arXiv subjects

Alexis de Villeroché

Publications and source records attributed to Alexis de Villeroché.

4 recordsLinked to original sources

arXiv2026

Shape optimization with inradius constraint and emergence of honeycomb structures

We maximize the average torsional rigidity among open subsets of $\mathbb{R}^2$ with fixed inradius whose complements consist of pairwise disjoint balls of fixed radius $\varepsilon$ with mutual distances bounded below. We prove that, provided $\varepsilon$ is sufficiently small, the optimal value is asymptotically attained by sequences of domains whose complements exhibit a regular honeycomb structure. The proof is reduced, via Delaunay triangulations, to the analysis of a mixed Dirichlet--Neumann problem on triangular domains with circular cutouts at their vertices. Our approach is quite general and extends to other variational energies, such as the Cheeger constant, for which we also derive an explicit bound of the ratio $\varepsilon/$ inradius.

math.OC↗

arXiv2026

Neural networks for spectral optimization

Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2\%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1\% mean relative error on the first ten eigenvalues, compared with 4\% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.

cs.LG↗

arXiv2026

Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue

Let $Ω\subset \mathbb{R}^d$ be an open set of finite measure and let $Θ$ be a disjoint union of two balls of half measure. We study the stability of the full Dirichlet spectrum of $Ω$ when its second eigenvalue is close to the second eigenvalue of $Θ$. Precisely, for every integer $k \ge 1$, we provide a quantitative control of the difference $|λ_k(Ω)-λ_k(Θ)|$ by the variation of the second eigenvalue $C(d,k)(λ_2(Ω)-λ_2(Θ))^α$, for a suitable exponent $α$ and a positive constant $C(d,k)$ depending only on the dimension of the space and the index $k$. We are able to find such an estimate for general $k$ and arbitrary $Ω$ with $α=α_d/(d+1)^2$ where $α_2 = 1/2$ and $0<α_d<1$ in higher dimensions. In the particular case where $λ_k(Ω)\ge λ_k(Θ)$, we can improve the inequality and find an estimate with the sharp exponent $α= 1/2$.

math.AP↗

arXiv2025

Optimal domains for the Cheeger inequality

In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(Ω)=\frac{λ_p^{1/p}(Ω)}{λ_q^{1/q}(Ω)},$$ where $1\le q<p\le+\infty$ and $λ_p(Ω)$ (respectively $λ_q(Ω)$) is the first eigenvalue of the $p$-Laplacian $-Δ_p$ (respectively $-Δ_q$) with Dirichlet boundary condition on $\partialΩ$. We study both the maximization and minimization problems for ${\mathcal{F}}_{p,q}$, and show the existence of optimal domains in ${\mathbb{R}}^d$, along with some of their qualitative properties. Surprisingly, the case of a bounded box $D$ constraint $$\max\Big\{λ_q(Ω)\ :\ Ω\subset D,\ λ_p(Ω)=1\Big\},$$ leads to a problem of different nature, for which the existence of a solution is shown by analyzing optimal capacitary measures. In the last section we list some interesting questions that, in our opinion, deserve to be investigated.

math.OC↗