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On page 135 of The Mathematical Theory of Finite Element Methods (Brenner and Scott), I encountered the following inequality:

$\left | \int_{\Gamma} \overline{v} - v \, ds \right | \leq |\Gamma |^{1/2} || \overline{v} - v ||_{L^2(\Gamma)}\,$.

The mean $\overline{v}$ is defined as

$\overline{v} = \frac{1}{\text{meas}(\Omega)} \int_{\Omega} v(x) \, dx \,$

and $\Gamma = \partial \Omega$.

The inequality is part of a proof that the bilinear form for the Poisson equation with Dirichlet boundary conditions is coercive.

My question is: could I replace the $\overline{v}-v$ on both sides of the inequality with some (more general) function? If so, what would the restrictions on the function be? I'm not sure where the inequality comes from.

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This is just Hoelder inequality. Let me denote $w:=\bar v -v$. Then $$ |\int_\Gamma w \ ds| \le \int_\Gamma |w|\cdot 1 \ ds \le \|1\|_{L^2(\Gamma)} \|w\|_{L^2(\Gamma)} = |\Gamma|^{1/2}\|w\|_{L^2(\Gamma)}. $$ This works if $w\in L^2(\Gamma)$ and $|\Gamma|<+\infty$.

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