arXiv subjects

Antoine Lemenant

Publications and source records attributed to Antoine Lemenant.

35 records · Page 2Linked to original sources

arXiv2017

Stationarity of the crack-front for the Mumford-Shah problem in 3D

In this paper we exhibit a family of stationary solutions of the Mumford-Shah functional in $\mathbb{R}^3$, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet \cite{b}. We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the crack-tip function for the Mumford-Shah problem in 2D is presented.

math.AP↗

arXiv2016

On a phase field approximation of the planar Steiner problem: existence, regularity, and asymptotic of minimizers

In this article, we consider and analyse a small variant of a functional originally introduced in \cite{BLS,LS} to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter $\varepsilon>0$ and resembles the (scalar) Ginzburg-Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as $\varepsilon\to0$, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.

math.AP↗

arXiv2016

Phase field models for thin elastic structures with topological constraint

This article is concerned with the problem of minimising the Willmore energy in the class of \emph{connected} surfaces with prescribed area which are confined to a small container. We propose a phase field approximation based on De Giorgi's diffuse Willmore functional to this variational problem. Our main contribution is a penalisation term which ensures connectedness in the sharp interface limit. The penalisation of disconnectedness is based on a geodesic distance chosen to be small between two points that lie on the same connected component of the transition layer of the phase field. We prove that in two dimensions, sequences of phase fields with uniformly bounded diffuse Willmore energy and diffuse area converge uniformly to the zeros of a double-well potential away from the support of a limiting measure. In three dimensions, we show that they converge $\mathcal{H}^1$-almost everywhere on curves. This enables us to show $Γ$-convergence to a sharp interface problem that only allows for connected structures. The results also imply Hausdorff convergence of the level sets in two dimensions and a similar result in three dimensions. We furthermore present numerical evidence of the effectiveness of our model. The implementation relies on a coupling of Dijkstra's algorithm in order to compute the topological penalty to a finite element approach for the Willmore term.

math.AP↗

arXiv2016

Regularity for the optimal compliance problem with length penalization

We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O§, plus its length. This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching the boundary of the domain only tangentially. Several new technical tools together with the classical ones are developed for this purpose.

math.OC↗

arXiv2016

Rectifiability of non Euclidean planar self-contracted curves

We prove that any self-contracted curve in R 2 endowed with a C 2 and strictly convex norm, has finite length. The proof follows from the study of the curve bisector of two points in R 2 for a general norm together with an adaptation of the argument used in [2].

math.MG↗

arXiv2014

Energy release rate for non smooth cracks in planar elasticity

This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.

math.AP↗

arXiv2014

A rigidity result for global Mumford-Shah minimizers in dimension three

We study global Mumford-Shah minimizers in $\R^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer $(u,K)$ in $\R^3$, if $K$ is contained in a half-plane and touching its edge, then it is the half-plane itself. This partially answers to a question of Guy David.

math.AP↗

arXiv2014

Approximation of length minimization problems among compact connected sets

In this paper we provide an approximation à la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the so-called Steiner Problem, but also the average distance problem, and finally a problem relying on the p-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of Gamma-convergence relies on technical tools from geometric measure theory, as for instance a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets.

math.MG↗

arXiv2013

A Modica-Mortola approximation for the Steiner Problem

In this note we present a way to approximate the Steiner problem by a family of elliptic energies of Modica-Mortola type, with an additional term relying on the weighted geodesic distance which takes care of the connexity constraint.

math.AP↗

arXiv2012

Rectifiability of Self-contracted curves in the Euclidean space and applications

It is hereby established that, in Euclidean spaces of finite dimension, bounded self-contracted curves have finite length. This extends the main result of Daniilidis, Ley, and Sabourau (J. Math. Pures Appl. 2010) concerning continuous planar self-contracted curves to any dimension, and dispenses entirely with the continuity requirement. The proof borrows heavily from a geometric idea of Manselli and Pucci (Geom. Dedicata 1991) employed for the study of regular enough curves, and can be seen as a nonsmooth adaptation of the latter, albeit a nontrivial one. Applications to continuous and discrete dynamical systems are discussed: continuous self-contracted curves appear as generalized solutions of nonsmooth convex foliation systems, recovering a hidden regularity after reparameterization, as consequence of our main result. In the discrete case, proximal sequences (obtained through implicit discretization of a gradient system) give rise to polygonal self-contracted curves. This yields a straightforward proof for the convergence of the exact proximal algorithm, under any choice of parameters.

math.CA↗

arXiv2012

Boundary regularity for the Poisson equation in reifenberg-flat domains

This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -Δu = f & \text{in} Ω u= 0 & \text{on} \partial Ω$$ where $f$ belongs to some $L^p(Ω)$ and $Ω$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $α\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally Hölder continuous provided that $Ω$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.

math.AP↗

arXiv2012

Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains

In this paper we establish new quantitative stability estimates with respect to domain perturbations for all the eigenvalues of both the Neumann and the Dirichlet Laplacian. Our main results follow from an abstract lemma stating that it is actually sufficient to provide an estimate on suitable projection operators. Whereas this lemma could be applied under different regularity assumptions on the domain, here we use it to estimate the spectrum in Lipschitz and in so-called Reifenberg-flat domains. Our argument also relies on suitable extension techniques and on an estimate on the decay of the eigenfunctions at the boundary which could be interpreted as a boundary regularity result.

math.AP↗

arXiv2012

On the extension property of Reifenberg-flat domains

We provide a detailed proof of the fact that any domain which is sufficiently flat in the sense of Reifenberg is also Jones-flat, and hence it is an extension domain. We discuss various applications of this property, in particular we obtain L^\infty estimates for the eigenfunctions of the Laplace operator with Neumann boundary conditions. We also compare different ways of measuring the "distance" between two sufficiently close Reifenberg-flat domains. These results are pivotal to the quantitative stability analysis of the spectrum of the Neumann Laplacian performed in another paper by the same authors.

math.AP↗

arXiv2011

The Stress-Intensity Factor for nonsmooth fractures in antiplane elasticity

Motivated by some questions arising in the study of quasistatic growth in brittle fracture, we investigate the asymptotic behavior of the energy of the solution $u$ of a Neumann problem near a crack in dimension 2. We consider non smooth cracks $K$ that are merely closed and connected. At any point of density 1/2 in $K$, we show that the blow-up limit of $u$ is the usual "cracktip" function $\sqrt{r}\sin(θ/2)$, with a well-defined coefficient (the "stress intensity factor" or SIF). The method relies on Bonnet's monotonicity formula \cite{b} together with $Γ$-convergence techniques.

math.NA↗

arXiv2008

On the homogeneity of global minimizers for the Mumford-Shah functional when K is a smooth cone

We show that if $(u,K)$ is a global minimizer for the Mumford-Shah functional in $R^N$, and if K is a smooth enough cone, then (modulo constants) u is a homogenous function of degree 1/2. We deduce some applications in $R^3$ as for instance that an angular sector cannot be the singular set of a global minimizer, that if $K$ is a half-plane then $u$ is the corresponding cracktip function of two variables, or that if K is a cone that meets $S^2$ with an union of $C^1$ curvilinear convex polygones, then it is a $P$, $Y$ or $T$.

math.AP↗

arXiv2008

Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets

Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with d_H(K; P) <epsilon, and if u is an energy minimizing function in B(0; 1)\K with restricted values on @B(0; 1)\K, then the normalized energy of u in B(0; r)\K is bounded by the normalized energy of u in B(0; 1)\K. We also prove the same result in R^3 when K is a epsilon-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type Y and T. Moreover, the result is still true for a further generalization of sets called (eps; eps_0)-minimal. This article is a preliminary study for a forthcoming paper where a regularity result for the singular set of the Mumford-Shah functional close to minimal cones in R^3 is proved by the same author.

math.AP↗

arXiv2008

Regularity of the singular set for Mumford-Shah minimizers in R^3 near a minimal cone

We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R^3, and if x, K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 degrees angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdorff distance in B(x; r), then K is C^1,alpha equivalent to the minimal cone in B(x; cr) where c < 1 is an universal constant.

math.AP↗