Rossella Bartolo

Convexity of Domains of Riemannian Manifolds

In this paper the problem of the geodesic connectedness and convexity ofincomplete Riemannian manifolds is analyzed. To this aim, a detailedstudy of the notion of convexity for the associated Cauchy boundary iscarried out. In particular, under widely discussed hypotheses,we prove the convexity of open domains (whose boundaries may benondifferentiable) of a complete Riemannian manifold. Variationalmethods are mainly used. Examples and applications are provided,including a result for dynamical systems on the existence oftrajectories with fixed energy.

Geodesics in Static Lorentzian Manifolds with Critical Quadratic Behavior

Abstract The aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉L) such that. if K is its irrotational timelike Killing vector field, the growth of β = -〈K, K〉L is (at most) quadratic. This is shown to be equivalent to a critical variational problem on a Riemannian manifold, where some geometrical interpretations yield the optimal results. Multiplicity of connecting geodesics is studied, especially in the timelike case, where the restrictions for the number of causal connecting geodesics are stressed. Extensions to the non-complete case, including discussions on pseudoconvexity, are also given.

Convex domains of Finsler and Riemannian manifolds

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop’s one in his classical result (Bishop, Indiana Univ Math J 24:169–172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.

Periodic trajectories with fixed energy on Riemannian and Lorentzian manifolds with boundary

We state some existence and multiplicity results for periodic solutions with prescribed energy of a Lagrangian system and for closed geodesics on Riemannian manifolds convex close to their boundary. As an application we get the existence and multiplicity of periodic trajectories with fixed energy on a class of physically important Lorentzian manifolds with boundary.

CONVEXITY CONDITIONS ON THE BOUNDARY OF A STATIONARY SPACETIME AND APPLICATIONS

We deal with the convexity of the boundary of a standard stationary spacetime L = M × ℝ. We obtain a characterization of this notion by means of Riemannian conditions involving a potential plus a magnetic field on M, where both are linked to the coefficients of the metric. Natural applications of our results concern geodesics having a prescribed parametrization proportional to the arc length, joining a point to a line and periodic, on non-complete manifolds, and in particular on Kerr spacetime.

Connectivity by geodesics on globally hyperbolic spacetimes with a lightlike Killing vector field

Given a globally hyperbolic spacetime endowed with a complete lightlike Killing vector field and a complete Cauchy hypersurface, we characterize the points which can be connected by geodesics. A straightforward consequence is the geodesic connectedness of globally hyperbolic generalized plane waves with a complete Cauchy hypersurface.

Multiplicity results for a class of asymptotically p-linear equations on ℝN

The aim of this paper is investigating the multiplicity of weak solutions of the quasilinear elliptic equation − Δpu + V (x)|u|p−2u = g(x,u) in ℝN, where 1 < p < +∞, the nonlinearity g behaves as |u|p−2u at infinity and V is a potential satisfying suitable assumptions so that an embedding theorem for weighted Sobolev spaces holds. Both the non-resonant and resonant cases are analyzed.

Trajectories Joining Two Submanifolds under the Action of Gravitational and Electromagnetic Fields on Static Spacetimes

In this paper we present existence and multiplicity results for orthogonal trajectories joining two submanifolds under the action of gravitational and electromagnetic fields on static spacetimes. These trajectories are critical points of unbounded functionals and they can be found by using a variant of the saddle point theorem and the relative category theory.

Existence of a Closed Geodesic on Non-compact Riemannian Manifolds with Boundary

Abstract A new result about the existence of a closed geodesic on a Riemannian manifold with boundary is given. A detailed comparison with previous results is carried out.

Infinitely Many Solutions for Quasilinear Elliptic Problems with Broken Symmetry

Abstract The aim of this paper is investigating the existence of solutions of the quasilinear elliptic Problem where Ω is an open bounded domain of RN with C2 boundary ∂Ω, Δpu = div(|∇u|p−2∇u), 1 < p < q < p∗, f ∈ C(Ω̅) and ϕ ∈ C2(Ω̅). By means of the so-called Bolle’s method in [3, 4], we extend a result in [10] where the authors consider Ω =]0, 1[N and u = 0 on ∂Ω.