73 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 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
ERP Modulation during Observation of Abstract Paintings by Franz Kline
The aim of this study was to test the involvement of sensorimotor cortical circuits during the beholding of the static consequences of hand gestures devoid of any meaning.In order to verify this hypothesis we performed an EEG experiment presenting to participants images of abstract works of art with marked traces of brushstrokes. The EEG data were analyzed by using Event Related Potentials (ERPs). We aimed to demonstrate a direct involvement of sensorimotor cortical circuits during the beholding of these selected works of abstract art. The stimuli consisted of three different abstract black and white paintings by Franz Kline. Results verified our experimental hypothesis showing the activation of premotor and motor cortical areas during stimuli observation. In addition, abstract works of art observation elicited the activation of reward-related orbitofrontal areas, and cognitive categorization-related prefrontal areas. The cortical sensorimotor activation is a fundamental neurophysiological demonstration of the direct involvement of the cortical motor system in perception of static meaningless images belonging to abstract art. These results support the role of embodied simulation of artist’s gestures in the perception of works of art
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
Supercritical biharmonic equations with power-type nonlinearity
The biharmonic supercritical equation , where and
, is studied in the whole space as well as in a
modified form with as right-hand-side with an additional
eigenvalue parameter in the unit ball, in the latter case together
with Dirichlet boundary conditions. As for entire regular radial solutions we
prove oscillatory behaviour around the explicitly known radial {\it singular}
solution, provided , where
is a further critical exponent, which was introduced in a recent work by
Gazzola and the second author. The third author proved already that these
oscillations do not occur in the complementing case, where .
Concerning the Dirichlet problem we prove existence of at least one singular
solution with corresponding eigenvalue parameter. Moreover, for the extremal
solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue
problem, we prove smoothness as long as
Analyticity and criticality results for the eigenvalues of the biharmonic operator
We consider the eigenvalues of the biharmonic operator subject to several
homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show
that simple eigenvalues and elementary symmetric functions of multiple
eigenvalues are real analytic, and provide Hadamard-type formulas for the
corresponding shape derivatives. After recalling the known results in shape
optimization, we prove that balls are always critical domains under volume
constraint.Comment: To appear on the proceedings of the conference "Geometric Properties
for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop" held in
Palinuro (Italy), May 25-29, 201
The Gel'fand problem for the biharmonic operator
We study stable solutions of a fourth order nonlinear elliptic equation, both
in entire space and in bounded domains
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