  This paper is dedicated to the classical Hadamard formula for asymptotics of
eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We
prove that the Hadamard formula still holds for $C^1$-domains with
$C^1$-perturbations. We also derive an optimal estimate for the remainder term
in the $C^{1,\alpha}$-case. Furthermore, if the boundary is merely Lipschitz,
we show that the Hadamard formula is not valid.
