arXiv subjects

Marc Pegon

Publications and source records attributed to Marc Pegon.

8 recordsLinked to original sources

arXiv2025

Fractional hypocoercivity in bounded domains in the anomalous diffusion limit

In this paper, we provide a result of exponential stability for several dissipative linear kinetic equations with heavy-tailed equilibria. The approach, inspired by the so-called $L^2$-hypocoercivity method, is robust enough to provide estimates that are uniform in the anomalous diffusion limit. Moreover, it is able to deal with bounded domains with periodic boundary condition or general Maxwell boundary condition (from the pure specular to the pure diffusive case). In addition, our framework accommodates linear collisional operators that act simultaneously on the velocity and spatial variables.

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arXiv2025

Symmetry breaking in the liquid drop model with screened interactions

We investigate liquid drop models with screened Riesz-type interactions, focusing in particular on truncated Coulomb and Yukawa potentials in three dimensions. While in the classical Gamow model with Coulomb interaction the only minimizers are balls, as recently proved by Chodosh and Gianocca (2026), we show that this is no longer true if the interaction is screened. In the case of truncated Coulomb and Yukawa potentials, we prove that for some values of the parameter, there exist connected minimizers which are not balls. For truncated Coulomb potentials, we further obtain genuine symmetry breaking by showing existence of non-radial connected minimizers. This gives the first evidence of minimizers which are not balls in three dimensions, and the first evidence of non-radial minimizers altogether, in the class of radial kernels. Our approach relies on a comparison of the energy-per-mass ratios of balls, core-shells and cylinders.

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arXiv2022

Uniform $C^{1,\alpha}$-regularity for almost-minimizers of some nonlocal perturbations of the perimeter

In this paper, we establish a $C^{1,\alpha}$-regularity theorem for almost-minimizers of the functional $\mathcal{F}_{\varepsilon,\gamma}=P-\gamma P_{\varepsilon}$, where $\gamma\in(0,1)$ and $P_{\varepsilon}$ is a nonlocal energy converging to the perimeter as $\varepsilon$ vanishes. Our theorem provides a criterion for $C^{1,\alpha}$-regularity at a point of the boundary which is uniform as the parameter $\varepsilon$ goes to $0$. Since the two terms in the energy are of the same order when $\varepsilon$ is small, we are considering here much stronger nonlocal interactions than those considered in most related works. As a consequence of our regularity result, we obtain that, for $\varepsilon$ small enough, volume-constrained minimizers of $\mathcal{F}_{\varepsilon,\gamma}$ are balls. For small $\varepsilon$, this minimization problem corresponds to the large mass regime for a Gamow-type problem where the nonlocal repulsive term is given by an integrable $G$ with sufficiently fast decay at infinity.

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arXiv2021

Compactness and structure of zero-states for unoriented Aviles-Giga functionals

Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional. We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.

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arXiv2021

Large mass rigidity for a liquid drop model in 2D with kernels of finite moments

Motivated by Gamow's liquid drop model in the large mass regime, we consider an isoperimetric problem in which the standard perimeter $P(E)$ is replaced by $P(E)-\gamma P_\varepsilon(E)$, with $0<\gamma<1$ and $P_\varepsilon$ a nonlocal energy such that $P_\varepsilon(E)\to P(E)$ as $\varepsilon$ vanishes. We prove that unit area minimizers are disks for $\varepsilon>0$ small enough. More precisely, we first show that in dimension $2$, minimizers are necessarily convex, provided that $\varepsilon$ is small enough. In turn, this implies that minimizers have nearly circular boundaries, that is, their boundary is a small Lipschitz perturbation of the circle. Then, using a Fuglede-type argument, we prove that (in arbitrary dimension $n\geq 2$) the unit ball in $\mathbb{R}^n$ is the unique unit-volume minimizer of the problem among centered nearly spherical sets. As a consequence, up to translations, the unit disk is the unique minimizer. This isoperimetric problem is equivalent to a generalization of the liquid drop model for the atomic nucleus introduced by Gamow, where the nonlocal repulsive potential is given by a radial, sufficiently integrable kernel. In that formulation, our main result states that if the first moment of the kernel is smaller than an explicit threshold, there exists a critical mass $m_0$ such that for any $m>m_0$, the disk is the unique minimizer of area $m$ up to translations. This is in sharp contrast with the usual case of Riesz kernels, where the problem does not admit minimizers above a critical mass.

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arXiv2020

Large mass minimizers for isoperimetric problems with integrable nonlocal potentials

This paper is concerned with volume-constrained minimization problems derived from Gamow's liquid drop model for the atomic nucleus, involving the competition of a perimeter term and repulsive nonlocal potentials. We consider a large class of potentials, given by general radial nonnegative kernels which are integrable on $\mathbb{R}^n$, such as Bessel potentials, and study the behavior of the problem for large masses (i.e., volumes). Contrarily to the small mass case, where the nonlocal term becomes negligible compared to the perimeter, here the nonlocal term explodes compared to it. However, using the integrability of those kernels, we rewrite the problem as the minimization of the difference between the classical perimeter and a nonlocal perimeter, which converges to a multiple of the classical perimeter as the mass goes to infinity. Renormalizing to a fixed volume, we show that, if the first moment of the kernels is smaller than an explicit threshold, the problem admits minimizers of arbitrarily large mass, which contrasts with the usual case of Riesz potentials. In addition, we prove that, any sequence of minimizers converges to the ball as the mass goes to infinity. Finally, we study the stability of the ball, and show that our threshold on the first moment of the kernels is sharp in the sense that large balls go from stable to unstable. A direct consequence of the instability of large balls above this threshold is that there exist nontrivial compactly supported kernels for which the problems admit minimizers which are not balls, that is, symmetry breaking occurs.

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arXiv2019

Partial regularity for fractional harmonic maps into spheres

This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.

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arXiv2019

Minimizing 1/2-harmonic maps into spheres

In this article, we improve the partial regularity theory for minimizing $1/2$-harmonic maps in the case where the target manifold is the $(m-1)$-dimensional sphere. For $m\geq 3$, we show that minimizing $1/2$-harmonic maps are smooth in dimension 2, and have a singular set of codimension at least 3 in higher dimensions. For $m=2$, we prove that, up to an orthogonal transformation, $x/|x|$ is the unique non trivial $0$-homogeneous minimizing $1/2$-harmonic map from the plane into the circle $\mathbb{S}^1$. As a corollary, each point singularity of a minimizing $1/2$-harmonic maps from a 2d domain into $\mathbb{S}^1$ has a topological charge equal to $\pm1$.

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