Let us start with the Poisson equation \begin{equation} -\Delta u=f \end{equation} on a domain $\Omega$. The classical regularity says if the the boundary of the domain is sufficiently smooth, then there exists some $p>2$, such that for all $f\in L^{q}$ for $q\in[p^*,p]$, where $p*$ is the conjugate of $p$, implies that $u\in W_0^{1,q}$. For a $C^2$ boundary, we even have $p\in(0,\infty)$. What I want to do now is to apply a similar result for a general elliptic operator $L$ with $Lu=f$. To do this, I want to apply the fixed point idea which was given in Gröger's paper. But $p>2$ is not enough good for three dimensional space, so in fact we need some $p>3$. To apply the idea given by Gröger, I need in fact an estimate $J_p$ which is given by \begin{equation} \|u_f\|_{W_0^{1,p}}\leq J_p \|f\|_{L^p}, \end{equation} where $u_f$ is the solution of the Poisson's equation. Such a constant exists due to Gröger's theorem, although we can only know that $J_p=J_{p^*}\geq J_2=1$. But to apply results for $p>3$, an optimal estimate for $J_3$ is needed.
The difficulties here are the lacking of test functions. Like in $L^p$ theory, a test function has the form $u^{p-1}$, but then its derivative contains also the term $(p-1)u^{p-2}$ due to chain rule. Another possibility is that we can in fact give the exact solution by using singular kernels, but then the estimates seem to be non optimal.
So my question is, is there any reference giving hope to this question? Any suggestion is very welcome!
