arXiv subjects

Vincent Boulard

Publications and source records attributed to Vincent Boulard.

4 recordsLinked to original sources

arXiv2026

Generic failure of uniform separation in planar Dirichlet spectra

Can a bounded planar domain have a simple Dirichlet spectrum with uniformly separated consecutive eigenvalues? Dimension two is critical: Weyl's law permits both uniform separation and arbitrarily small gaps. We prove that uniform separation is nevertheless exceptional in a natural rough-domain setting, even after multiplicities are removed. Let $D\subset\mathbb{R}^2$ be a bounded domain and, for $\ell\geq 1$, let $\mathcal{C}_\ell(D)$ be the space of nonempty connected open sets $\Omega\subset D$ such that $\overline{D}\setminus\Omega$ has at most $\ell$ connected components, endowed with the complementary-Hausdorff topology. We prove that $\mathcal{C}_\ell(D)$ is completely metrizable and Baire, and that smooth domains are dense in it. If $0<\nu_1(\Omega)<\nu_2(\Omega)<\cdots$ are the distinct Dirichlet eigenvalues, our main result states that \[ \left\{\Omega\in\mathcal{C}_\ell(D):\inf_{m\geq 1}\bigl(\nu_{m+1}(\Omega)-\nu_m(\Omega)\bigr)=0\right\} \] is residual. This statement requires no simplicity assumption. Combining it with our transfer of Micheletti's classical generic-simplicity theorem to $\mathcal{C}_\ell(D)$ shows that a generic domain has simple spectrum and consecutive gaps with zero lower limit. The proof uses \v{S}ver\'ak's planar spectral continuity theorem and a local surgery that implants an arbitrarily high pair of close consecutive distinct eigenvalues.

math.AP↗

arXiv2026

Optimal geometric barriers for weighted observability of heat semigroups on metric measure spaces

Weighted integrated observability inequalities for heat equations usually involve a small-time factor of the form $e^{-\gamma/t}$. We prove that this scale is not an artefact of Carleman or spectral methods: it is forced by the geometry of the observation set. Let $A$ be a nonnegative self-adjoint operator on sections of a finite-rank Euclidean vector bundle over a doubling metric measure space, satisfying ultracontractivity, Davies-Gaffney estimates (equivalently, finite speed of propagation for the wave equation) and a pointwise local Weyl law. If a weighted integrated observability inequality holds on a measurable set $\omega$, for a fixed horizon $T\in(0,+\infty]$ and an admissible weight $h$, then, for every $0<\kappa<\frac{1}{2}$, $$ h(t)\leq A_{T,\kappa}\exp\left(-\kappa\frac{\mathcal{L}(\omega)^2}{t}\right),\qquad 0<t<T, $$ where $\mathcal{L}(\omega)$ is the essential maximal distance to $\omega$, replaced by any finite radius when $\mathcal{L}(\omega)=+\infty$. Thus, for $h(t)=e^{-\gamma/t}$, necessarily $\gamma\geq\mathcal{L}(\omega)^2/2$. This settles, with the optimal threshold, the maximal-distance lower bound for the infinite-time constant left open in earlier work. In the control-norm convention, the fast-control rate is at least $\mathcal{L}(\omega)^2/4$, recovering Miller's bound. The proof rests on the spectral packet $(\cosh(r\sqrt A)-1)e^{-tA}$. A pointwise Weyl law gives its sharp lower growth, while finite propagation speed and a weak-kernel Kannai transmutation formula make it exponentially small on $\omega$. Without kernel continuity or compact resolvent, we develop pointwise spectral measures and weak wave kernels. The framework covers Laplace-type operators on compact Riemannian manifolds, coupled heat systems, Schr\"odinger operators on $\mathbb{R}^d$, equiregular sub-Laplacians and Grushin models, and $\delta'$-coupled Laplacians on metric graphs.

math.AP↗

arXiv2026

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

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.

math.AP↗

arXiv2026

F-equivalence for parabolic systems and applications to the stabilization of nonlinear PDE

We consider the $F$-equivalence problem for parabolic systems: under which conditions a control system, governed by a parabolic operator $A$ and a control operator $B$, can be made equivalent to an exponentially stable system with arbitrarily large decay rate through an appropriate control feedback law? While this problem has been resolved for finite-dimensional systems fifty years ago, good conditions for infinite-dimensional systems remain a challenge, especially for systems in spatial dimension larger than one. Our main result establishes optimal conditions for the existence of an $F$-equivalence pair $(T,K)$ for a given parabolic control system $(A,B)$. We introduce an extended framework for $F$-equivalence of parabolic operators, addressing key limitations of existing approaches, and we prove that the pair $(T,K)$ is unique if and only if $(A,B)$ is approximately controllable. As a consequence, this provides a method to construct feedback operators for the rapid stabilization of semilinear parabolic systems, possibly multi-dimensional in space. We provide several illustrative examples, including the rapid stabilization of the heat equation, the Kuramoto-Sivashinsky equation, the Navier-Stokes equations and the quasilinear heat equation.

math.AP↗