I would like to prove that there exists $C>0$ such that
$$\| u \|^2_{L^2(B(0,1))} \leq C \left ( \| \nabla u \|^2_{L^2(B(0,1))} + \| u \|^2_{L^2(\partial B(0,1))} \right )$$
for every $u \in C^\infty(\overline{B(0,1)})$, where $B(0,1)$ is the unit ball in $\mathbb{R}^2$.
To me this looks similar to a Poincare inequality except that it compensates for the fact that $u$ may not be zero on the boundary. With this in mind, I tried manipulating the version of the Poincare inequality where we subtract off the average so that it would look like this situation. With that I get a $C>0$ such that
$$\| u \|^2_{L^2(B(0,1))} \leq C \left ( \overline{u}^2 + \| \nabla u \|^2_{L^2(B(0,1))} \right )$$
where $\overline{u}$ is the average of $u$ over $B(0,1)$. Now I get stuck trying to relate $\overline{u}$ to $u$ on the boundary. From here I would want to show that $\overline{u}^2 \leq C \| u \|^2_{L^2(\partial B(0,1))}$. But that can't work, because we could have a nonzero function which is zero on the boundary yet strictly positive on the interior. Of course, in this case the result follows from the Poincare inequality for $H^1_0$. So this result seems to be somewhat like an interpolation between the Poincare inequality for $H^1_0$ and the Poincare inequality for $H^1$ functions with mean zero.
Any suggestions?
Argue by contradiction: For every $k$ there exists $u_k$ with $\| u_k\|_{L^2(B)}=1$ and $$ \| \nabla u_k \|_{L^2(B)} + \| u_k\|_{L^2(\partial B)} \leq 1/k. $$ Then $u_k\to 0$ weakly in $H^1$, this contradicts the fact that the $u_k$ have unit $L^2$ norm by Rellich's compactness theorem.