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This is on page 542 of Evans PDE book. The last inequality states that

$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$

Where is this coming from? I think this is just young's inequality and then holder applied to $|Du|$ (since $u$ is assumed to be in $H_0^1[U]$) but why write it in such a weird way?

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So we have the inequality $ \int_U |Du |^2 + \mu |u|^2 dx \leqslant \frac{1}{2} \int_U |Du|^2 dx + C \int_U |u|^2 + 1 dx .$ Now assume that $\mu$ is sufficiently large, for example let $\mu > C + \frac{1}{2} $ . Then this inequality becomes \begin{eqnarray*} \frac{1}{2}\int_U |Du|^2 dx &\leqslant& (C - \mu ) \int_U |u|^2 dx + C \int_U dx \\ &\leqslant & - \frac{1}{2} \int_U |u|^2 dx + C \int_U dx \end{eqnarray*} or $ \frac{1}{2} \int_U |Du|^2 + |u|^2 dx \leqslant C \int_U dx .$ Thus we can see that $ \| u \|_{H_0^1 (U)} = \int_U |Du|^2 + |u|^2 dx \leqslant 2C \int_U dx \leqslant C' < \infty $ holds.

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    EXCELLENT! Thank you I understand now.2012-12-13