Q:

estimation with Sobolev Space $L^2(0,T,H^1)$

if we have the relation 
$$
\dfrac{1}{2} ||w||^2_{L^2(\Omega)} + ||\nabla w||^2_{L^2((0,T);L^2(\Omega))} \leq ||f||_{L^2((0,T);L^2(\Omega))} ||w||_{L^2((0,T);L^2(\Omega))} 
$$
where $f$ is data, how we prouve that 
$$
||w||_{L^2((0,T),H^1(\Omega))} \leq R,
$$
where $R$ is constant? 
Thank's in advance to the help.

A:

This is a nice application for Young's inequality
$$
a b \le \frac12 a^2 + \frac12 b^2
\qquad\forall a,b\ge 0.$$
(Note that this can be proven by expanding $(a-b)^2$).
Now, by replacing $a$ with $a \, \sqrt\varepsilon$ and $b$ with $b/\sqrt\varepsilon$,
we find
$$
a b \le \frac{\varepsilon}2 a^2 + \frac1{2\,\varepsilon} b
\qquad\forall a,b \ge 0, \varepsilon > 0.$$
This is very useful, since the coefficient in front of $a^2$ can be made arbitrarily small.
You just need to apply this inequality with $a = \|w\|_{L^2(L^2)}$, $b = \|f\|_{L^2(L^2)}$ and $\varepsilon = 1/2$.

