# Optimal assumption on H^2 regularity

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!

One cannot weaken the regularity hypothesis on $$A$$ to $$C^{\alpha}$$. Consider for example the 1D case: when $$f = 0$$ the solution $$u$$ satisfies $$u'(x) = \frac{const.}{A(x)},$$ which is not in $$H^1$$ for a generic choice of $$C^{\alpha}$$ coefficient.