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Berardo Ruffini

Publications and source records attributed to Berardo Ruffini.

At least 19 recordsLinked to original sources
arXiv2026

Nonexistence for screened Hartree energies

We develop an anti-concentration method for volume-constrained nonlocal shape optimization. Precisely, we apply it to prove the nonexistence of minimizers of an optimal design problem driven by a Hartree-type energy in the large mass regimes. The proof refines and extends the ideas from [Lu-Otto, CPAM 2014], where Lu and Otto dealt with nonexistence of minimizers for the Thomas-Fermi-Dirac-von Weizs\"acker energy. We also establish the asymptotic behavior of the associated isoperimetric profile of the energy.

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arXiv2026

Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem

We consider a shape optimization problem for a hybrid energy combining local confinement and nonlocal Coulomb repulsion. Specifically, for any open set $\Omega \subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $\Omega$, defined as a linear combination of its homogeneous $\dot{H}^1$ and $\dot{H}^{-1}$ seminorms. We show that in the small mass regime, volume-constrained minimizers of this geometric functional exist and are $C^{2,\alpha}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $\Gamma$-convergence, elliptic PDE theory, and one-phase free boundary regularity. A key novelty of this paper lies in the treatment of the Coulombic repulsive term: unlike standard competitive models, the lack of (a priori) sign constraints on the optimal functions forces the nonlocal term to exhibit two natures: it acts both as a scattering and an homogenizing force.

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arXiv2025

A strong quantitative form of the fractional isoperimetric inequality

We show a strong version of the fractional quantitative isoperimetric inequality, in which the isoperimetric deficit controls not only the Fraenkel asymmetry but also a sort of oscillation of the boundary. This generalizes the local result by Fusco and Julin in \cite{FJ}. The proof follows a regularization process as in \cite{FJ} but it is quite different in its spirit. Then, as a consequence of the quantitative inequality, we prove some stability estimates for a fractional Cheeger inequality.

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arXiv2025

Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

We establish the convergence of threshold dynamics-type approximation schemes to propagating fronts evolving according to an anisotropic mean curvature motion in the presence of a forcing term depending on both time and position, thus generalizing the consistency result obtained in [Ishii-Pires-Souganidis, 1999] by extending the results obtained in [Caffarelli-Souganidis, 2010] for $\alpha \in [1,2)$ to anisotropic kernels and in the presence of a driving force. The limit geometric evolution is of a variational type and can be approximated, at a large scale, by eikonal-type equations modeling dislocations dynamics. We prove that it preserves convexity under suitable convexity assumptions on the forcing term and that convex evolutions of compact sets are unique. If the initial set is bounded and sufficiently large, and the driving force is constant, then the corresponding generalized front propagation is asymptotically similar to the Wulff shape.

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arXiv2024

A charged liquid drop model with Willmore energy

We consider a variational model of electrified liquid drops, involving competition between surface tension and charge repulsion. Since the natural model happens to be ill-posed, we show that by adding to the perimeter a Willmore-type energy, the problem turns back to be well-posed. We also prove that for small charge the droplets is spherical.

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arXiv2024

An optimal design problem for a charge qubit

In this paper we introduce a simple variational model describing the ground state of a superconducting charge qubit. The model gives rise to a shape optimization problem that aims at maximizing the number of qubit states at a given gating voltage. We show that for small values of the charge optimal shapes exist and are $C^{2,\alpha}$-nearly spherical sets. In contrast, we prove that balls are not minimizers for large values of the charge and conjecture that optimal shapes do not exist, with the energy favoring disjoint collections of sets.

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arXiv2023

From energy bounds to dimensional estimates in a branched transport model for type-I superconductors

We consider a branched transport type problem which describes the magnetic flux through type-I superconductors in a regime of very weak applied fields. At the boundary of the sample, deviation of the magnetization from being uniform is penalized through a negative Sobolev norm. It was conjectured by S. Conti, F. Otto and S. Serfaty that as a result, the trace of the magnetization on the boundary should be a measure of Hausdorff dimension $8/5$. We prove that this conjecture is equivalent to the proof of local energy bounds with an optimal exponent. We then obtain local bounds which are however not optimal. These yield improved lower bounds on the dimension of the irrigated measure but unfortunately does not improve on the trivial upper bound. In order to illustrate the dependence of this dimension on the choice of penalization, we consider in the last part of the paper a toy model where the boundary energy is given by a Wasserstein distance to Lebesgue. In this case minimizers are finite graphs and thus the trace is atomic.

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arXiv2023

On a Cheeger--Kohler-Jobin inequality

We discuss the minimization of a Kohler-Jobin type scale-invariant functional among open, convex, bounded sets, namely $\min T_2(Ω) ^{\frac{1}{N+2}}h_1(Ω)$ among open convex bounded sets $Ω\subset \mathbb R^N$, where $T_2(Ω)$ denotes the torsional rigidity of a set $Ω$ and $h_1(Ω)$ its Cheeger constant. We prove the existence of an optimal set and we conjecture that the ball is the unique minimizer. We provide a sufficient condition for the validity of the conjecture, and an application of the conjecture to prove a quantitative inequality for the Cheeger constant. We also show lack of existence for the problem above among several other classes of sets. As a side result we discuss the equivalence of the several definitions of Cheeger constants present in the literature and show a quite general class of sets for which those are equivalent.

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arXiv2022

Rigidity of the ball for an isoperimetric problem with strong capacitary repulsion

We consider a variational problem involving competition between surface tension and charge repulsion. We show that, as opposed to the case of weak (short-range) interactions where we proved ill-posedness of the problem in a previous paper, when the repulsion is stronger the perimeter dominates the capacitary term at small scales. In particular we prove existence of minimizers for small charges as well as their regularity. Combining this with the stability of the ball under small $C^{1,γ}$ perturbations, this ultimately leads to the minimality of the ball for small charges. We cover in particular the borderline case of the $1-$capacity where both terms in the energy are of the same order.

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arXiv2021

A quantitative stability inequality for fractional capacities

The aim of this work is to show a non-sharp quantitative stability version of the fractional isocapacitary inequality. In particular, we provide a lower bound for the isocapacitary deficit in terms of the Fraenkel asymmetry. In addition, we provide the asymptotic behaviour of the $s$-fractional capacity when $s$ goes to $1$ and the stability of our estimate with respect to the parameter $s$.

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arXiv2021

A spectral shape optimization problem with a nonlocal competing term

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.

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arXiv2020

Conducting flat drops in a confining potential

We study a geometric variational problem arising from modeling two-dimensional charged drops of a perfectly conducting liquid in the presence of an external potential. We characterize the semicontinuous envelope of the energy in terms of a parameter measuring the relative strength of the Coulomb interaction. As a consequence, when the potential is confining and the Coulomb repulsion strength is below a critical value, we show existence and partial regularity of volume-constrained minimizers. We also derive the Euler--Lagrange equation satisfied by regular critical points, expressing the first variation of the Coulombic energy in terms of the normal $\frac12$-derivative of the capacitary potential.

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arXiv2018

A Spectral Characterization Of Geodesic Balls In Non-Compact Rank One Symmetric Spaces

In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is known in this setting. In this paper we prove that, in a non-compact rank one symmetric space, the geodesic balls uniquely maximize the first Steklov eigenvalue among the domains of fixed volume, extending to this context a result of Brock in the Euclidean space. Then we show that a stability version of the ensuing Brock-Weinstock inequality holds. The idea behind the proof is to exploit a suitable weighted isoperimetric inequality which we prove to hold true, as well as in a stability form, on harmonic manifolds. Eventually we show that, in general, the geodesic balls are not global maximizers on the standard sphere.

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arXiv2018

On equilibrium shapes of charged flat drops

Equilibrium shapes of two-dimensional charged, perfectly conducting liquid drops are governed by a geometric variational problem that involves a perimeter term modeling line tension and a capacitary term modeling Coulombic repulsion. Here we give a complete explicit solution to this variational problem. Namely, we show that at fixed total charge a ball of a particular radius is the unique global minimizer among all sufficiently regular sets in the plane. For sets whose area is also fixed, we show that balls are the only minimizers if the charge is less than or equal to a critical charge, while for larger charge minimizers do not exist. Analogous results hold for drops whose potential, rather than charge, is fixed.

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arXiv2017

Equilibrium shapes of charged droplets and related problems: (mostly) a review

We review some recent results on the equilibrium shapes of charged liquid drops. We show that the natural variational model is ill-posed and how this can be overcome by either restricting the class of competitors or by adding penalizations in the functional. The original contribution of this note is twofold. First, we prove existence of an optimal distribution of charge for a conducting drop subject to an external electric field. Second, we prove that there exists no optimal conducting drop in this setting.

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arXiv2017

On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

We study a variational problem modeling the behavior at equilibrium of charged liquid drops under convexity constraint. After proving well-posedness of the model, we show C 1,1-regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.

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