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Antoine Lemenant

Publications and source records attributed to Antoine Lemenant.

At least 19 recordsLinked to original sources
arXiv2026

Good access to the crack and integrability of the full gradient for Griffith almost-minimizers in the plane

We show that, for almost-minimizers of the Griffith energy in the plane, the complement of the crack is locally covered by a finite number of John domains where the displacement satisfies Hölder-type estimates. As a consequence, we derive the integrability of the full gradient and the finiteness of the traces along the crack. In particular, we show that any Griffith almost-minimizer in dimension two locally belongs to the space SBV^2.

math.AP↗
arXiv2025

Finite number of traces for Mumford-Shah minimizers in dimension 2

In this short note, we answer a question raised by E. De Giorgi, showing that a Mumford-Shah minimizer in dimension 2 can only admit three maximum limit values as approaching the singular set. This result stems from tools developed in the early 2000's by G. David, A. Bonnet, and J.-C. Léger.

math.AP↗
arXiv2025

Reilly inequality for Varifolds

The famous Reilly inequality gives an upper bound for the first eigenvalue of the Laplacian defined on compact submanifolds of the Euclidean space in terms of the $L^2$-norm of the mean curvature vector. In this paper, we generalize this inequality in a Varifold context. In particular we generalize it for the class of $H(2)$ varifolds and for polygons and we analyse the equality case.

math.DG↗
arXiv2025

A field-road system with a rectifiable set

The aim of this paper is to define a field-road system in 2D where the road is a merely 1D-rectifiable set. For this purpose we introduce a general setting in order to define a parabolic problem onto a rectifiable set, which is coupled with another more classical parabolic problem outside this set, with transmission conditions.

math.AP↗
arXiv2025

Phase field approximation for Plateau's problem: a curve geodesic distance penalty approach

This work focuses on a phase field approximation of Plateau's problem. Inspired by Reifenberg's point of view, we introduce a model that combines the Ambrosio-Torterelli energy with a geodesic distance term, which can be considered as a generalization of the approach developed by Bonnivard, Lemenant and Santambrogio to approximate solutions to Steiner's problem. First, we present a Gamma-convergence analysis of this model in the simple case of a single curve located on the edge of a cylinder. In a numerical section, we detail the numerical optimisation schemes used to minimize this energy for numerous examples, for which good approximations of solutions to Plateau's problem are found.

math.OC↗
arXiv2025

Quantitative regularity properties for the optimal design problem

In this paper we slightly improve the regularity theory for the so called optimal design problem. We first establish the uniform rectifiability of the boundary of the optimal set, for a larger class of minimizers, in any dimension. As an application, we improve the bound obtained by Larsen in dimension~2 about the mutual distance between two connected components. Finally we also prove that the full regularity in dimension 2 holds true provided that the ratio between the two constants in front of the Dirichlet energy is not larger than 4, which partially answers to a question raised by Larsen.

math.OC↗
arXiv2025

Boundary regularity for the polyharmonic Dirichlet problem

In this paper we prove that any solution of the $m$-polyharmonic Poisson equation in a Reifenberg-flat domain with homogeneous Dirichlet boundary condition, is $\mathscr{C}^{m-1,α}$ regular up to the boundary. To achieve this result we extend the Nirenberg method of translations to operators of arbitrary order, and then use some Mosco-convergence tools developped in a previous paper.

math.AP↗
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

The biharmonic optimal support problem

We establish a $Γ$-convergence result for $h\to 0$ of a thin nonlinearly elastic 3D-plate of thickness $h>0$ which is assumed to be glued to a support region in the 2D-plane $x_3=0$ over the $h$-2D-neighborhood of a given closed set $K$. In the regime of very small vertical forces we identify the $Γ$-limit as being the bi-harmonic energy, with Dirichlet condition on the gluing region $K$, following a general strategy by Friesecke, James, and Müller that we have to adapt in presence of the glued region. Then we introduce a shape optimization problem that we call "optimal support problem" and which aims to find the best glued plate. In this problem the bi-harmonic energy is optimized among all possible glued regions $K$ that we assume to be connected and for which we penalize the length. By relating the dual problem with Griffith almost-minimizers, we are able to prove that any minimizer is $C^{1,α}$ regular outside a set of Hausdorff dimension strictly less then one.

math.AP↗
arXiv2024

Asymptotic limit of linear parabolic equations with spatio-temporal degenerated potentials

In this paper, we observe how the heat equation in a non-cylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as a parabolic version of a previous work by the first and last authors, concerning the stationary case. We provide a strong convergence result for the solution by use of energetic methods and $Γ$-convergence technics. Then, we establish an exponential decay estimate coming from an adaptation of an argument due to B. Simon.

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

Epsilon-regularity for Griffith almost-minimizers in any dimension under a separating condition

In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is $ε$-close to a plane in some ball B $\subset$ R N while separating the ball B in two big parts, then K is C 1,$α$ in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of [4], with a different and more sophisticate approach inspired by [23, 24], using also [20] in order to adapt a part of the argument to Griffith minimizers.

math.AP↗
arXiv2023

Stable domains for higher order elliptic operators

This paper is devoted to prove that any domain satisfying a $(δ_0,r_0)-$capacity condition of first order is automatically $(m,p)-$stable for all $m\geqslant 1$ and $p\geqslant 1$, and for any dimension $N\geqslant 1$. In particular, this includes regular enough domains such as $\mathscr{C}^1-$domains, Lipchitz domains, Reifenberg flat domains, but is weak enough to also includes cusp points. Our result extends some of the results of Hayouni and Pierre valid only for $N=2,3$, and extends also the results of Bucur and Zolesio for higher order operators, with a different and simpler proof.

math.OC↗
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↗
arXiv2021

Regularity for the planar optimal p-compliance problem

In this paper we prove a partial $C^{1,α}$ regularity result in dimension $N=2$ for the optimal $p$-compliance problem, extending for $p\not = 2$ some of the results obtained by A. Chambolle, J. Lamboley, A. Lemenant, E. Stepanov (2017). Because of the lack of good monotonicity estimates for the $p$-energy when $p\not = 2$, we employ an alternative technique based on a compactness argument leading to a $p$-energy decay at any flat point. We finally obtain that every optimal set has no loop, is Ahlfors regular, and $C^{1,α}$ at $\mathcal{H}^1$-a.e. point for every $p \in (1 ,+\infty)$.

math.OC↗
arXiv2019

Partial regularity for the crack set minimizing the two-dimensional Griffith energy

In this paper we prove a $\mathcal C^{1,α}$ regularity result for minimizers of the planar Griffith functional arising from a variational model of brittle fracture. We prove that any isolated connected component of the crack, the singular set of a minimizer, is locally a $\mathcal C^{1,α}$ curve outside a set of zero Hausdorff measure.

math.AP↗