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Consider an elliptic differential equation of the form $$-\nabla.(A\nabla u)=f~~\mbox{ in } \Omega \tag{1}\label{1}$$ with Dirichlet boundary conditions, where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary and $A$ is a matrix with $L^\infty$ coefficients and is elliptic. By standard Lax-Milgram Lemma, we have a solution in $H^1_0(\Omega)$ and by further regularity, in $H^2(\Omega)$.

By other regularity results of De Giorgi, Moser, Nash etc., the solution, $u$ is, in fact, Hölder continuous.

  • What can be said about the weak derivative of the solution $\nabla u$? Is it bounded? Hölder continuous? I know, that by writing down an equation (system) for the gradient of $u$, one can conclude higher regularity of $\nabla u$. But this would require the coefficients of the equation \eqref{1} to be $W^{1,\infty}$. I do not want to increase the regularity of $A$.
  • Further, if not the solution, could we say something better about the regularity of the eigenfunctions of \eqref{1}, without changing the regularity of the coefficients?
  • Is there a version of Schauder Theory for merely bounded coefficients?
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    $\begingroup$ If you assume $A \in C^{0,\alpha}$ then you can get Hölder estimates for $\nabla u$ - this is near the end of Chapter 8 of Gilbarg & Trudinger. I believe this is essentially necessary but don't have an counterexample offhand. For the Schauder theory you can weaken the regularity to Dini continuity, but I think that's it. $\endgroup$ – Anthony Carapetis Jul 24 '17 at 0:06
  • $\begingroup$ Yeah, Ladyzhenskaya and Uraltseva, too. $\endgroup$ – Tanuj Dipshikha Jul 24 '17 at 4:37

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