arXiv subjects

Raphaël Prunier

Publications and source records attributed to Raphaël Prunier.

5 recordsLinked to original sources

arXiv2026

Unstable free boundary problems in optimal control theory: existence and regularity

We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form ``maximise $\int ψ(Θ_m)-c\int m$ where $-ΔΘ_m=mΘ_m+B(x,Θ_m)$, under the constraint $0\leq m\leq 1$ a.e.", the solution $m^*$ is bang-bang, in the sense that $m^*=χ_{E^*}$, and that $\partial E^*$ is smooth up to a $(d-2)$-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem ``maximise $\int ψ(Θ_m)$ where $-ΔΘ_m=mΘ_m+B(x,Θ_m)$, under the constraint $0\leq m\leq 1$ a.e and $\int m=m_0$" are bang-bang in the sense that $m^*=χ_{E^*}$ and that, in the two-dimensional case, $\partial E^*$ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau \& Weiss and Chanillo, Kenig \& To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary.

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

Some optimal control and shape optimisation problems for bulk-surface cooperative systems

The goal of this paper is to address some optimal control and shape optimisation problems arising from bulk-surface cooperative systems. The basic model under consideration is the following: letting $Ω$ be a fixed domain, we assume that a population (with density $u$) lives inside $Ω$ and can access some resources $f$, while a second population (with density $v$) lives on the boundary $\partial Ω$ and can access other resources $g$. These two populations are coupled in a cooperative manner by a constant exchange rate at the boundary, leading to a non-standard PDE system that has already been studied in previous works by Bogosel, Giletti and Tellini, for its connection with road-field models. Building on the considerations of the aforementioned previous works, we have two main objectives here: first, investigate the question of optimal resources distribution inside the domain $Ω$ and on the surface $\partial Ω$, i.e. how to spread resources in order to guarantee an optimal survival of the two species. We establish rigid Talenti inequalities and comparison results when $Ω$ is a ball, extending in particular the results of J. J. Langford on symmetrisation for Neumann and Robin problems. Second, when the resources distribution $f$ and $g$ are constant, we provide a partial analysis of the natural shape optimisation problem: which shape $Ω$ maximises the survival rate of the two species? Namely, we show that in certain regimes there can be no optimal shape and, by computing second-order shape derivatives, we investigate the local optimality of the ball.

math.AP↗

arXiv2024

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↗

arXiv2023

Fuglede-type arguments for isoperimetric problems and applications to stability among convex shapes

This paper is concerned with stability of the ball for a class of isoperimetric problems under convexity constraint. Considering the problem of minimizing $P+\varepsilon R$ among convex subsets of $\mathbb{R}^N$ of fixed volume, where $P$ is the perimeter functional, $R$ is a perturbative term and $\varepsilon>0$ is a small parameter, stability of the ball for this perturbed isoperimetric problem means that the ball is the unique (local, up to translation) minimizer for any $\varepsilon$ sufficiently small. We investigate independently two specific cases where $Ω\mapsto R(Ω)$ is an energy arising from PDE theory, namely the capacity and the first Dirichlet eigenvalue of a domain $Ω\subset\mathbb{R}^N$. While in both cases stability fails among all shapes, in the first case we prove (non-sharp) stability of the ball among convex shapes, by building an appropriate competitor for the capacity of a perturbation of the ball. In the second case we prove sharp stability of the ball among convex shapes by providing the optimal range of $\varepsilon$ such that stability holds, relying on the \emph{selection principle} technique and a regularity theory under convexity constraint.

math.OC↗