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?