Let $Omegasubsetmathbb{R}^3$ be a bounded smooth domain. In general, for a Poincare inequality of the type

$$|u|_{L^2}le C |nabla u|_{L^2}$$

to hold for all $uin Xsubset H^1(Omega)$ and $C$ independent of $u$, then $X$ needs to be such that it doesn’t contain constant translates. That is, if we consider $u+M$ for large $M>0$, the left hand side of the inequality increases indefinitely while the right hand side is unchanged, so we need some extra constraint in the definition of $X$. So common choices are $X=H^1_0(Omega)$ or $X={uin H^1(Omega)| int_Omega u,dx=0}$.

Here’s my question. Suppose, we’d like to say that there exists $C$ such that for all $uin X={uin H^2(Omega)|(partial_n u+u)_{|partialOmega}=0}subset H^2(Omega)$ we have

$$|u|_{L^2}le C|nabla u|_{L^2}.$$

First, is this true? If so, how does one prove such a statement? Essentially the requirement that $u$ satisfies the homogeneous Robin condition $(partial_n u+u)_{|partialOmega}=0$ should at least formally rule out constant translates, since $(partial_n u+u)_{|partialOmega}=0$ is not invariant under translation of $u$ by constants.

My guess is that it IS true, however, the usual proof I know of such statements usually relies on some compactness argument. For example, if $X$ were simply $H_0^1(Omega)$, then for the sake of contradiction, if we assume that there exists a sequence $u_nin H_0^1$ such that

$$|u_n|_{L^2}ge n|nabla u_n|_{L^2}$$

then, defining $v_n=u_n/|u_n|_{L^2}$, we have

$$frac{1}{n}ge |nabla v_n|_{L^2}.$$

Thus we have a bounded sequence in $H^1$ and a subsequence that converges strongly in $L^2$ and weakly in $H^1$ to some $vin H^1$. Because $|nabla v_n|_{L^2}to 0$, $v$ is constant. And since the trace map is continuous (and weakly continuous) from $H^1(Omega)$ to $H^frac{1}{2}(partialOmega)$ we have that $v$ is in fact in $H^1_0(Omega)$ and therefore $v=0$. Then we have a contradiction because $|v_n|_{L^2}=1$ for each $n$ implies that $|v|_{L^2}=1$.

Now this argument doesn’t work for Robin boundary conditions because now the relevant (Robin) trace operator is continuous and weakly continuous from $H^2(Omega)$ to $H^frac{1}{2}(partialOmega)$. In particular, if $v$ is the weak $H^1$ limit of a sequence $v_nin H^2$, then $v$ could be in $H^1$ but not $H^2$ and thus the notion of the normal derivative $(partial_n v)_{|partialOmega}$ may not even make sense for $v$. And without being able to say $(partial_n v+v)_{|partialOmega}=0$, we can’t necessarily say that $v=0$ like we did in the previous paragraph. So this is where I’m stuck. Any help would be appreciated.