# Integral Inequality with L-2 Norm

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.

## 1 Answer

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 $$|\Gamma|<+\infty$$.