arXiv subjects

Jimmy Lamboley

Publications and source records attributed to Jimmy Lamboley.

At least 19 recordsLinked to original sources

arXiv2025

Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load

We study a higher order analogue to the Alt-Caffarelli functional that arises in several shape optimization problems, among which the minimization of the critical buckling load of a clamped plate of fixed area. We obtain several regularity results up to the boundary in two dimensions, in particular we prove the full regularity of the boundary (analytic outside angles of opening $\approx 1.43\pi$) near any point of density less than 1 of the optimal shape. These results are based on the monotonicity formula discovered by Dipierro, Karakhanyan, and Valdinoci, which we improve with a new epiperimetric inequality.

math.AP↗

arXiv2025

Sharp quantitative Talenti's inequality in particular cases

In this paper, we focus on the famous Talenti's symmetrization inequality, more precisely its $L^p$ corollary asserting that the $L^p$-norm of the solution to $-\Delta v=f^\sharp$ is higher than the $L^p$-norm of the solution to $-\Delta u=f$ (we are considering Dirichlet boundary conditions, and $f^\sharp$ denotes the Schwarz symmetrization of $f:\Omega\to\mathbb{R}_+$). We focus on the particular case where functions $f$ are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the $L^p$-Talenti inequality with the sharp exponent 2.

math.AP↗

arXiv2023

Sharp Quantitative Stability of the Dirichlet spectrum near the ball

Let $\Omega\subset\mathbb{R}^n$ be an open set with the same volume as the unit ball $B$ and let $\lambda_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator of $\Omega$ with Dirichlet boundary conditions on $\partial\Omega$. In this work, we answer the following question: if $\lambda_1(\Omega)-\lambda_1(B)$ is small, how large can $|\lambda_k(\Omega)-\lambda_k(B)|$ be ? We establish quantitative bounds of the form $|\lambda_k(\Omega)-\lambda_k(B)|\le C (\lambda_1(\Omega)-\lambda_1(B))^\alpha$ with sharp exponents $\alpha$ depending on the multiplicity of $\lambda_k(B)$. We first show that such an inequality is valid with $\alpha=1/2$ for any $k$, improving previous known results and providing the sharpest possible exponent. Then, through the study of a vectorial free boundary problem, we show that one can achieve the better exponent $\alpha=1$ if $\lambda_{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when $\lambda_{k}(B)$ is multiple, thus providing a complete answer to the question above. As a consequence of these results, we obtain the persistence of the ball as the minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue on the one hand, and a full reverse Kohler-Jobin inequality on the other hand, solving an open problem formulated by M. Van Den Berg, G. Buttazzo and A. Pratelli.

math.AP↗

arXiv2022

Regularity in shape optimization under convexity constraint

This paper is concerned with the regularity of shape optimizers of a class of isoperimetric problems under convexity constraint. We prove that minimizers of the sum of the perimeter and a perturbative term, among convex shapes, are C 1,1-regular. To that end, we define a notion of quasi-minimizer fitted to the convexity context and show that any such quasi-minimizer is C 1,1-regular. The proof relies on a cutting procedure which was introduced to prove similar regularity results in the calculus of variations context. Using a penalization method we are able to treat a volume constraint, showing the same regularity in this case. We go through some examples taken from PDE theory, that is when the perturbative term is of PDE type, and prove that a large class of such examples fit into our C 1,1-regularity result. Finally we provide a counterexample showing that we cannot expect higher regularity in general.

math.OC↗

arXiv2021

Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains

In this paper we prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either E f ($\Omega$), the Dirichlet energy of the Laplacian in the domain $\Omega$, or $\lambda$ 1 ($\Omega$), the first eigenvalue of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other constraints to ensure existence of an optimal shape) one allows to consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate the convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional usually was the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f or $\lambda$ 1 , for shapes that are smooth and convex. The main novelty in the present paper is indeed the proof of a weak convexity property of E f and $\lambda$ 1 when it is only known that the shape is planar and convex, which means we have to consider rather non-smooth shapes. This work involves new computations and estimates of the second order shape derivative of E f and $\lambda$ 1 that are interesting in themselves.

math.OC↗

arXiv2020

Free boundary problems involving singular weights

In this paper we initiate the investigation of free boundary minimization problems ruled by general singular operators with $A_2$ weights. We show existence and boundedness of minimizers. The key novelty is a sharp $C^{1+γ}$ regularity result for solutions at their singular free boundary points. We also show a corresponding non-degeneracy estimate.

math.AP↗

arXiv2019

Existence and regularity of Faber Krahn minimizers in a Riemannian manifold

In this paper, we study the minimization of $λ_{1}(Ω)$, the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets $Ω$ of fixed volume in a Riemmanian manifold $(M,g)$. In the Euclidian setting (when $(M,g)=(\mathbb{R}^n,e)$), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds $(M,g)$. In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

math.AP↗

arXiv2018

Stability in shape optimization with second variation

We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.

math.OC↗

arXiv2016

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalueproblem in a fixed box $Ω$ and we consider the general case of Robin boundary conditions on $\partialΩ$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be parametrized by the characteristic function of the subset $E$ of $Ω$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to $E$, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.

math.AP↗

arXiv2016

Regularity of Minimizers of Shape Optimization Problems involving Perimeter

We prove existence and regularity of optimal shapes for the problem$$\min\Big\{P(Ω)+\mathcal{G}(Ω):\ Ω\subset D,\ |Ω|=m\Big\},$$where $P$ denotes the perimeter, $|\cdot|$ is the volume, and the functional $\mathcal{G}$ is either one of the following:\textless{}ul\textgreater{}\textless{}li\textgreater{} the Dirichlet energy $E\_f$, with respect to a (possibly sign-changing) function $f\in L^p$;\textless{}/li\textgreater{}\textless{}li\textgreater{}a spectral functional of the form $F(λ\_{1},\dots,λ\_{k})$, where $λ\_k$ is the $k$th eigenvalue of the Dirichlet Laplacian and $F:\mathbb{R}^k\to\mathbb{R}$ is Lipschitz continuous and increasing in each variable.\textless{}/li\textgreater{}\textless{}/ul\textgreater{}The domain $D$ is the whole space $\mathbb{R}^d$ or a bounded domain. We also give general assumptions on the functional $\mathcal{G}$ so that the result remains valid.

math.OC↗

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↗

arXiv2015

Estimates of First and Second Order Shape Derivatives in Nonsmooth Multidimensional Domains and Applications

In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDE's around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform exterior ball condition. We prove rather sharp continuity results for these shape derivatives with respect to Sobolev norms of the boundary-traces of the displacements. With respect to previous results of this kind, the approach is quite different and is valid in any dimension $N\geq 2$. It is based on sharp regularity results for Poisson-type equations in such nonsmooth domains. We also enlarge the class of functionals and PDEs for which these estimates apply. Applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement.

math.OC↗

arXiv2014

An extremal eigenvalue problem for the Wentzell-Laplace operator

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $Ω$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

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

New examples of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in a Riemannian manifold with boundary

We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally.

math.DG↗

arXiv2011

Sharp bounds for the p-torsion of convex planar domains

We obtain some sharp estimates for the $p$-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on the behaviour of the inner parallel sets of convex polygons. As an application of our isoperimetric inequalities, we consider the shape optimization problem which consists in maximizing the $p$-torsion among polygons having a given number of vertices and a given area. A long-standing conjecture by Pólya-Szegö states that the solution is the regular polygon. We show that such conjecture is true within the subclass of polygons for which a suitable notion of "asymmetry measure" exceeds a critical threshold.

math.OC↗

arXiv2011

Regularity and singularities of Optimal Convex shapes in the plane

We focus here on the analysis of the regularity or singularity of solutions $\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$ J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ Ω\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\rightarrow\R$ is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results: i) under a suitable convexity property of the functional $J$, we prove that $Ω_0$ is a $W^{2,p}$-set, $p\in[1,\infty]$. This result applies, for instance, with $p=\infty$ when the shape functional can be written as $ J(Ω)=R(Ω)+P(Ω), $ where $R(\Om)=F(|\Om|,E_{f}(\Om),\la_{1}(\Om))$ involves the area $|Ω|$, the Dirichlet energy $E_f(Ω)$ or the first eigenvalue of the Laplace-Dirichlet operator $λ_1(Ω)$, and $P(Ω)$ is the perimeter of $\Om$, ii) under a suitable concavity assumption on the functional $J$, we prove that $Ω_{0}$ is a polygon. This result applies, for instance, when the functional is now written as $ J(\Om)=R(Ω)-P(Ω), $ with the same notations as above.

math.OC↗

arXiv2011

Optimal convex shapes for concave functionals

Motivated by a long-standing conjecture of Polya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetriclike inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Polya-Szegö problem.

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