86 research outputs found
Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models
We consider least energy solutions to the nonlinear equation posed on a class of Riemannian models of dimension
which include the classical hyperbolic space as well as manifolds
with unbounded sectional geometry. Partial symmetry and existence of least
energy solutions is proved for quite general nonlinearities , where
denotes the geodesic distance from the pole of
Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions
We model the roadway of a suspension bridge as a thin rectangular plate and
we study in detail its oscillating modes. The plate is assumed to be hinged on
its short edges and free on its long edges. Two different kinds of oscillating
modes are found: longitudinal modes and torsional modes. Then we analyze a
fourth order hyperbolic equation describing the dynamics of the bridge. In
order to emphasize the structural behavior we consider an isolated equation
with no forcing and damping. Due to the nonlinear behavior of the cables and
hangers, a structural instability appears. With a finite dimensional
approximation we prove that the system remains stable at low energies while
numerical results show that for larger energies the system becomes unstable. We
analyze the energy thresholds of instability and we show that the model allows
to give answers to several questions left open by the Tacoma collapse in 1940.Comment: 33 page
On some strong Poincaré inequalities on Riemannian models and their improvements
We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed
Poincaré and Hardy Inequalities on Homogeneous Trees
We study Hardy-type inequalities on infinite homogeneous trees. More precisely, we derive optimal Hardy weights for the combinatorial Laplacian in this setting and we obtain, as a consequence, optimal improvements for the Poincaré inequality
On the stability of a nonlinear nonhomogeneous multiply hinged beam
The paper deals with a nonlinear evolution equation describing the dynamics of a nonhomogeneous multiply hinged beam, subject to a nonlocal restoring force of displacement type. First, a spectral analysis for the associated weighted stationary problem is performed, providing a complete system of eigenfunctions. Then, a linear stability analysis for bimodal solutions of the evolution problem is carried out, with the final goal of suggesting optimal choices of the density and of the position of the internal hinged points in order to improve the stability of the beam. The analysis exploits both analytical and numerical methods; the main conclusion of the investigation is that nonhomogeneous density functions improve the stability of the structure
The fractional porous medium equation on the hyperbolic space
We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual Lp spaces or to larger (weighted) spaces determined either in terms of a ground state of the laplacian, or of the (fractional) Green’s function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative L1- L∞ estimates. To the best of our knowledge, this seems the first time in which the fractional porous medium equation has been treated on non-compact, geometrically non-trivial examples
The role of aerodynamic forces in a mathematical model for suspension bridges
In a fish-bone model for suspension bridges studied by us in a previous paper
we introduce linear aerodynamic forces. We numerically analyze the role of
these forces and we theoretically show that they do not influence the onset of
torsional oscillations. This suggests a new explanation for the origin of
instability in suspension bridges: it is a combined interaction between
structural nonlinearity and aerodynamics and it follows a precise pattern. This
gives an answer to a long-standing question about the origin of torsional
instability in suspension bridges
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