Dirichlet eigenfunctions with nonzero mean value

We consider Laplacian eigenfunctions on a domain $\Omega \subset \mathbb{R}^d$. Under Neumann boundary conditions, the first eigenfunction is constant and the others have mean value 0. The situation is different for Dirichlet boundary conditions: on `generic' domains, one would expect that every eigenfunction has nonzero mean value. The other extreme is the ball in $\mathbb{R}^d$, where among the first $n$ eigenfunctions only $\sim n^{1/d}$ have a mean value different from zero. We prove that this rate is sharp in \textit{any} smooth domain, up to a logarithmic factor: in any smooth domain~$\Omega$, among the first $n$ Dirichlet eigenfunctions at least $(\log{n})^{-1/2} \cdot n^{1/d}$ have a nonzero mean.