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Let $\Omega\subset\mathbb{R^2},$ $\phi\in L_2, t\in(0,T), T>0$ and $\gamma>0.$

$$ |\xi|_{H^1(\Omega)}^2(t)\leq\gamma\int_{0}^T \left|\frac{\partial\phi}{\partial t}\right|_{L_2(\Omega)}^2dt$$

If we multiply with $(T-t)$ and then integrate from 0 to T for t, we have, $$ \int_0^T(T-t)|\xi|_{H^1(\Omega)}^2(t)dt\leq\frac{1}{2}\gamma\int_{0}^T (T-t)^2\left|\frac{\partial\phi}{\partial t}\right|_{L_2(\Omega)}^2dt$$

Can anybody help me with the derivation for the right hand side of the last equation?

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1 Answer 1

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I assume that your first inequality is of the form \begin{align*} |\xi|_{H^1(\Omega)}^2(t)\leq\gamma\int_{0}^t \left|\frac{\partial\phi}{\partial s}\right|_{L_2(\Omega)}^2ds. \end{align*} Then, with integration by parts, \begin{align*} \int_0^T(T-t)|\xi|_{H^1(\Omega)}^2(t)dt &\le \gamma\int_0^T(T-t) dt\int_{0}^t \left|\frac{\partial\phi}{\partial s}\right|_{L_2(\Omega)}^2ds\\ &=\frac{1}{2}\gamma\int_{0}^T (T-t)^2\left|\frac{\partial\phi}{\partial t}\right|_{L_2(\Omega)}^2dt. \end{align*}

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