Gagliardo-Nirenberg inequality with special boundary condition

Question

For simple connected domain $\Omega\subset \mathbb{R }^3 $ with smooth boundary, we have Gagliardo-Nirenberg inequality for $u\in H^1(\Omega)$: $$\left\|u \right\|_{L^p(\Omega )}\le C_1 \left\| u \right\|_{L^2(\Omega )}^{\frac{6-p}{2p} }\left\|\nabla u \right\|_{L^2(\Omega )}^{\frac{3p-6}{2p} } +C_2\left\| u \right\|_{L^2(\Omega )} , p\in [2,6]. $$ My problem is: if we require $u\cdot n _{|\partial \Omega }=0 $ or $u\times n _{|\partial \Omega }=0 $, can we let $C_2=0$? Or replace the last term by boundary term, just like following formulation? $$\left\|u \right\|_{L^p(\Omega )}\le C_1 \left\| u \right\|_{L^2(\Omega )}^{\frac{6-p}{2p} }\left\|\nabla u \right\|_{L^2(\Omega )}^{\frac{3p-6}{2p} } +C_2\left\| u \right\|_{L^1(\partial \Omega )} , p\in [2,6] . $$