Annali di Matematica Pura ed Applicata最新文献

In this paper, the orbital instability theorem is introduced for Hamiltonian partial differential equation (PDE) systems. Our specific focus lies on the Schrödinger system featuring quadratic nonlinearity, and we apply the theorem to analyze its behavior. Our theorem establishes the abstract instability theorem for a specific class of Hamiltonian PDE systems. We consider the energy functional to be of class (C^2) rather than (C^3), particularly when the second derivative of the energy exhibits multiple degenerate kernels. Using this theorem, we provide a comprehensive classification of the stability and instability of the semitrivial solution within the Hamiltonian PDE system featuring quadratic nonlinearity. This classification resolves an open problem previously posed by Colin et al. (Ann Inst Henri Poincaré Anal Non Linéaire 26:2211–2226, 2009), specifically in cases of homogeneous nonlinearity. Additionally, we present proof of instability results for synchronous solutions of Hamiltonian PDE systems. We believe that this abstract theorem constitutes a novel contribution with potential applicability in various situations beyond those specifically discussed in this paper.

In this paper, we first study the minimality of triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms under the assumption that the shape operator is diagonalizable. Then, we prove that such nonminimal hypersurfaces have constant scalar curvature. As its applications, we estimate the constant scalar curvature and the constant mean curvature.

Let G be a group, p a prime and (H_p(G)) the subgroup of G generated by the elements of order different from p. In 1957, D. R. Hughes conjectured that either (H_p(G)=1), (H_p(G)=G), or ([G:H_p(G)]=p). In this paper, we prove this conjecture for finite extraspecial p-groups (where (p>2)), finite minimal non-abelian p-groups and finite non-abelian p-groups having cyclic maximal subgroup. Moreover, we give some sufficient conditions for 2-generated finite non-abelian p-groups which guarantee the existence of the Hughes conjecture.

We study an extension problem for continuous linear maps in the setting of (LB)-spaces. More precisely, we characterize the pairs (E, Z), where E is a locally complete space with a fundamental sequence of bounded sets and Z is an (LB)-space, such that for every exact sequence of (LB)-spaces

the map

is surjective, meaning that each continuous linear map (X rightarrow E) can be extended to a continuous linear map (Y rightarrow E) via (iota ), under some mild conditions on E or Z (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24].

This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfaces in the affine plane (mathbb {C}^2) satisfying interpolation and hitting conditions. We also show that every compact Riemann surface contains a Cantor set whose complement admits a proper holomorphic embedding in (mathbb {C}^2), and every connected domain in (mathbb {C}^2) admits complete, everywhere dense, injectively immersed complex discs. The focal point of the paper is a lemma saying for every compact bordered Riemann surface, M, closed discrete subset E of (mathring{M}=Msetminus bM), and compact subset (Ksubset mathring{M}setminus E) without holes in (mathring{M}), any (mathscr {C}^1) embedding (f:Mhookrightarrow mathbb {C}^2) which is holomorphic in (mathring{M}) can be approximated uniformly on K by holomorphic embeddings (F:Mhookrightarrow mathbb {C}^2) which map (Ecup bM) out of a given ball and satisfy some interpolation conditions.

Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space ((V,Vert cdot Vert )) equipped with a particular Jordan structure, called a (textrm{JB}^*)-triple. We identify the horofunctions in the metric compactification of ((V,Vert cdot Vert )) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space (V^*). Finally, we show that the exponential map (exp _0 :V longrightarrow D) at (0in D) extends to a homeomorphism between the metric compactifications of ((V,Vert cdot Vert )) and ((D,rho )), preserving the geometric structure, where (rho ) is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of ((V,Vert cdot Vert )).

We extend the estimates for maximal Fourier restriction operators proved by Müller et al. (Rev Mat Iberoam 35:693–702, 2019) and Ramos (Proc Am Math Soc 148:1131–1138, 2020) to the case of arbitrary convex curves in the plane, with constants uniform in the curve. The improvement over Müller, Ricci, and Wright and Ramos is given by the removal of the ({mathcal {C}}^2) regularity condition on the curve. This requires the choice of an appropriate measure for each curve, that is suggested by an affine invariant construction of Oberlin (Michigan Math J 51:13–26, 2003). As corollaries, we obtain a uniform Fourier restriction theorem for arbitrary convex curves and a result on the Lebesgue points of the Fourier transform on the curve.

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs (m, n) the locus of polarised abelian surfaces of type (1, d) that contain two complementary elliptic curve of exponents m, n, denoted (mathcal {E}_d(m,n)) is non-empty. We show that if d is square-free, the locus (mathcal {E}_d(m,n)) is an irreducible surface (if non-empty). We also show that the loci (mathcal {E}_d(d,d)) can have many components if d is an odd square. As an application, we show that for a genus 3 curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings (f_i:Crightarrow E_i, i=1,2,3) satisfy ({{,textrm{lcm},}}(deg (f_1),deg (f_2))={{,textrm{lcm},}}(deg (f_1),deg (f_3))={{,textrm{lcm},}}(deg (f_2),deg (f_3))).

From a fixed acute triangular base (Delta ABC), all possible tetrahedra in three-dimensional real space are considered. The possible angles at the additional vertex P are shown to be bounded by certain inequalities, mostly linear inequalities. Together, these inequalities provide fairly tight bounds on the possible angle combinations at P. Four sets of inequalities are used for this purpose, though the inequalities in the first set are rather trivial. The inequalities in the second set can be established quickly, but do not seem to be known. The third and fourth set of inequalities are proved by studying scalar and vector fields on toroids. The first three sets of inequalities are linear in the angles at P, but the last set involves cosines of these angles. A generalization of the last two sets of inequalities is also proved, using the Poincaré–Hopf Theorem. Extensive testing of these results has been done using Mathematica and C++. The C++ code for this is listed in an appendix. While it has been demonstrated that the inequalities bound the possible combinations of angles at P, the results also reveal that additional inequalities, in particular linear inequalities, exist that would provided tighter bounds.

We propose the novel spectral properties of the Neumann Laplacian in a two-dimensional bounded domain perturbed by an infinite number of compact sets with zero Lebesgue measure, so-called curved holes. These holes consist of segments or parts of curves enclosed in small spheres such that the diameters of holes tend to zero as the number of holes approaches infinity. Specifically, we rigorously demonstrate that the spectrum of the Neumann Laplacian on the perturbed domain converges to that of the original operator on the domain without holes under specific geometric assumptions and an appropriate selection of hole sizes. Furthermore, we derive sophisticated estimates on the convergence rate in terms of operator norms and estimate the Hausdorff distance between the spectra of the Laplacians.