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By De Giorge, Nash and Moser solutions of \begin{equation} \operatorname{div} (A(x) Du) = 0 \end{equation} where $Du$ denotes the gradient of $u$ and $A$ is a $\lambda,\Lambda$ elliptic matrix. That is, \begin{equation} \lambda|\xi|^2\le\langle A_\pm(x)\,\xi,\xi\rangle \le \Lambda|\xi|^2, \end{equation} for constants $0< \lambda \le \Lambda$ can be only Hölder continuous. So the classical strong maximum principle, see for instance Fanghua Lin, enter image description here can not be applyed because the solutin can not be sufficiently smooth. If the equation enter image description here

was satisfyied only in the weak sense, the SMP istill holds?

I know that already exists similar questions here but the different context difficult the undertand.

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    $\begingroup$ In this context the SMP follows from Harnack inequality. If $m$ is the infimum of $u$, then $v=u-m$ has infimum 0. If $v(x_0)=0$, then $\sup_B v \leq C \inf_B v=0$ for any ball $B$ containing $x_0$. The set of zeros of $v$ is then open and closed and $v=0$. $\endgroup$
    – Giorgio Metafune
    Commented May 18, 2023 at 17:09

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