Lincei Mat Research Articles

Minimal extension for the $\alpha$-Manhattan norm

Let \partial\mathcal{Q} be the boundary of a convex polygon in \mathbb{R}^2 , e_\alpha=(\cos\alpha,\sin\alpha) and e_{\alpha}^{\bot}=(-\sin\alpha,\cos\alpha) a basis of \mathbb{R}^2 for some \alpha\in[0,2\pi) and \varphi:\partial\mathcal{Q}\to\mathbb{R}^2 a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism v:\mathcal{Q}\to\mathbb{R}^2 coinciding with \varphi on \partial\mathcal{Q} such that the following property holds: |\langle Dv, e_{\alpha}\rangle|(\mathcal{Q}) (resp., \langle Dv, e_{\alpha}^{\bot}\rangle|(\mathcal{Q}) ) is as close as we want to \inf|\langle Du, e_{\alpha}\rangle|(\mathcal{Q}) (resp., \inf|\langle Du, e_{\alpha}^{\bot}\rangle|(\mathcal{Q}) ) where the infimum is meant over the class of all BV homeomorphisms u extending \varphi inside \mathcal{Q} . This result extends that already proven by Pratelli and the third author in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 511–555] in the shape of the domain.

Quasi-periodic motions in generic nearly-integrable mechanical systems

In this note, we present and briefly discuss results, which include as a particular case the theorem announced in [Rend. Lincei Mat. Appl. 26 (2015), 423–432], concerning the typical behavior of nearly-integrable mechanical systems with generic analytic potentials.

Interacting particle systems with long-range interactions: Approximation by tagged particles in random fields

In this paper, we continue the study of the derivation of different types of kinetic equations which arise from scaling limits of interacting particle systems. We began this study in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 32 (2021), 335–377]. More precisely, we consider the derivation of the kinetic equations for systems with long-range interaction. Particular emphasis is put on the fact that all the kinetic regimes can be obtained approximating the dynamics of interacting particle systems, as well as the dynamics of Rayleigh gases, by a stochastic Langevin-type dynamics for a single particle. We will present this approximation in detail and we will obtain precise formulas for the diffusion and friction coefficients appearing in the limit Fokker–Planck equation for the probability density of the tagged particle f(x,v,t) , for three different classes of potentials. The case of interaction potentials behaving as Coulombian potentials at large distances will be considered in detail. In particular, we will discuss the onset of the the so-called Coulombian logarithm.

Analytic and rational sections of relative semi-abelian varieties

The hyperbolicity statements for subvarieties and complements of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings (and Vojta for the semi-abelian case). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme over an affine algebraic curve. These sections form a group; while the group of rational sections (the Mordell-Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory.

An asymptotic analysis for a generalized Cahn\u2013Hilliard system with fractional operators

In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers A^{2r} and B^{2sigma } (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space L^2(Omega ), for some bounded and smooth domain Omega subset {{mathbb {R}}}^3, and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter sigma appearing in the operator B^{2sigma } decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.

Positive solution to Schrödinger equation with singular potential and double critical exponents

In this paper, we consider the following Schrödinger equation with singular potential and double critical exponents: -\Delta u + \frac{A}{|x|^{\alpha}}u = |u|^{2^{*}-2}u + |u|^{p-2}u + \lambda |u|^{2_{\alpha}^{*}-2}u,\quad x\in \mathbb{R}^{N}, where N\geq 3 , \alpha\in(0,2) , p\in(\frac{2N-4+2\alpha}{N-2},2^{*}) , and A,\lambda>0 are two real constants, and 2^{*}=\frac{2N}{N-2} is the Sobolev critical exponent, and 2_{\alpha}^{*}=2+\frac{4\alpha}{2N-2-\alpha} is the critical exponent with respect to the parameter \alpha . First, using the refined Sobolev inequality, we establish the Lions type theorem. Second, we prove that any nonnegative weak solutions of above equation satisfy Pohozaev type identity. Finally, by using perturbation method, Pohozaev type identity and Lions type theorem, we show the existence of positive solution to above equation. We point out that the double critical exponents is an new phenomenon, and we are the first to consider it. Our result partial extends the results in Badiale and Rolando [Rend. Lincei Mat. Appl. 17 (2006)], and Su, Wang and Willem [Commun. Contemp. Math. 9 (2007)].

Heat conservation for generalized Dirac Laplacians on manifolds with boundary

We consider a notion of conservation for the heat semigroup associated with a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero-order piece (Weitzenbock potential) of the Dirac Laplacian, and on the endomorphism defining the mixed boundary condition, we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property for the heat semigroup which follows from an extension to this setting of a Feynman–Kac formula recently proved by the author de Lima (Pac J Math 292(1):177–201, 2018) in the context of differential forms. When applied to the Hodge Laplacian acting on differential forms satisfying absolute boundary conditions, this extends previous results by Vesentini (Ann Math Pura Appl 182:1–19, 2003) and Masamune (Atti Accad Naz Lincei Rend Lincei Mat Appl 18(4):351–358, 2007) in the boundaryless case. Along the way, we also prove a vanishing result for $$L^2$$ harmonic sections in the broader context of generalized (not necessarily Dirac) Laplacians. These results are further illustrated with applications to the Dirac Laplacian acting on spinors and to the Jacobi operator acting on sections of the normal bundle of a free boundary minimal immersion.

Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].

A Differential Model for Growing Sandpiles on Networks

We consider a system of differential equations of Monge--Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton--Jacobi equations on networks introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535--545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.

Dynamic of rotating fluid layers: $$L^2$$ L 2 -absorbing sets and onset of convection

Via the longtime behavior of the perturbations to thermal conduction solution $$m_0$$ , the nonlinear longtime behavior of Navier–Stokes fluid mixtures filling horizontal rotating layers uniformly heated from below and salted by one salt—either from above or below—is investigated. Via the existence of $$L^2$$ -absorbing sets, it is shown that the perturbations to $$m_0$$ are ultimately bounded. The onset of steady or oscillatory convection is analyzed. Via a Linearization Principle (Rionero in Rend Lincei Mat Appl 25:1–44, 2014) it is shown that the linear theory captures completely the physics of the problem since the linear stability implies the nonlinear global asymptotic stability in the $$L^2$$ -norm.

On a separation and irreducibility problem of polynomials arising from the nonlinear Schrödinger equation

We discuss an algebraic problem (Separation and Irreducibility Conjecture) which arises from the study of the nonlinear Schrödinger equation (NLS for short). This problem is about separation and irreducibility (over the ring of integers) of the characteristic polynomials of the graphs, describing blocks of a normal form for the NLS. For the cubic NLS the problem has been completely solved (see [C. Procesi, M. Procesi and B. Van Nguyen, The energy graph of the nonlinear Schrödinger equation, Rend. Lincei Mat. Appl. 24(2) (2013) 229–301]), meanwhile for higher degree NLS it is still open, even in small dimensions (see [C. Procesi, The energy graph of the non-linear Schrödinger equation, open problems. Int. J. Algebra Comput. 23(4) (2013) 943–962]). In this work, the author will give a partial answer for this problem, in particular, the author will prove Separation and Irreducibility Conjecture for NLS of arbitrary degree on one-dimensional and two-dimensional tori.

Porous MHD convection: stabilizing effect of magnetic field and bifurcation analysis

MHD convection in a horizontal porous-layer filled by a plasma, imbedded in a transverse magnetic field and heated from below, is investigated. The critical Rayleigh number is found and, in simple algebraic closed forms, conditions necessary and sufficient for the onset of steady and oscillatory convection are obtained. It is shown that the stabilizing effect of the magnetic field grows with $$Q^2$$ , Q being the Chandrasekhar number. The linearization principle (Rionero, Rend Lincei Mat Appl 25:368, 2014): “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” continues to hold also in the case at stake and allows to recover for the global nonlinear stability the conditions of linear stability.

Convection in multi-component rotating fluid layers via the auxiliary system method

The onset of thermal convection in a uniformly rotating horizontal layer filled by a Navier–Stokes multi-component fluid mixture, heated from below and salted partly from above and partly from below, is investigated via the new approach named auxiliary system method Rionero (Rend Lincei Mat Appl 25:1–44, 2014). In the free–free case, via the generalization of the Rionero Linearization Principle: “Decay of linear energy for any initial data implies decay of nonlinear energy at any instant” [given in Rionero (Rend Lincei Mat Appl 25:1–44, 2014) in the absence of rotation], it is shown that conditions guaranteeing linear stability of thermal conduction solution guarantee also absence of subcritical instabilities and global exponential asymptotic nonlinear stability. The classical Benard problem is investigated via a procedure different from the celebrated one given in Chandrasekhar (Hydrodynamic and hydromagnetic stability, 1981).

$${\mathbb P^r}$$ -scrolls arising from Brill–Noether theory and K3-surfaces

In this paper we study examples of \({\mathbb P^r}\)-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill–Noether theory of the general curve in the primitive linear system on S and from some results of Lazarsfeld. We show that such scrolls form an open dense subset of a component \({\mathcal H}\) of their Hilbert scheme; moreover, we study some properties of \({\mathcal H}\) (e.g. smoothness, dimensional computation, etc.) just in terms of \({\mathfrak F_g}\), the moduli space of such K3’s, and Mv(S), the moduli space of semistable torsion-free sheaves of a given rank on S. One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a very general K3 as well as Brill–Noether theory of vector-bundles on suitable degenerations of projective curves. We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces introduced and studied in Calabri et al. (Rend Lincei Mat Appl 17(2):95–123, 2006) and Calabri et al. (Rend Circ Mat Palermo 57(1):1–32, 2008).

Pathological solutions to elliptic problems in divergence form with continuous coefficients

We construct a function u ∈ W loc 1 , 1 ( B ( 0 , 1 ) ) which is a solution to div ( A ∇ u ) = 0 in the sense of distributions, where A is continuous and u ∉ W loc 1 , p ( B ( 0 , 1 ) ) for p > 1 . We also give a function u ∈ W loc 1 , 1 ( B ( 0 , 1 ) ) such that u ∈ W loc 1 , p ( B ( 0 , 1 ) ) for every p < ∞ , u satisfies div ( A ∇ u ) = 0 with A continuous but u ∉ W loc 1 , ∞ ( B ( 0 , 1 ) ) . This answers questions raised by H. Brezis (On a conjecture of J. Serrin, Rend. Lincei Mat. Appl. 19 (2008) 335–338). To cite this article: T. Jin et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Curves of minimal action over metric spaces

Given a metric space X, we consider a class of action functionals, generalizing those considered in Brancolini et al. (J Eur Math Soc 8:415–434, 2006) and Ambrosio and Santambrogio (Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei Mat Appl 18: 23–37, 2007), which measure the cost of joining two given points x0 and x1, by means of an absolutely continuous curve. In the case X is given by a space of probability measures, we can think of these action functionals as giving the cost of some congested/concentrated mass transfer problem. We focus on the possibility to split the mass in its moving part and its part that (in some sense) has already reached its final destination: we consider new action functionals, taking into account only the contribution of the moving part.