arXiv subjects

Antoine Henrot

Publications and source records attributed to Antoine Henrot.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative Kr\"{o}ger inequalities for Neumann eigenvalues of convex domains

Refining the sharp upper bounds $\mu_{k,d}^* $ obtained by Kr\"oger (1999) for the $k$-th Neumann eigenvalue of a convex domain $\Omega \subset \mathbb{R}^d$, we prove the following inequalities: for any $k\in \mathbb{N}$ there exists a constant $C(k,d) >0$ such that $$D_{\Omega}^2 \mu_k(\Omega) \leq \mu_{k,d}^* - C(k,d) a_2(\Omega)^2/D_{\Omega}^2$$ where $D_{\Omega}$ is the diameter of $\Omega$ and $a_2(\Omega)$ is the second largest semiaxis of the John ellipsoid of $\Omega$. In the planar case, for $k=1$ we also give an explicit value of the constant $C(1,2)$.

math.AP↗

arXiv2025

Optimal sets for the quantitative isoperimetric inequality in the plane with the barycentric distance

In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $δ(K)$ and the barycentric distance $λ_0(K)$ for sets $K\subset \mathbb{R}^N$ with given diameter $D$ and measure. In this work we are interested in the optimal sets for this inequality in the plane, i.e. sets that minimize the ratio $δ(K)/λ_0(K)^2$. We prove existence of optimal sets (at least when $D$ is large enough), regularity and express the optimality conditions. Moreover, we prove that the optimal sets have exactly two connected components and their boundary does not contain any arc of circle.

math.OC↗

arXiv2025

The diagram $(λ_1,μ_1)$

In this paper, we are interested in the possible values taken by the pair $(λ_1(Ω), μ_1(Ω))$ the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain $Ω$. We prove that, without any particular assumption on the class of open sets $Ω$, the two classical inequalities (the Faber-Krahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product $λ_1 μ_1$. We plot the so-called Blaschke--Santaló diagram and give some conjectures.

math.OC↗

arXiv2024

Minimization of the first eigenvalue for the Lamé system

In this article, we address the problem of determining a domain in $\mathbb{R}^N$ that minimizes the first eigenvalue of the Lamé system under a volume constraint. We begin by establishing the existence of such an optimal domain within the class of quasi-open sets, showing that in the physically relevant dimensions $N = 2$ and $3$, the optimal domain is indeed an open set. Additionally, we derive both first and second-order optimality conditions. Leveraging these conditions, we demonstrate that in two dimensions, the disk cannot be the optimal shape when the Poisson ratio is below a specific threshold, whereas above this value, it serves as a local minimizer. We also extend our analysis to show that the disk is nonoptimal for Poisson ratios $ν$ satisfying $ν\leq 0.4$.

math.AP↗

arXiv2024

Is the Faber-Krahn inequality true for the Stokes operator?

The goal of this paper is to investigate the minimisation of the first eigenvalue of the (vectorial) incompressible Dirichlet-Stokes operator. After providing an existence result, we investigate optimality conditions and we prove the following surprising result: while the ball satisfies first and second-order optimality conditions in dimension 2, it does not in dimension 3, so that the Faber-Krahn inequality for the Stokes operator is probably true in $\mathbb{R}^2$, but does not hold in $\mathbb{R}^3$. The multiplicity of the first eigenvalue of the Dirichlet-Stokes operator in the ball in $\mathbb{R}^3$ plays a crucial role in the proof of that claim.

math.AP↗

arXiv2024

Optimization of Neumann Eigenvalues under convexity and geometric constraints

In this paper we study optimization problems for Neumann eigenvalues $μ_k$ among convex domains with a constraint on the diameter or the perimeter. We work mainly in the plane, though some results are stated in higher dimension. We study the existence of an optimal domain in all considered cases. We also consider the case of the unit disk, giving values of the index $k$ for which it can be or cannot be extremal. We give some numerical examples for small values of $k$ that lead us to state some conjectures.

math.AP↗

arXiv2023

Two extremum problems for Neumann eigenvalues

Neumann eigenvalues being non-decreasing with respect to domain inclusion, it makes sense to study the two shape optimization problems $\min\{μ_k(Ω):Ω\mbox{ convex},Ω\subset D, \}$ (for a given box $D$) and $\max\{μ_k(Ω):Ω\mbox{ convex},ω\subset Ω, \}$ (for a given obstacle $ω$). In this paper, we study existence of a solution for these two problems in two dimensions and we give some qualitative properties. We also introduce the notion of {\it self-domains} that are domains solutions of these extremal problems for themselves and give examples of the disk and the square. A few numerical simulations are also presented.

math.SP↗

arXiv2023

About the Blaschke-Santalo diagram of area, perimeter and moment of inertia

We study the Blaschke-Santaló diagram associated to the area, the perimeter, and the moment of inertia. We work in dimension 2, under two assumptions on the shapes: convexity and the presence of two orthogonal axis of symmetry. We discuss topological and geometrical properties of the diagram. As a by-product we address a conjecture by Pólya, in the simplified setting of double symmetry.

math.OC↗

arXiv2023

Isoperimetric sets for weighted twisted eigenvalues

In tis paper we prove an isoperimetric inequality for the first twisted eigenvalue $λ_{1,γ}^T(Ω)$ of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space $H_0^1(Ω,dγ)$ and have weighted mean value equal to zero in $Ω$. We are interested in positive measures $dγ=γ(x) dx$ for which we are able to identify the isoperimetric sets, namely, the sets that minimize $λ_{1,γ}^T(Ω)$ among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of the isoperimetric sets (for the weighted perimeter with respect to the weighted measure).

math.AP↗

arXiv2022

Optimal bounds for Neumann eigenvalues in terms of the diameter

In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues in two situations (that are closely related). First we consider a one-dimensional Sturm-Liouville eigenvalue problem where the density is a function $h(x)$ whose some power is concave. We prove existence of a maximizer for $μ_k(h)$ and we completely characterize it. Then we consider the Neumann eigenvalues (for the Laplacian) of a domain $Ω\subset \mathbb{R}^d$ of given diameter and we assume that its profile function (defined as the $d-1$ dimensional measure of the slices orthogonal to a diameter) has also some power that is concave. This includes the case of convex domains in $\mathbb{R}^d$, containing and generalizing previous results by P. Kröger. On the other hand, in the last section, we give examples of domains for which the upper bound fails to be true, showing that, in general, $\sup D^2(Ω)μ_k(Ω)= +\infty$.

math.AP↗

arXiv2022

An isoperimetric problem with two distinct solutions

In this paper we prove that among all convex domains of the plane with two axis of symmetry, the maximizer of the first non trivial Neumann eigenvalue $μ_1$ with perimeter constraint is achieved by the square and the equilateral triangle. Part of the result follows from a new general bound on $μ_1$ involving the minimal width over the area. Our main result partially answers to a question addressed in 2009 by R. S. Laugesen, I. Polterovich, and B. A. Siudeja.

math.AP↗

arXiv2022

A comparison between Neumann and Steklov eigenvalues

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue $|Ω| μ_1(Ω)$ for a Lipschitz open set $Ω$ in the plane, and the normalized first (non-trivial) Steklov eigenvalue $P(Ω) σ_1(Ω)$. More precisely, we study the ratio $F(Ω):=|Ω| μ_1(Ω)/P(Ω) σ_1(Ω)$. We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets $Ω$. Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke-Santaló diagrams $(x,y)=\left(|Ω| μ_1(Ω), P(Ω) σ_1(Ω) \right)$.

math.AP↗

arXiv2022

A Poincaré type inequality with three constraints

In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$\inf_{u\in\mathcal{W}}\frac{\displaystyle\int_{-π}^π[(u')^2-u^2]dθ}{\displaystyle\left[\int_{-π}^π|u| dθ\right]^2}$$where $u\in \mathcal{W}$ is a $H^1(-π,π)$ periodic function, with zero average on $(-π,π)$ and orthogonal to sine and cosine.

math.OC↗

arXiv2021

On the quantitative isoperimetric inequality in the plane with the barycentric distance

In this paper we study the following quantitative isoperimetric inequality in the plane: $λ_0^2(Ω) \leq C δ(Ω)$ where $δ$ is the isoperimetric deficit and $λ_0$ is the barycentric asymmetry. Our aim is to generalize some results obtained by B. Fuglede in \cite{Fu93Geometriae}. For that purpose, we consider the shape optimization problem: minimize the ratio $δ(Ω)/λ_0^2(Ω)$ in the class of compact connected sets and in the class of convex sets.

math.OC↗

arXiv2021

The missing (A, D, r) diagram

In this paper we are interested in "optimal" universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term "optimal" is to be understood in the following sense: we tackle the issue of minimizing/maximizing the Lebesgue measure of a convex body among all convex sets of given diameter and inradius. The minimization problem in the two-dimensional case has been solved in a previous work, by M. Hernandez-Cifre and G. Salinas. In this article, we provide a generalization to the n-dimensional case based on a different approach, as well as the complete solving of the maximization problem in the two-dimensional case. This allows us to completely determine the so-called 2-dimensional Blaschke-Santal{ó} diagram for planar convex bodies with respect to the three magnitudes area, diameter and inradius in euclidean spaces, denoted (A, D, r). Such a diagram is used to determine the range of possible values of the area of convex sets depending on their diameter and inradius. Although this question of convex geometry appears quite elementary, it had not been answered until now. This is likely related to the fact that the diagram description uses unexpected particular convex sets, such as a kind of smoothed nonagon inscribed in an equilateral triangle.

math.MG↗

arXiv2021

Body of constant width with minimal area in a given annulus

In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in \cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons.

math.MG↗

arXiv2020

A Blaschke-Lebesgue Theorem for the Cheeger constant

In this paper we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the $p$-Laplacian for any $p\in (1,+\infty)$ (the current paper covers the case $p=1$ whereas the case $p=+\infty$ was already known).

math.AP↗

arXiv2020

Asymptotic behaviour of the Steklov problem on dumbbell domains

We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.

math.AP↗