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(This question is a duplicate from here)

We start with the linear elliptic PDE

$$ -\operatorname{div}(A\nabla u)=f \quad\text{in}\ \Omega,\\ u=0 \quad\text{on}\ \partial\Omega $$ We assume that $\Omega\subset\mathbb{R}^3$ is a smooth domain, that $A\in C^\alpha(\overline{\Omega})$ (for some $\alpha>0$) is a symmetric, positive definite matrix-valued function (there is $c_A>0$ such that $A(x)\xi\cdot\xi>c_\alpha|\xi|^2$ for all $\xi\in\mathbb{R}^3$ and all $x\in\Omega$), and a function $f\in L^\infty(\Omega)$.

For some $\delta>0$ and some symmetric $B\in C^\alpha(\overline{\Omega})$, we also look at the perturbed problem

$$ -\operatorname{div}((A+\delta B)\nabla v)=f \quad\text{in}\ \Omega,\\ v=0 \quad\text{on}\ \partial\Omega. $$

My question: If I know, that I have bounds on the gradient of $u$, like

$$ \|\nabla u\|_{L^\infty(\Omega)}\leq C(C_P,|\Omega|,c_\alpha,\|f\|_{L^\infty(\Omega)}), $$

can I conclude something similar for $v$ (for sufficiently small $\delta$), like

$$ \|\nabla v\|_{L^\infty(\Omega)}\leq C(C_P,|\Omega|,c_\alpha,\|f\|_{L^\infty(\Omega)},\delta, \|B\|_{C^\alpha(\overline{\Omega})})? $$ In these estimates, $C_P$ denotes the Poincare inequality constant.

I tried to prove this estimate via difference quotients and cut-offs but somehow got stuck. If it holds, what technique would you use to prove this? And if it is not possible, why could this estimate fail?

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    $\begingroup$ What method do you use to estimate the gradient for $u$? Is the problem a result of the lack of symmetry in $B$? $\endgroup$
    – sorrymaker
    Commented Mar 25, 2023 at 3:07
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    $\begingroup$ This will be very hard to say since your assumptions are in general not strong enough to give bounded solution gradients for any matrix $A$ and data $f$ with the stated properties, so you either need to strengthen these accordingly (see e.g. Gilbarg/Trudinger Chapter 8.11) or you need to tell about the particularities of the given $A$ and $f$. $\endgroup$
    – Hannes
    Commented Mar 26, 2023 at 11:57
  • $\begingroup$ @sorrymaker symmetry is not the issue for me, I am fine with assuming it (I have updated the question accordingly) $\endgroup$
    – Muschkopp
    Commented Mar 27, 2023 at 7:44
  • $\begingroup$ @Hannes I was thinking of an a priori argument (assuming you have bounds for $u$ do they also hold for $v$) but I have now updated the question to ensure the regularity for $u$ $\endgroup$
    – Muschkopp
    Commented Mar 27, 2023 at 7:47
  • $\begingroup$ @Muschkopp But now the assumptions for $B$ are also so strong that your desired estimate follows from the Gilbarg/Trudinger Chapter anyway, no? :-) $\endgroup$
    – Hannes
    Commented Mar 28, 2023 at 14:54

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