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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!

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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.

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  • $\begingroup$ Thanks! So the regularity of second order is strictly correlated to the derivative of A $\endgroup$
    – Simmetrico
    Commented Jan 15, 2023 at 14:53
  • $\begingroup$ Yes, but this depends on the divergence form of the operator. If the operator is in not in divergence form, the interior estimates are true in case of continuous coefficeints. $\endgroup$
    – Giorgio Metafune
    Commented Jan 16, 2023 at 16:33

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