Dirichlet Eigenfunctions with Nonzero Mean Value

We consider Laplacian eigenfunctions on a domain S2 subset of Rd. 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 Rd, where among the first n eigenfunctions only similar to n1/d have a mean value different from zero. We prove that this rate is sharp in any smooth domain, up to a logarithmic factor: in any smooth domain S2, among the first n Dirichlet eigenfunctions at least (log n)-1/2 n1/d have a nonzero mean.