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Motion of discrete interfaces on the triangular lattice
Full text linkWe study the motion of discrete interfaces driven by ferromagnetic interactions on the two-dimensional triangular lattice by coupling
the Almgren, Taylor and Wang minimizing movements approach and a discrete-to-continuum analysis, as introduced by Braides, Gelli and Novaga in the pioneering case of the square lattice. We examine the motion of origin-symmetric convex ''Wulff-like" hexagons, i.e. origin-symmetric convex hexagons with sides having the same orientations as those of the hexagonal Wulff shape related to the density of the anisotropic perimeter Gamma-limit of the ferromagnetic energies as the lattice spacing vanishes. We compare the resulting limit motion with the corresponding evolution by crystalline curvature with natural mobility
Non-local approximation of the Griffith functional
Full text linkAn approximation, in the sense of Γ-convergence and in any dimension d ≥ 1, of Griffith-type functionals, with p-growth (p > 1) in
the symmetrized gradient, is provided by means of a sequence of nonlocal integral functionals depending on the average of the symmetrized gradients on small balls
A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension
Full text linkIn 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
Invertibility of Orlicz–Sobolev Maps
No full textWe extend the global invertibility result (Henao et al., Adv Calculus Var 14(2):207–230, 2021) to a class of orientation-preserving Orlicz–Sobolev maps with an integrability just above n − 1, whose traces on the boundary are also Orlicz–Sobolev and which do not present cavitation in the interior or at the boundary. As an application, we prove the existence of a.e. injective minimizers within this class for functionals in nonlinear elasticity
Chirality Transitions in Frustrated Ferromagnetic Spin Chains: A Link with the Gradient Theory of Phase Transitions
Full text linkWe study chirality transitions in frustrated ferromagnetic spin chains, in view of a possible connection with the theory of Liquid Crystals. A variational approach to the study of these systems has been recently proposed by Cicalese and Solombrino, focusing close to the helimagnet/ferromagnet transition point corresponding to the critical value of the frustration parameter alpha = 4. We reformulate
this problem for any alpha>0 in the framework of surface energies in nonconvex discrete systems with nearest neighbours ferromagnetic and next-to-nearest neighbours antiferromagnetic interactions and we link it to the gradient theory of phase transitions, by showing a uniform equivalence by Gamma-convergence on [0; 4] with Modica-Mortola type functionals
Lower semicontinuity in GSBD for nonautonomous surface integrals
Full text linkWe provide a sufficient condition for lower semicontinuity of nonautonomous noncoercive surface energies defined on the space of functions,
whose dependence on the -variable is or even : the notion of \emph{nonautonomous symmetric joint convexity}, which extends the analogous definition devised for autonomous integrands in \cite{FPS} where the conservativeness of the approximating vector fields is assumed. This condition allows to extend to our setting a nonautonomous chain formula in obtained in \cite{ACDD}, and this is a key tool in the proof of the lower semicontinuity result. This new joint convexity can be checked explicitly for some classes of surface energies arising from variational models of fractures in inhomogeneous materials
Motion of discrete interfaces in periodic media
No full textWe study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional periodic environment by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. The case of a homogeneous environment has been recently treated by Braides, Gelli and Novaga, showing that the effective continuous motion is a flat motion related to the crystalline perimeter obtained by Gamma-convergence from the ferromagnetic energies, with an additional discontinuous dependence on the curvature, giving in particular a pinning threshold. In this paper we give an example showing that in general the motion does not depend only on the Gamma-limit, but also on geometrical features that are not detected in the static description. In particular we show how the pinning threshold is influenced by the microstructure and that the effective motion is described by a new homogenized velocity
Nucleation and backward motion of discrete interfaces
Full text linkWe use a discrete approximation of the motion by crystalline curvature to define an evolution of sets from a single point (nucleation) following a criterion of "maximization" of the perimeter, formally giving a backward version of the motion by crystalline curvature. This evolution depends on the approximation chosen
Partial Regularity for Minimizers of Discontinuous Quasiconvex Integrals with general growth
Full text linkWe prove the partial Hölder continuity for minimizers of quasiconvex functionals
F(u):=∫Ωf(x,u,Du)dx,
where f satisfies a uniform VMO condition with respect to the x-variable and is continuous with respect to u. The growth condition with respect to the gradient variable is assumed a general one
Nucleation and growth of lattice crystals
Full text linkA variational lattice model is proposed to define an evolution of sets from a single point (nucleation) following a criterion of “maximization” of the perimeter. At a discrete level, the evolution has a “checkerboard” structure and its shape is affected by the choice of the norm defining the dissipation term. For every choice of the scales, the convergence of the discrete scheme to a family of expanding sets with constant velocity is proved
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