In many text book (Evans, Gilbarg-Trudinger for example) there is a classical result of interior regularity for weak solutions to a elliptic divergence problem $\rm{div}(A(x)u)=f$ in $\Omega\subset\mathbb{R}^n$ (a bounded domain), where $A$ is an uniformly elliptic matrix whit coefficients in $L^\infty$, $f\in L^2(\Omega)$ and $u\in H^1(\Omega)$ is a weak solution to the problem.

If we also suppose that the coefficients of $A$ are Lipschitz (i.e. $A\in C^{0,1}(\Omega)$), then for every $\Omega'\subset\subset\Omega$ ( $\Omega'$ compactly contained in $\Omega$) it follows that $u\in H^2(\Omega')$.

I wonder if it is possible to weaken the assumptions on the matrix $A$, in the specific case I assume $\alpha$-Hölder continuity of $A$ with $0<\alpha<1$, to obtain the same result.

I know that in the Hölder case unique continuation principle does not work.

If anyone know any references please send them to me! Thank you all!