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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!

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  • $\begingroup$ What exactly are your assumptions on the coefficients in $L$ now? Or are you looking for a classification of coefficients for which your desired inequality holds true? $\endgroup$ – Hannes Mar 14 '17 at 16:09
  • $\begingroup$ The assumptions should at least be that the coefficients are Lipschitz continuous. I already found a result given by Giaquinta and Martinazzi in his book "An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs". $\endgroup$ – Peter Mar 17 '17 at 10:53
  • $\begingroup$ @Hannes forgot to at you. $\endgroup$ – Peter Mar 17 '17 at 17:08
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I think you could give a look to the following paper of mine  Estimates for Divergence Form Elliptic Equations with Discontinuous Coefficients. You can get it from research gate.

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  • $\begingroup$ Iif you are still in trouble feel free to ask $\endgroup$ – Giuseppe Di Fazio Dec 14 '16 at 21:30
  • $\begingroup$ Yes, I think the problem I have is that I may need an optimal estimate for the inverse operator norm, which is $c$ in your work, for some kind of fixed point theorem, but I think the optimality is not given in your work. And also, I think it is very hard to give an optimal estimate for the case $p>2$, since we need many estimates in between and this makes it difficult. Maybe I should find another way to do this problem. $\endgroup$ – Peter Dec 15 '16 at 9:47
  • $\begingroup$ Could you give a reference of the paper in which the problem has been solved for the operator \Delta? $\endgroup$ – Giuseppe Di Fazio Dec 15 '16 at 14:39
  • $\begingroup$ In fact it is not solved, so I asked for reference, but now I think this can be hard done, and it was also mentioned in Grisvard's book "elliptic problems in non smooth domains" that the inverse norm is very non optimal for $p>2$. For the case $p=2$, an optimal estimate can be obtained by using Mikhlin's theorem and Plancherel's equality, so was mentioned by his book, but i can not find the source anymore. $\endgroup$ – Peter Dec 15 '16 at 15:47

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