arXiv subjects

Mickaël Nahon

Publications and source records attributed to Mickaël Nahon.

14 recordsLinked to original sources

arXiv2026

Optimal regularity for vectorial, higher order and non-minimizing Bernoulli problems

We prove the Lipschitz regularity of solutions for a wide class of generalized Bernoulli free boundary problems, in vectorial, higher order, and non-minimizing settings. Our method does not rely on notions of viscosity solutions or comparison methods, which allows us to reach the optimal regularity for Bernoulli-type problems associated to elliptic operators which do not satisfy any maximum principle, namely the elasticity, biharmonic and Stokes equation. With a similar method we obtain the optimal Lipschitz regularity for stationary, non-minimizing solutions of the standard Bernoulli problem in two dimensions.

math.AP↗

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 $\approx 1.43π$) 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 Stability of the Dirichlet spectrum near the ball

Let $Ω\subset\mathbb{R}^n$ be an open set with the same volume as the unit ball $B$ and let $λ_k(Ω)$ be the $k$-th eigenvalue of the Laplace operator of $Ω$ with Dirichlet boundary conditions on $\partialΩ$. In this work, we answer the following question: if $λ_1(Ω)-λ_1(B)$ is small, how large can $|λ_k(Ω)-λ_k(B)|$ be ? We establish quantitative bounds of the form $|λ_k(Ω)-λ_k(B)|\le C (λ_1(Ω)-λ_1(B))^α$ with sharp exponents $α$ depending on the multiplicity of $λ_k(B)$. We first show that such an inequality is valid with $α=1/2$ 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 $α=1$ if $λ_{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when $λ_{k}(B)$ 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↗

arXiv2025

Computation of harmonic functions on higher genus surfaces

We extend a classical approximation result of harmonic functions in planar domains due to Bernstein and Walsch to the setting of harmonic functions in Riemann surfaces. This result gives an exact characterization of the rate at which a harmonic function in a subdomain of a compact Riemann surface may be approached by globally defined harmonic functions with prescribed poles. We illustrate the effectiveness and the impact of the method solving general boundary value Laplace problems in subdomains of the surface; we lay the groundwork for this numerical method in Riemann surfaces represented by a gluing of hyperbolic polygons. In particular, we give a general approximation procedure that computes this basis efficiently with arbitrary precision.

math.NA↗

arXiv2025

Spherical caps do not always maximize Neumann eigenvalues on the sphere

We prove the existence of an open set $Ω\subset\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.

math.AP↗

arXiv2023

A free discontinuity approach to optimal profiles in Stokes flows

In this paper we study obstacles immerged in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD). In dimension 2, we prove that the solution is classical.

math.AP↗

arXiv2022

Sharp inequalities for Neumann eigenvalues on the sphere

We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version, involving the harmonic mean of the eigenvalues of order $2$ to $n$, and extends to densities. A (surprising) consequence occurs on the maximality of a geodesic ball for the first nontrivial eigenvalue under the volume constraint: the hemisphere inclusion condition of the Ashbaugh-Benguria result can be relaxed to a weaker one, namely empty intersection with a geodesic ball of the prescribed volume. Although we do not prove that this last inclusion result is sharp, for a mass less than the half of the sphere, we numerically identify a density with higher first eigenvalue than the corresponding geodesic ball and with support equal to the full sphere $\mathbb{S}^2$.

math.AP↗

arXiv2022

Existence and regularity of optimal shapes for spectral functionals with Robin boundary conditions

We establish the existence and find some qualitative properties of open sets that minimize functionals of the form $ F(λ_1(Ω;β),\dots,λ_k(Ω;β))$ under measure constraint on $Ω$, where $λ_i(Ω;β)$ designates the $i$-th eigenvalue of the Laplace operator on $Ω$ with Robin boundary conditions of parameter $β>0$. Moreover, we show that minimizers of $λ_k(Ω;β)$ for $k\geq 2$ verify the conjecture $λ_k(Ω;β)=λ_{k-1}(Ω;β)$ in dimension three and more.

math.AP↗

arXiv2022

Shape optimization of a thermal insulation problem

We study a shape optimization problem involving a solid $K\subset\mathbb{R}^n$ that is maintained at constant temperature and is enveloped by a layer of insulating material $Ω$ which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $(K,Ω)$ with prescribed measure for $K$ and $Ω$, and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $\partialΩ$) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.

math.AP↗

arXiv2022

Boundary behavior of Robin problems in non-smooth domains

We analyze strict positivity at the boundary for nonnegative solutions of Robin problems in general (non-smooth) domains, e.g. open sets with rectifiable topological boundaries having finite Hausdorff measure. This question was raised by Bass, Burdzy and Chen in 2008 for harmonic functions, in a probabilistic context. We give geometric conditions such that the solutions of Robin problems associated to general elliptic operators of $p$-Laplacian type, with a positive right hand side, are globally or locally bounded away from zero at the boundary. Our method, of variational type, relies on the analysis of an isoperimetric profile of the set and provides quantitative estimates as well.

math.AP↗

arXiv2021

Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces

We prove stability estimates for the isoperimetric inequalities for the first and the second nonzero Laplace eigenvalues on surfaces, both globally and in a fixed conformal class. We employ the notion of eigenvalues of measures and show that if a normalized eigenvalue is close to its maximal value, the corresponding measure must be close in the Sobolev space $W^{-1,2}$ to the set of maximizing measures. In particular, this implies a qualitative stability result: metrics almost maximizing the normalized eigenvalue must be $W^{-1,2}$-close to a maximal metric. Following this approach, we prove sharp quantitative stability of the celebrated Hersch's inequality for the first eigenvalue on the sphere, as well as of its counterpart for the second eigenvalue. Similar results are also obtained for the precise isoperimetric eigenvalue inequalities on the projective plane, torus, and Klein bottle. The square of the $W^{-1,2}$ distance to a maximizing measure in these stability estimates is controlled by the difference between the normalized eigenvalue and its maximal value, indicating that the maxima are in a sense nondegenerate. We construct examples showing that the power of the distance can not be improved, and that the choice of the Sobolev space $W^{-1,2}$ is optimal.

math.DG↗

arXiv2020

Degenerate free discontinuity problems and spectral inequalities in quantitative form

We introduce a new geometric-analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with negative sign. This is motivated by searching quantitative inequalities for best constants of Sobolev-Poincaré inequalities with trace terms in $\mathbb{R}^n$ which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.

math.AP↗

arXiv2020

Stability and instability issues of the Weinstock inequality

Given two planar, conformal, smooth open sets $Ω$ and $ω$, we prove the existence of a sequence of smooth sets $Ω_n$ which geometrically converges to $Ω$ and such that the (perimeter normalized) Steklov eigenvalues of $Ω_n$ converge to the ones of $ω$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur.

math.AP↗

arXiv2017

A new continuum theory for incompressible swelling materials

Swelling media (e.g. gels, tumors) are usually described by mechanical constitutive laws (e.g. Hooke or Darcy laws). However, constitutive relations of real swelling media are not well known. Here, we take an opposite route and consider a simple packing heuristics, i.e. the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets while in the parallel direction, it is determined by a Laplace-Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.

math.AP↗