Annali di Matematica Pura ed Applicata最新文献

We study a quasilinear elliptic problem (-text {div} (nabla Phi (nabla u))+V(x)N'(u)=f(u)) with anisotropic convex function (Phi ) on the whole (mathbb {R}^n). To prove existence of a nontrivial weak solution we use the mountain pass theorem for a functional defined on anisotropic Orlicz–Sobolev space ({{{,mathrm{textbf{W}},}}^1}{{,mathrm{textbf{L}},}}^{{Phi }} (mathbb {R}^n)). As the domain is unbounded we need to use Lions type lemma formulated for Young functions. Our assumptions broaden the class of considered functions (Phi ) so our result generalizes earlier analogous results proved in isotropic setting.

We first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. Among the applications is a new proof of the main result of Catanese and Cesarano (Electron Res Arch 29(6):4315–4325, 2021) that, for a general pair (A, X) of an (ample) Hypersurface X in an Abelian Variety A, the canonical map (Phi _X) of X is birational onto its image if the polarization given by X is not principal. The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions.

We consider a parabolic equation whose coefficients are Log-Lipschitz continuous in t and Lipschitz continuous in x. Combining a recent conditional stability result with a well posed variational problem, we reconstruct the initial condition of an unknown solution from a rough measurement at the final time.

In this paper, we study the fractional Schrödinger–Poisson system with a general nonlinearity as follows:

where (frac{1}{2}<tle s<1), the potential (lin C(mathbb {R}^{2},mathbb {R}^{+})) and (fin C(mathbb {R},mathbb {R})) does not require the classical (AR)-condition. When (l(x)equiv mu >0) is a parameter, by establishing new estimates for the fractional Laplacian, we find two positive solutions, depending on the range of (mu ). As a result, a positive ground state solution with negative energy exists for the non-autonomous system without any symmetry on l(x). When l(x) is radially symmetric, we show that the symmetry breaking phenomenon can occur, and that a non-radial ground state solution with negative energy exists. Furthermore, under additional assumptions on l(x), three positive solutions are found. The intrinsic differences between the planar SP system and the planar fSP system are analyzed.

Let (lambda ) be a general length function for modules over a Noetherian ring R. We use (lambda ) to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of (lambda ). We show that the leading term (mu ) of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for R[X]-modules. Similar to algebraic entropy, (mu ) in general is not additive for exact sequences of R[X]-modules: we demonstrate how to adapt certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

A result of Chernoff gives sufficient condition for an (L^2)-function on ({mathbb { R}}^n) to be quasi-analytic, in the sense that the function and all its derivatives cannot vanish at a point. This is a generalization of the classical Denjoy–Carleman theorem on ({mathbb { R}}) and of the subsequent works on ({mathbb { R}}^n) by Bochner and Taylor. In this note we endeavour to obtain an exact analogue of the result of Chernoff for (L^p, pin [1,2]) functions on the Riemannian symmetric spaces of noncompact type. No restriction on the rank of the symmetric spaces and no condition on the symmetry of the functions is assumed.

In this paper we consider steady inviscid three-dimensional stratified water flows of finite depth with a free surface and an interface. The interface plays the role of an internal wave that separates two layers of constant and different density. We study two cases separately: when the free surface and the interface are functions of one variable and when the free surface and the interface are functions of two variables. In both cases, considering effects of surface tension, we prove that the bounded solutions to the three-dimensional equations are essentially two-dimensional. More specifically, assuming that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other we prove that their third coordinate vanishes in both layers. Also we prove that the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion.

In the setting of the thin-shell approximation of the Euler equations in spherical coordinates for oceanic flows with variable density on the spinning Earth, we study a vorticity equation for a pseudo stream function (psi ), whereby the assumption of incompressibility allows us to express the density as a function of (psi ). Via an elliptic comparison argument, we show that, under certain assumptions, the (explicit) solution in the case of zero rate of rotation (i.e., on a fixed sphere) in a bounded region with smooth boundary contained either in the Northern or in the Southern Hemisphere is an approximation, in a suitable sense, of the corresponding solution of the equation with positive rate of rotation in the same region. This provides new insight into the dynamics of ocean gyres.

In this paper, we obtain the Gehring–Hayman type theorem on smoothly bounded pseudoconvex domains of finite type in (mathbb {C}^2). As an application, we provide a quantitative comparison between global and local Kobayashi distances near a boundary point for these domains.

In this paper we show that every connected extremal Kähler submanifold of a complex projective space has a natural extension which is a complete Kähler manifold and admits a holomorphic isometric immersion into the same ambient space. We also give an application to study the scalar curvatures of extremal Hypersurfaces of complex projective spaces.