Young's inequality with duality bracket

Question

Let $\Omega$ be an open set in $R^n$ with smooth boundary $\Gamma$. let $u$ be in $H_0^1(\Omega ) \cap {H^2}(\Omega )$, ${{\partial _\nu }}$ denote the normal derevative. I have the following integarle$$\int\limits_\Gamma {{\partial _\nu }u.vd\Gamma } $$ I want to apply Young's inequalty $ab \leqslant \frac{{{a^2}}}{{4\varepsilon }} + \varepsilon {b^2}$ to estimate eatch terme alone. with $v$ $ \in $ $L^2({{\partial \Omega }})$. Is that makes sense? thank you.