Given a domain $\Omega\subset\mathbb{R}^d$, suppose that the boundary is sufficiently regular. Then by Gröger's regularity theory we know that the following operator \begin{equation} \nabla\cdot\nabla u(v):=\int_{\Omega}\nabla u\cdot\nabla vdx \end{equation} in fact defines an invertible isomorphism between $W_0^{1,p}(\Omega)$ and $W^{1,p}(\Omega)$ for some $p>2$. If the boundary of the domain is sufficiently smooth, for instance $C^1$, we know that $p$ can be an arbitrary number in $[2,\infty)$. More generally, one can even show this for mixed boundary function spaces. My question now is, is there any analogous theory for elliptic system, i.e for the operator \begin{equation} \mathrm{A}u(v):=\int_{\Omega}\mu(x)\epsilon(u):\epsilon(v)dx, \end{equation} where $\mu$ is some tensor of fourth order (say fulfilling some kind of ellipticity) and $\epsilon$ denotes the stain tensor by $\epsilon(u)=\frac{1}{2}(\nabla u+\nabla u^T)$ for $u\in W^{1,p}(\Omega,\mathbb{R}^d)$ such that $u_{\Gamma_D}=0$ for some $\Gamma_D\subset \partial\Omega$. It was shown that such an operator defines an isomorphism for some $p>2$ in the paper by HALLERDINTELMANN, JONSSON, KNEES, REHBERG. But for my problem, this is not enough, I need at least that $p>3$, since I have to deal with problems in three dimension and for some kind of Sobolev embeddings, $p>3$ is necessary. Is there any research involving this phenomenon?
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$\begingroup$ Did you mean this paper? mathematik.unikassel.de/~dknees/publications/… $\endgroup$ – Todd Trimble♦ Nov 14 '16 at 15:20

$\begingroup$ @ToddTrimble yes, I meant it. $\endgroup$ – Yongyong Nov 14 '16 at 15:21

$\begingroup$ Not sure I caught all typos. Should "stain" be "strain"? Also, does "isomorphism like previous" refer to the one between "$W_0^{1,p}(\Omega)$ and $W^{1,p}(\Omega)$ for some $p>2$"? $\endgroup$ – Todd Trimble♦ Nov 14 '16 at 15:27

$\begingroup$ I don't really have an answer to your problem regarding systems, but the isomorphy property you look for is pretty difficult already for a scalar equation. My goto reference (in fact, from the same "larger circle" of authors) would be Optimal Sobolev Regularity for Linear SecondOrder Divergence Elliptic Operators Occurring in RealWorld Problems. Do you maybe need $p$ larger than the space dimension only for the embedding into a Hölder space? $\endgroup$ – Hannes Nov 14 '16 at 15:27

$\begingroup$ @ToddTrimble yes, thank you for pointing me the typos... $\endgroup$ – Yongyong Nov 14 '16 at 15:29