arXiv subjects

Dario Mazzoleni

Publications and source records attributed to Dario Mazzoleni.

At least 19 recordsLinked to original sources

arXiv2026

Some remarks on nonexistence in the Gamow liquid drop model

In this paper we deal with the celebrated Gamow liquid drop model, namely, for $m>0$, $N\geq 2$ and $α\in (0,N)$, we study the problem \[ \min\Big\{P(Ω)+ V_α(Ω) : Ω\subset \mathbb{R}^N,\;|Ω|=m\Big\}\,, \] and focus mostly on the nonexistence of minimizers for large masses. First we show that any possible minimizer in the asymptotic regime must satisfy an optimal thickness property. Eventually, in the simplified setting of sets made up as a tiling of cubes, we show that nonexistence occurs for large masses for the whole range of parameters $N\geq 2$ and $α\in (0,N)$.

math.AP↗

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 Weizsäcker energy. We also establish the asymptotic behavior of the associated isoperimetric profile of the energy.

math.AP↗

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 $Ω\subseteq \mathbb{R}^3$ of prescribed volume, we consider the ground state energy of an $L^2$-normalized function supported in $Ω$, 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 $C^{2,α}$ perturbations of a ball. The proof relies on a combination of surgery techniques, $Γ$-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.

math.AP↗

arXiv2026

Free boundary regularity for a spectral optimal partition problem with volume and inclusion constraints

This paper is devoted to a complete characterization of the free boundary of all solutions to the following spectral $k$-partition problem with measure and inclusion constraints: \[ \inf \left\{\sum_{i=1}^k λ_1(ω_i)\; : \; ω_i \subset Ω\mbox{ are nonempty open sets for all } i=1,\ldots, k,\; ω_i \cap ω_j = \emptyset \: \text{for all}\: i \not=j \mbox{ and } \sum_{i=1}^{k}|ω_i| = a \right\}, \] where $Ω$ is a bounded domain of $\mathbb{R}^N$, $a\in (0,|Ω|)$. In particular, we prove free boundary conditions, classify contact points, characterize the regular and singular part of the free boundary (including branching points), and describe the interaction of the partition with the fixed boundary $\partial Ω$. The proof is based on a perturbed version of the problem, combined with monotonicity formulas, blowup analysis and classification of blowups, suitable deformations of optimal sets and eigenfunctions, as well as the improvement of flatness of [Russ-Trey-Velichkov, CVPDE 58, 2019] for the one-phase points, and of [De Philippis-Spolaor-Velichkov, Invent. Math. 225, 2021] at two-phase points.

math.AP↗

arXiv2025

On a degenerate second order traffic model: existence of discrete evolutions, deterministic many-particle limit and first order approximation

We propose and analyse a new microscopic second order Follow-the-Leader type scheme to describe traffic flows. The main novelty of this model consists in multiplying the second order term by a nonlinear function of the global density, with the intent of considering the attentiveness of the drivers in dependence on the amount of congestion. Such term makes the system highly degenerate; indeed, coherently with the modellistic viewpoint, we allow for the nonlinearity to vanish as soon as consecutive vehicles are very close to each other. We first show existence of solutions to the degenerate discrete system. We then perform a rigorous discrete-to-continuum limit, as the number of vehicles grows larger and larger, by making use of suitable piece-wise constant approximations of the relevant macroscopic variables. The resulting continuum system turns out to be described by a degenerate pressure-less Euler-type equation, and we discuss how this could be considered an alternative to the groundbreaking Aw-Rascle-Zhang traffic model. Finally, we study the singular limit to first order dynamics in the spirit of a vanishing-inertia argument. This eventually validates the use of first order macroscopic models with nonlinear mobility to describe a congested traffic stream.

math.AP↗

arXiv2025

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 $C^{2,α}$-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.

math.AP↗

arXiv2024

Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem

We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $Ω\subset \mathbb{R}^{N}$, $N\ge2$, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of $D$ tends to zero, the unique maximum point of $u$, $P\in \partial Ω$, tends to a point of maximal mean curvature of $\partial Ω$. Furthermore, we show that $D$ is the intersection with $Ω$ of a $C^{1,1}$ nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of $D$. These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework.

math.AP↗

arXiv2023

On the dimension of the singular set in optimization problems with measure constraint

In this paper, we prove estimates on the dimension of the singular part of the free boundary for solutions to shape optimization problems with measure constraints. The focus is on the heat conduction problem studied by Aguilera, Caffarelli, and Spruck and the one-phase Bernoulli problem with measure constraint introduced by Aguilera, Alt and Caffarelli. To estimate the Hausdorff dimension of the singular set, we introduce a new formulation of the notion of stability for the one-phase problem along volume-preserving variations, which is preserved under blow-up limits. Finally, the result follows by applying the program developed in [Buttazzo et al. 2022] to this class of domain variation.

math.AP↗

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.

math.AP↗

arXiv2022

Regularity of the optimal sets for a class of integral shape functionals

We prove {the first} regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain $Ω$ is obtained as the integral of a cost function $j(u,x)$ depending on the solution $u$ of a certain PDE problem on $Ω$. The main feature of these functionals is that the minimality of a domain $Ω$ cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions $j(u,x)=-g(x)u+Q(x)$, where $u$ is the solution of the PDE $-Δu=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal $u$ from the inwards/outwards optimality of $Ω$ and then we use the stability of $Ω$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function $u$. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose $\partialΩ$ into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partialΩ$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove $C^\infty$ regularity of the regular part of the free boundary when the data are smooth.

math.AP↗

arXiv2022

$L^2$-Gradient Flows of Spectral Functionals

We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the eigenvalues of Schrödinger potentials $V$ for a wide class of differential operators associated to closed, symmetric, and coercive bilinear forms, including the case of all the Dirichlet forms (as for second order elliptic operators in Euclidean domains or Riemannian manifolds). We suppose that $\mathcal F$ arises as the sum of a $-θ$-convex functional $\mathcal K$ with proper domain $\mathbb{K}\subset L^2$ forcing the admissible potentials to stay above a constant $V_{\rm min}$ and a term $\mathcal H(V)=φ(λ_1(V),\cdots,λ_J(V))$ which depends on the first $J$ eigenvalues associated to $V$ through a $C^1$ function $φ$. Even if $\mathcal H$ is not a smooth perturbation of a convex functional (and it is in fact concave in simple important cases as the sum of the first $J$ eigenvalues) and we do not assume any compactness of the sublevels of $\mathcal K$, we prove the convergence of the Minimizing Movement method to a solution $V\in H^1(0,T;L^2)$ of the differential inclusion $V'(t)\in -\partial_L^-\mathcal F(V(t))$, which under suitable compatibility conditions on $φ$ can be written as \[ V'(t)+\sum_{i=1}^J\partial_iφ(λ_1(V(t)),\dots, λ_J(V(t)))u_i^2(t)\in -\partial_F^-\mathcal K(V(t)) \] where $(u_1(t),\dots, u_J(t))$ is an orthonormal system of eigenfunctions associated to the eigenvalues $(λ_1(V(t)), ,\dots,λ_J(V(t)))$.

math.AP↗

arXiv2021

Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems

We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain $Ω\subset \mathbb{R}^{N}$, within a suitable class of sign-changing weights. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of $D $ tends to zero. We show that, when the measure of $D$ is sufficiently small, $u $ has a unique local maximum point lying on the boundary of $Ω$ and $D$ is connected. Furthermore, the boundary of $D$ intersects the boundary of the box $Ω$, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial Ω)\ge C|D|^{(N-1)/N} $ for some universal constant $C>0$. Though widely expected, these properties are still unknown if the measure of $D$ is arbitrary.

math.AP↗

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.

math.AP↗

arXiv2021

Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures

In this paper we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schrödinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $γ$-convergence.

math.AP↗

arXiv2020

Regularity of the optimal sets for the second Dirichlet eigenvalue

This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $Ω$ minimizes the functional \[ \mathcal F_Λ(Ω)=λ_2(Ω)+Λ|Ω|, \] among all subsets of a smooth bounded open set $D\subset \mathbb{R}^d$, where $λ_2(Ω)$ is the second eigenvalue of the Dirichlet Laplacian on $Ω$ and $Λ>0$ is a fixed constant, then $Ω$ is equivalent to the union of two disjoint open sets $Ω_+$ and $Ω_-$, which are $C^{1,α}$-regular up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in the one-phase free boundaries $D\cap \partialΩ_+\setminus\partialΩ_-$ and $D\cap\partialΩ_-\setminus\partialΩ_+$.

math.AP↗

arXiv2019

On principal frequencies, volume and inradius in convex sets

We provide a sharp double-sided estimate for Poincaré-Sobolev constants on a convex set, in terms of its inradius and $N-$dimensional measure. Our results extend and unify previous works by Hersch and Protter (for the first eigenvalue) and of Makai, Pólya and Szegő (for the torsional rigidity), by means of a single proof.

math.SP↗

arXiv2019

Asymptotic spherical shapes in some spectral optimization problems

We study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box $Ω\subset\mathbb{R}^N$, which arises in the investigation of the survival threshold in population dynamics. When trying to minimize such eigenvalue with respect to the weight, one is lead to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures). We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. We prove that, for suitable choices of the box $Ω$, the optimal shapes for this second problem are indeed spherical; moreover, for general $Ω$, we show that small volume spectral drops are asymptotically spherical, with center at points of $\partialΩ$ having large mean curvature.

math.AP↗

arXiv2019

Quantitative analysis of a singularly perturbed shape optimization problem in a polygon

We carry on our study of the connection between two shape optimization problems with spectral cost. On the one hand, we consider the optimal design problem for the survival threshold of a population living in a heterogenous habitat $Ω$; this problem arises when searching for the optimal shape and location of a shelter zone in order to prevent extinction of the species. On the other hand, we deal with the spectral drop problem, which consists in minimizing a mixed Dirichlet-Neumann eigenvalue in a box $Ω$. In a previous paper arXiv:1811.01623 we proved that the latter one can be obtained as a singular perturbation of the former, when the region outside the refuge is more and more hostile. In this paper we sharpen our analysis in case $Ω$ is a planar polygon, providing quantitative estimates of the optimal level convergence, as well as of the involved eigenvalues.

math.AP↗