Someone, could you please tell me suitable references on how the constant $C$ appearing in the a priori estimate for the elliptic Dirichlet problem depends on the domain $\Omega$ ? More precisely, we consider Lemma 9.17 in the book by Gilbarg-Trudinger and concentrate there on the dependency of the constant $C$ on $\Omega$. Let $\Omega_{\delta} := \{ x : {\rm dist}(x, \Omega)<\delta \}$ with $\delta > 0$. Then, I believe that the constant $C_\delta$ satisfies that $C_\delta \sim C_\Omega$ for $\delta > 0$ small. Is it true ? I have not yet find any paper or book about ensuring it.
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$\begingroup$ I do not know a place where this is written, but I think you are wright with the $\Omega_\delta$ question. Partitions of unities and local coordinates are uniform with respect to small $\delta$...but one should fill the details. $\endgroup$– Giorgio MetafuneCommented Sep 2, 2023 at 14:09
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$\begingroup$ Dear Giorgio, Thank you for your comment. I understand what you mean. I agree with it. We need to make a detail proof for it. $\endgroup$– kichrCommented Sep 2, 2023 at 14:57
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$\begingroup$ Maybe it is written somewhere, I hope that somebody else knows it! $\endgroup$– Giorgio MetafuneCommented Sep 2, 2023 at 15:07
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