontinuous coefficients in generalized Morrey spaces, European Journal of Mathematics (2017) 3: 728. DOI: 10.1007/s40879-017-0168-y

and the references therein. As a quick summary the paper says in the introduction $$a_{ij} \in W^{1,n}(\Omega)$$ is enough to guarantee $$Lu \in L^2(\Omega)$$ for $$u\in W^{2,p}$$, with $$p$$ within some $$\epsilon$$ of $$2$$.

The main thrust of the paper as I understand it is that $$VMO\cap L^\infty$$ (Sarason class VMO of functions with vanishing mean oscillation) is the "right" space for leading coefficients coefficients and for the lower coefficients it is more complicated (hence the Morrey spaces)