Poincaré inequality with special boundary condition?

Question

For simple connected domain $\Omega\subset \mathbb{R }^3 $ with smooth boundary, we have Poincaré inequality for $u\in H^1_0(\Omega)$(equivalent say, $u\in H^1(\Omega),u_{|\partial \Omega }=0 $ ): $$\left\|u \right\|_{L^2(\Omega )}\le C\left\|\nabla u \right\|_{L^2(\Omega )}. $$ My problem is: does it hold still if we replace $u_{| \partial \Omega }=0 $ with $u\cdot n _{|\partial \Omega }=0 $ or $u\times n _{|\partial \Omega }=0 $? Or some extra term should be added, just like following formulation? $$\left\|u \right\|_{L^2(\Omega )}\le C\left\|\nabla u \right\|_{L^2(\Omega )}+C\left\| u \right\|_{L^1(\partial \Omega ) } . $$

Answer 1

No it does not hold if you assume merely $u\cdot n=0$. Consider indeed the case $\Omega=\{x_3>0\}$, and take for integer $k$, $$u_k(x)=(f_k(x_1)f_k(x_2)f_k(x_3),0,0)$$ where $f_k$ is the piecewise affine and continuous function such that $f_k(s)=1$ for $|s|\le k$, $f_k(s)=0$ for $|s|\ge k+1$. Then $u_k\cdot n=0$ on $\partial\Omega$, and as $f_k$ is larger than the indicator function of $[0,k]$, we have $$\|u_k\|^2_{L^2(\Omega)}=\left(\int_{\mathbb{R}}f_k^2(s)ds\right)^3\ge k^3$$ while since $f_k$ is smaller that the indicator function of $[0,k+1]$, and $|f'_k|\le 1$, $$\|\nabla u_k\|^2_{L^2(\Omega)}=3\left(\int_{\mathbb{R}}{f'_k}^2(s)ds\right)\left(\int_{\mathbb{R}}f_k^2(s)ds\right)^2\le 3(k+1)^2.$$ Since there is no real $C$ such that $k^3\le C\times 3(k+1)^2$ for all $k$, we have a counter example (assuming that I did not make an error computing norms, which should however not change the nature of the result). I believe you can build a similar example for the vectorial product.