arXiv subjects

Antoine Henrot

Publications and source records attributed to Antoine Henrot.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2020

Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations

Minimizing the so-called "Dirichlet energy" with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.

math.OC↗

arXiv2020

Maximization of the Steklov eigenvalues with a diameter constraint

In this paper, we address the problem of maximizing the Steklov eigenvalues with a diameter constraint. We provide an estimate of the Steklov eigenvalues for a convex domain in terms of its diameter and volume and we show the existence of an optimal convex domain. We establish that balls are never maximizers, even for the first non-trivial eigenvalue that contrasts with the case of volume or perimeter constraints. Under an additional regularity assumption, we are able to prove that the Steklov eigenvalue is multiple for the optimal domain. We illustrate our theoretical results by giving some optimal domains in the plane thanks to a numerical algorithm.

math.SP↗

arXiv2020

From Steklov to Neumann via homogenisation

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.

math.AP↗

arXiv2020

The double queen Dido's problem

This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with three different weights, one for the hyperplane and one for each of the two open half-spaces in which $\mathbb{R}^N$ gets partitioned. We then consider the problem of characterizing the sets $Ω$ that minimize this weighted perimeter functional under the additional constraint that the volumes of the portions of $Ω$ in the two half-spaces are given. It is shown that the problem admits two kinds of minimizers, which will be called type I and type II, respectively. These minimizers are made of the union of two spherical domes whose angle of incidence satisfies some kind of \textquotedblleft Snell's law\textquotedblright. Finally, we provide a complete classification of the minimizers depending on the various parameters of the problem.

math.DG↗

arXiv2019

Hyperbolic solutions to Bernoulli's free boundary problem

Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and hyperbolic solutions. Elliptic solutions are ``stable'' solutions and tractable by variational methods and maximum principles, while hyperbolic solutions are ``unstable'' solutions of which the qualitative behavior is less known. We introduce a new implicit function theorem based on the parabolic maximal regularity, which is applicable to problems with loss of derivatives. Clarifying the spectral structure of the corresponding linearized operator by harmonic analysis, we prove the existence of foliated hyperbolic solutions as well as elliptic solutions in the same regularity class.

math.AP↗

arXiv2019

Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$

math.CA↗

arXiv2018

Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle

Inside a fixed bounded domain $Ω$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $λ_1(Ω\setminus K)$. We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and, in some cases non symmetry results, identifying the solution. We choose to work with the outer Minkowski content as the "good" notion of perimeter. Therefore, we are led to prove some new properties for it as its lower semicontinuity with respect to the Hausdorff convergence and the fact that the outer Minkowski content is equal to the Hausdorff lower semicontinuous envelope of the classical perimeter.

math.AP↗

arXiv2018

Maximization of the second non-trivial Neumann eigenvalue

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of R N with prescribed measure m attains its maximum on the union of two disjoint balls of measure m 2. As a consequence, the P{ó}lya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

math.AP↗

arXiv2017

On two functionals involving the maximum of the torsion function

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider $T(Ω)/(M(Ω)|Ω|)$ and $M(Ω)λ_1(Ω) $, where $Ω$ is a bounded open set of $\mathbb{R}^d$ with finite Lebesgue measure $|Ω|$, $M(Ω)$ denotes the maximum of the torsion function, $T(Ω)$ the torsion, and $λ_1(Ω)$ the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

math.AP↗

arXiv2016

Elasticae and inradius

The elastic energy of a planar convex body is defined by $E(\Om)=\frac 12\,\int\_{\partial\Om} k^2(s)\,ds$where $k(s)$ is the curvature of the boundary. In this paper we are interested in the minimization problemof $E(\Om)$ with a constraint on the inradius of $\Om$. By contrast with all the other minimization problemsinvolving this elastic energy (with a perimeter, area, diameter or circumradius constraints) for which thesolution is always the disk, we prove here that the solution of this minimization problem is not the disk and we completely characterizeit in terms of elementary functions.

math.AP↗

arXiv2015

On The Quantitative Isoperimetric Inequality In The Plane

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $Ω$, different from a ball, which minimizes the ratio $δ(Ω)/λ^2(Ω)$, where $δ$ is the isoperimetric deficit and $λ$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

math.MG↗

arXiv2014

A new isoperimetric inequality for the elasticae

For a smooth curve $γ$, we define its elastic energy as $E(γ)= \frac 12 \int_γ k^2 (s) ds$ where $k(s)$ is the curvature. The main purpose of the paper is to prove that among all smooth, simply connected, bounded open sets of prescribed area in $\mathbb{R}^2$, the disc has the boundary with the least elastic energy. In other words, for any bounded simply connected domain $Ω$, the following isoperimetric inequality holds: $E^2(\partial Ω)A(Ω)\geq π^3$. The analysis relies on the minimization of the elastic energy of drops enclosing a prescribed area, for which we give as well an analytic answer.

math.OC↗

arXiv2014

On the local minimizers of the Mahler volume

We focus on the analysis of local minimizers of the Mahler volume, that is to say the local solutions to the problem $$\min\{ M(K):=|K||K^\circ|\;/\;K\subset\R^d\textrm{ open and convex}, K=-K\}, $$ where $K^\circ:=\{ξ\in\R^d ; \forall x\in K, x\cdotξ<1\}$ is the polar body of $K$, and $|\cdot|$ denotes the volume in $\R^d$. According to a famous conjecture of Mahler the cube is expected to be a global minimizer for this problem. We express the Mahler volume in terms of the support functional of the convex body, which allows us to compute first and second derivatives, and leads to a concavity property of the functional. As a consequence, we prove first that any local minimizer has a Gauss curvature that vanishes at any point where it is defined. Going more deeply into the analysis in the two-dimensional case, we also prove that any local minimizer must be a parallelogram. We thereby retrieve and improve an original result of Mahler, who showed that parallelograms are global minimizers in dimension 2, and also the case of equality of Reisner, who proved that they are the only global minimizers.

math.OC↗

arXiv2014

Elastic energy of a convex body

In this paper a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set $$\mathcal{E}:=\left\{(x,y)\in \R^2, x=\frac{4πA(Ω)}{P(Ω)^2},y=\frac{E(Ω)P(Ω)}{2π^2},\,Ω\mbox{convex} \right\},$$ where $A$ is the area, $P$ is the perimeter and $E$ is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem: $$\min_{Ω\in\mathcal{C}}\{E(Ω)+μA(Ω)\},$$ where $\mathcal{C}$ is the class of convex bodies with fixed perimeter and $μ\ge 0$ is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

math.OC↗

arXiv2014

Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift

This paper deals with the eigenvalue problem for the operator $L=-Δ-x\cdot \nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $λ_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\in \mathbb{N} $ the following minimization problem $$ \min\left\{λ_k(Ω): \> Ω\>\mbox{quasi-open} \>\mbox{set}, \> \int_Ωe^{|x|^2/2}dx\le c\right\} $$ has a solution.

math.AP↗

arXiv2014

On the minimization of total mean curvature

In this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partialΩ} H\,dA \geq \sqrt{4πA(\partialΩ)}$ valid for any regular open set $Ω\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature and $A$ the area. We prove that this inequality holds true for axisymmetric domains which are convex in the direction orthogonal to the axis of symmetry. We also show that this inequality cannot be true in more general situations. However we prove that $\int_{\partialΩ} |H|\,dA \geq \sqrt{4πA(\partialΩ)}$ remains true for any axisymmetric domain.

math.DG↗

arXiv2013

An overdetermined problem with non constant boundary condition

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and regularity of a solution. Moreover several geometric properties of the solution are shown.

math.AP↗