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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 $k,p,q,r$?

Wolf
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2 Answers2

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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.

Shuhao Cao
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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)

Bill Lionheart
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