181 research outputs found
A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension
In this paper we study the vanishing inertia and viscosity limit of a second
order system set in an Euclidean space, driven by a possibly nonconvex
time-dependent potential satisfying very general assumptions. By means of a
variational approach, we show that the solutions of the singularly perturbed
problem converge to a curve of stationary points of the energy and characterize
the behavior of the limit evolution at jump times. At those times, the left and
right limits of the evolution are connected by a finite number of heteroclinic
solutions to the unscaled equation
Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach
We perform a convergence analysis of a discrete-in-time minimization scheme
approximating a finite dimensional singularly perturbed gradient flow. We allow
for different scalings between the viscosity parameter and the
time scale . When the ratio diverges, we
rigorously prove the convergence of this scheme to a (discontinuous) Balanced
Viscosity solution of the quasistatic evolution problem obtained as formal
limit, when , of the gradient flow. We also characterize the
limit evolution corresponding to an asymptotically finite ratio between the
scales, which is of a different kind. In this case, a discrete interfacial
energy is optimized at jump times
Delayed loss of stability in singularly perturbed finite-dimensional gradient flows
In this paper we study the singular vanishing-viscosity limit of a gradient
flow in a finite dimensional Hilbert space, focusing on the so-called delayed
loss of stability of stationary solutions. We find a class of time-dependent
energy functionals and initial conditions for which we can explicitly calculate
the first discontinuity time of the limit. For our class of functionals,
coincides with the blow-up time of the solutions of the linearized system
around the equilibrium, and is in particular strictly greater than the time
where strict local minimality with respect to the driving energy gets
lost. Moreover, we show that, in a right neighborhood of , rescaled
solutions of the singularly perturbed problem converge to heteroclinic
solutions of the gradient flow. Our results complement the previous ones by
Zanini, where the situation we consider was excluded by assuming the so-called
transversality conditions, and the limit evolution consisted of strict local
minimizers of the energy up to a negligible set of times
Functionals defined on piecewise rigid functions: Integral representation and -convergence
We analyze integral representation and -convergence properties of
functionals defined on \emph{piecewise rigid functions}, i.e., functions which
are piecewise affine on a Caccioppoli partition where the derivative in each
component is constant and lies in a set without rank-one connections. Such
functionals account for interfacial energies in the variational modeling of
materials which locally show a rigid behavior. Our results are based on
localization techniques for -convergence and a careful adaption of the
global method for relaxation (Bouchitt\'e et al. 1998, 2001) to this new
setting, under rather general assumptions. They constitute a first step towards
the investigation of lower semicontinuity, relaxation, and homogenization for
free-discontinuity problems in spaces of (generalized) functions of bounded
deformation
Mean-Field Optimal Control
We introduce the concept of {\it mean-field optimal control} which is the
rigorous limit process connecting finite dimensional optimal control problems
with ODE constraints modeling multi-agent interactions to an infinite
dimensional optimal control problem with a constraint given by a PDE of
Vlasov-type, governing the dynamics of the probability distribution of
interacting agents. While in the classical mean-field theory one studies the
behavior of a large number of small individuals {\it freely interacting} with
each other, by simplifying the effect of all the other individuals on any given
individual by a single averaged effect, we address the situation where the
individuals are actually influenced also by an external {\it policy maker}, and
we propagate its effect for the number of individuals going to infinity. On
the one hand, from a modeling point of view, we take into account also that the
policy maker is constrained to act according to optimal strategies promoting
its most parsimonious interaction with the group of individuals. This will be
realized by considering cost functionals including -norm terms penalizing
a broadly distributed control of the group, while promoting its sparsity. On
the other hand, from the analysis point of view, and for the sake of
generality, we consider broader classes of convex control penalizations. In
order to develop this new concept of limit rigorously, we need to carefully
combine the classical concept of mean-field limit, connecting the finite
dimensional system of ODE describing the dynamics of each individual of the
group to the PDE describing the dynamics of the respective probability
distribution, with the well-known concept of -convergence to show that
optimal strategies for the finite dimensional problems converge to optimal
strategies of the infinite dimensional problem.Comment: 31 page
Pseudohermitian Hamiltonians, time-reversal invariance and Kramers degeneracy
A necessary and sufficient condition in order that a (diagonalizable)
pseudohermitian operator admits an antilinear symmetry T such that T^{2}=-1 is
proven. This result can be used as a quick test on the T-invariance properties
of pseudohermitian Hamiltonians, and such test is indeed applied, as an
example, to the Mashhoon-Papini Hamiltonian.Comment: 6 page
Hemihelical local minimizers in prestrained elastic bi-strips
We consider a double layered prestrained elastic rod in the limit of
vanishing cross section. For the resulting limit Kirchoff-rod model with
intrinsic curvature we prove a supercritical bifurcation result, rigorously
showing the emergence of a branch of hemihelical local minimizers from the
straight configuration, at a critical force and under clamping at both ends. As
a consequence we obtain the existence of nontrivial local minimizers of the
-d system.Comment: 16 pages, 2 figure
Chirality transitions in frustrated -valued spin systems
We study the discrete-to-continuum limit of the helical XY -spin
system on the lattice . We scale the interaction parameters in
order to reduce the model to a spin chain in the vicinity of the
Landau-Lifschitz point and we prove that at the same energy scaling under which
the -model presents scalar chirality transitions, the cost of every
vectorial chirality transition is now zero. In addition we show that if the
energy of the system is modified penalizing the distance of the field
from a finite number of copies of , it is still possible to prove the
emergence of nontrivial (possibly trace dependent) chirality transitions
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