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 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.

In this article, we improve the partial regularity theory for minimizing 1/2-harmonic maps of [30,33] in the case where the target manifold is the (m − 1)-dimensional sphere. For m 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 S 1 . As a corollary, each point singularity of a minimizing 1/2-harmonic maps from a 2d domain into S 1 has a topological charge equal to ±1.

Large mass rigidity for a liquid drop model in 2D with kernels of finite momentsTome 9 (2022), p. 63-100. © Les auteurs, 2022. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L'accès aux articles de la revue « Journal de l'École polytechnique -Mathématiques » (http://jep.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l'édition scientifique ouverte www.centre-mersenne.org

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