Dirichlet eigenfunctions with nonzero mean value

We consider Laplacian eigenfunctions on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$. 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 $\sim n^{1\mathrm{/}d}$ have a mean value different from zero. We prove that this rate is sharp in $\mathit{\text{any}}$ smooth domain, up to a logarithmic factor: in any smooth domain~$\mathrm{\Omega }$, among the first $n$ Dirichlet eigenfunctions at least $(\log n{)}^{-1\mathrm{/}2}\cdot n^{1\mathrm{/}d}$ have a nonzero mean.