arXiv subjects

João Miguel Machado

Publications and source records attributed to João Miguel Machado.

4 recordsLinked to original sources

arXiv2026

On the effective sharp stability of the Faber-Krahn inequality

We prove the sharp quantitative Faber--Krahn inequality with a computable dimensional constant, resolving a problem posed by Brasco and De Philippis~\cite[Open Problem~2]{brasco2017spectral}. In contrast with the known proof of the sharp estimate, our argument avoids indirect compactness arguments and selection principles. It runs through a new level-set proof of the Saint--Venant inequality, using that the torsional deficit controls both the average isoperimetric deficit and the oscillation of the gradient magnitude along the boundaries of level sets. An effective local strong isoperimetric estimate for confined sets then yields the result.

math.AP↗

arXiv2026

Quantitative stability for the Brascamp-Lieb inequality and moment measures

We develop a quantitative stability theory for moment measures based on a new sharp uniform stability principle for the Brascamp-Lieb variance inequality in terms of the $L^1$-distance. Our results yield structural stability estimates for solutions of the moment-measure problem that are uniform over a natural class of convex functions, thereby addressing several questions that have been open in this direction. A key novelty of our approach is that the Brascamp-Lieb stability bound is not only sharp in its stability exponent, but also uniform across a broad class of convex potentials. This uniformity is absent from previous results in the literature and, beyond its intrinsic mathematical interest, it is the mechanism that allows stability of the Brascamp-Lieb inequality to transfer to nonlinear variational problems such as the moment-measure problem. We moreover show that the $L^1-$nature of this stability estimate is sharp, in the sense that such a uniform estimate cannot hold in any $L^p$-metric, with $p>1$.

math.FA↗

arXiv2026

Quantitative Stability for Minkowski's problem

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $μ,ν$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form \[ \inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_μ, x + E_ν) \le C \mathrm{d_C}(μ,ν)^{\frac{1}{n-1}}, \quad α(E_μ, E_ν)^2 \le C \mathrm{d_C}(μ,ν)^{1 + \frac{1}{n-1}}, \] where $\mathrm{d_H}$ denotes the Hausdorff distance, $α$ denotes the Fraenkel asymmetry and $\mathrm{d_C}$ is the dual-convex distance of probability measures on the sphere. Our arguments are based on a variational problem whose optimizers are Minkowski bodies, for which we can obtain strong-concavity properties with the quantitative Brunn-Minkowski and isoperimetric inequalities. While the exponent in the Hausdorff distance is sharp, the exponent in the Fraenkel asymmetry is optimal in dimension $2$.

math.AP↗

arXiv2026

From Nash to Cournot--Nash equilibria via $Γ$-convergence

This work addresses the issue of the convergence of an $N$-player game towards a limit model involving a continuum of players, as the number of agents $N$ goes to infinity. More precisely, we investigate the convergence of Nash equilibria to a Cournot--Nash equilibrium of the limit model. When the cost function of the players is the first variation of some potential function, equilibria can be characterized by a stationarity condition, satisfied in particular by the minimizers of the potential. We demonstrate such a characterization under low regularity assumptions. Then we focus on the case where the players interact in a pairwise fashion; in this case we show that the original sequence of $N$-player games also admit a potential structure and prove that their corresponding potential functions converge in the sense of $Γ$-convergence to the potential function of the limit game.

math.AP↗