Dirichlet eigenfunctions have non-zero mean for generic domains

Abstract

Let $D \subset \mathbb{R}^d$ ($d \geq 2$) be a bounded connected open set of class $C^m$ with $m \geq 3$. We prove that, for a generic domain $Ω\in \mathcal{O}^m(D)$ in the sense of Baire category for the Micheletti topology, every Dirichlet-Laplacian eigenfunction has nonzero mean, $\int_Ωφ_n \neq 0$ for all $n \geq 1$. This answers a question raised by Steinerberger and Venkatraman. The proof rests on a shape-derivative computation whose degeneracy is identified with an overdetermined elliptic problem of Schiffer type. Rather than resolving this problem, we bypass it through a Baire-category argument based on the meagerness of domains with analytic boundary. As a consequence, we obtain that for a generic domain the Dirichlet heat equation is approximately controllable, and rapidly stabilizable, i.e. stabilizable at any prescribed exponential decay rate, by a single spatially homogeneous scalar control, via the Fattorini-Hautus criterion.