Generic non-vanishing of Dirichlet eigenfunction averages and absence of symmetries

Abstract

We prove that, for a generic $C^m$-domain $\Omega \subset \mathbb{R}^d$ (with $m\geq3$, $d \geq 2$) in the sense of Baire category for the Micheletti topology, every Dirichlet-Laplacian eigenfunction has nonzero average. This gives a standalone spectral-geometric answer to a question raised by Steinerberger and Venkatraman. The proof is based on a direct shape-perturbation argument. A key ingredient, which appears to be of independent interest, is a Baire-category theorem for shapes: domains with real-analytic boundary form a meager subset of the space of $C^m$ domains. It is this input that allows us to bypass, rather than resolve pointwise, the overdetermined elliptic configurations in which the first shape derivative of the mean degenerates. We then derive two consequences. First, from a geometric viewpoint, a generic domain has trivial Euclidean isometry group. Second, in control theory, a generic domain makes the Dirichlet heat equation approximately controllable and rapidly stabilizable, that is, stabilizable at any prescribed exponential decay rate, by means of a single spatially homogeneous scalar control.