# Isomorphism between Sobolev space and its dual for elliptic system

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 HALLER-DINTELMANN, 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?

• Did you mean this paper? mathematik.uni-kassel.de/~dknees/publications/… – Todd Trimble Nov 14 '16 at 15:20
• @ToddTrimble yes, I meant it. – Yongyong Nov 14 '16 at 15:21
• 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 $p>2$"? – Todd Trimble Nov 14 '16 at 15:27
• 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 go-to reference (in fact, from the same "larger circle" of authors) would be Optimal Sobolev Regularity for Linear Second-Order Divergence Elliptic Operators Occurring in Real-World Problems. Do you maybe need $p$ larger than the space dimension only for the embedding into a Hölder space? – Hannes Nov 14 '16 at 15:27
• @ToddTrimble yes, thank you for pointing me the typos... – Yongyong Nov 14 '16 at 15:29