Rendiconti dell’Istituto di Matematica dell’Università di Trieste: an International Journal of Mathematics vol.57 (2025)

We classify all closed non-orientable P2-irreducible 3- manifolds obtained by identifying the faces of a cube, i.e. those with cubic-complexity one. We show that they are the four flat ones.

In this paper, we produce explicit examples of mean curvature flow of (2m − 1)-dimensional submanifolds which converge to (2m − 2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in Cm with a U(m)-invariant Kähler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them.

We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole RN. A quasilinear fractional Schr¨odinger equation, where the leading operator is the N s - fractional Laplacian, is coupled with a higher-order and possibly fractional Poisson equation. For both operators the dimension N ≥ 2 corresponds to the limiting case of the Sobolev embedding, hence we consider nonlinearities with exponential growth. Since standard variational tools cannot be applied due to the sign-changing logarithmic Riesz kernel of the Poisson equation, we employ a variational approximating procedure for an auxiliary Choquard equation, where the Riesz kernel is uniformly approximated by polynomial kernels. Qualitative properties of solutions such as symmetry, regularity and decay are also established. Our results extend and complete the analysis carried out in the planar case in [13].

Hecke modifications of vector bundles have played a significant role in several areas of mathematics. They appear in subjects ranging from number theory to complex geometry. This article intends to be a friendly introduction to the subject. We give an overview of how Hecke modifications appear in the literature, explain their origin and their importance in number theory and classical algebraic geometry. Moreover, we report the progress made in describing Hecke modifications explicitly and why these explicit descriptions are important. We describe all the Hecke modifications of the trivial rank 2 vector bundle over a fixed point of degree 5 on the projective line, as well as all the vector bundles over a certain elliptic curve, which admit a rank 2 and degree 0 trace bundle as a Hecke modification. This result is not present in existing literature.

We consider a class of perturbed (p, q)-eigenvalue problems. Using the Nehari method, we show that for all small values of the parameter λ > 0, the problem has at least three nontrivial bounded solutions all with sign information (positive, negative and nodal).

We give a brief and concise guide for the analysis of the local behavior of the elements of local and nonlocal homogeneous De Giorgi classes: local boundedness, local H¨older continuity and Harnacktype inequalities. In the local case, we promote a simplified itinerary in the classic theory, propaedeutic for the successive part; while in the nonlocal case, we gather recent new developments into an unitary and concise framework. Employing a suitable definition of De Giorgi classes, we show a new proof of the Harnack inequality, way easier than in the local case, that bypasses any sort of Krylov-Safonov argument or cube decomposition.

In this paper we study autonomous systems whose Lagrangian function is the combination of several homogeneous terms with respect to positions and velocities. We show that, assuming certain relations between the degrees of homogeneity of such terms, the systems considered possess (in addition to energy) a further first integral that provides information about their solutions. A new feature of these results is the use of the theory of nonlocal constants, which finds useful constants using one-parameter perturbed motions.

Let ΔG be a sublaplacian on a Carnot group, and let μ be a local measure on the open set Ω ⊂ G. If u ∈ L1l oc(Ω) is such that −ΔGu = μ, u ≥ 0 on Ω, then μc ≥ 0, where μc is the concentrated component of μ with respect to the G-capacity. This extends to the Carnot group setting a result contained in [9].

We prove advanced regularity results for solutions to a sixth order equation arising in the mechanical Kirchhoff-Love’s type model of the static equilibrium of a nanoplate in bending. Such regularity properties play a crucial role in the treatment, among others, of the inverse problem consisting in the determination of the Winkler coefficient of a nanoplate.

We provide a detailed proof that the Nonlinear Fermi Golden Rule coefficient that appears in our recent proof of the asymptotic stability of ground states for the pure power Nonlinear Schr¨odinger equations in R with exponent 0 < |p − 3| ≪ 1 is nonzero.

For the third order Galerkin approximation of the Navier- Stokes equations under Navier boundary conditions in a cube we prove global existence and qualitative behaviour of the solution. By modifying properly the signs of the resulting ODEs system and using the test function technique developed by Mitidieri-Pohoˇzaev we prove, instead, finite time blow-up.

These notes aim to develop a tool for constructing polynomial differential p-forms vanishing on prescribed loci through syzygies of homogeneous ideals. Examples are provided through implementing this method in Macaulay2, particularly examples of instanton bundles of charges 4 and 5 on P3 that arise in this construction.

We classify involutions acting on spherical 3-manifolds up to conjugacy. Our geometric approach provides insights into numerous topological properties of these involutions.

We investigate the existence of some sporadic rank-r ⩾ 1 Ulrich vector bundles on suitable 3-fold scrolls X over the Hirzebruch surface F0, which arise as tautological embeddings of projectivization of very-ample vector bundles on F0 that are uniform in the sense of Brosius and Aprodu–Brinzanescu, cf. [11] and [4] respectively. Such Ulrich bundles arise as deformations of “iterative” extensions by means of sporadic Ulrich line bundles which have been contructed in our former paper [30] (where instead higher-rank sporadic bundles were not investigated therein). We explicitely describe irreducible components of the corresponding sporadic moduli spaces of rank r ⩾ 1 vector bundles which are Ulrich with respect to the tautological polarization on X. In some cases, such irreducible components turn out to be a singleton, in some other such components are generically smooth, whose positive dimension has been computed and whose general point turns out to be a slope-stable, indecomposable vector bundle.

The paper concerns universal a priori estimates for positive solutions to a large class of elliptic quasilinear equations and systems involving the p-Laplacian operator on arbitrary domains of RN and a convective term in the reaction. Our main theorems, new even for the Laplacian operator, extend previous estimates by Pol´aˇcik, Quitter and Souplet in [38] to very general nonlinearities admitting solely a lower bound, yielding a curious dichotomy. The main ingredients are a key doubling property, a rescaling argument, different from the classical blow-up technique of Gidas and Spruck, and Liouville-type theorems for inequalities. A discussion on the sharpness of the exponent in the power type term is also included.

This research project considers the d-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain, d = 2, 3 with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, the actuators will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space eB2−2/p q,p (Ω) × eB2−2/p q,p (Ω), 1 < p < 2q 2q−1 , q > d, of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to L3(Ω) for d = 3). It is known that such Besov space does not recognize compatibility conditions at the boundary, yet it provides a “minimal” level of regularity necessary to handle the nonlinear terms. In this paper we provide a solution of the first step: uniform stabilization of the linearized MHD. Showing maximal Lp-regularity up to T = ∞ for the feedback stabilized linearized system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem. The solution of the nonlinear stabilization problem is to be given in a successive paper [29].

Let D ⊆ Rn, n ≥ 3, be a bounded open set and let x0 ∈ D. Assume that the Newtonian potential of D is proportional outside D to the Newtonian potential of a mass concentrated at {x0}. Then D is a Euclidean ball centered at x0. This Theorem, proved by Aharonov, Schiffer and Zalcman in 1981, was extended to the caloric setting by Suzuki and Watson in 2001. In this note, we show that Suzuki–Watson Theorem is a particular case of a more general rigidity result related to a class of Kolmogorov-type PDEs.

Via the global bifurcation theorem due to Rabinowitz, the paper shows bifurcation properties of the solutions of the following nonlinear Dirichlet problem, involving a double phase operator, that is ( −Δap u − νΔmu = λa(x)|u|m−2u + f(x, u) in Ω, u = 0 on ∂Ω, where 1 < m < p < N, p/m < 1 + 1/N and λ, ν ∈ R.

We classify completely the equivalence groups and the classical Lie point symmetry groups of generalized radial Liouville-Bratu- Gelfand problems.