1,720,981 research outputs found
Elastic networks, statics and dynamics
No full textWe consider planar networks minimizing the elastic energy, we state an existence and regularity result, and we discuss some geometric properties of minimal configurations. We also consider the evolution of networks by the gradient flow of the energy, and we give a well-posedness result in the case of natural boundary conditions
Motion by Curvature of Networks with Two Triple Junctions
No full textWe show that any strictly mean convex translator of dimension n ≥ 3 which admits a cylindrical estimate and a corresponding gradient estimate is rotationally symmetric. As a consequence, we deduce that any translating solution of the mean curvature flow which arises as a blow-up limit of a two-convex mean curvature flow of compact immersed hypersurfaces of dimension n ≥ 3 is rotationally symmetric. The proof is rather robust, and applies to a more general class of translator equations. As a particular application, we prove an analogous result for a class of flows of embedded hypersurfaces which includes the flow of twoconvex hypersurfaces by the two-harmonic mean curvature
Coarsening phenomena in the network flow
Full text linkIn this short note we summarize recent results on the asymptotic behaviour of the network flow and we give indications of an expected coarsening-type behaviour for the network flow past singularities. The paper is complemented with a discussion on critical points and local minimizers of the length functional.In questa breve nota riassumiamo alcuni risultati sul comportamento asintotico del moto per curvatura di network, focalizzandoci sugli indizi di comportamenti di tipo coarsening. La nota contiene anche una discussione sui punti critici e minimi locali del funzionale lunghezza
Degenerate elastic networks
Full text linkWe minimize a linear combination of the length and the L2-norm of the curvature among networks in Rd belonging to a given class determined by the number of curves, the order of the junctions, and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of d=2 we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization
A survey of the elastic flow of curves and networks
Full text linkWe collect and present in a unified way several results in recent years about
the elastic flow of curves and networks, trying to draw the state of the art of
the subject. In particular, we give a complete proof of global existence and
smooth convergence to critical points of the solution of the elastic flow of
closed curves in . In the last section of the paper we also
discuss a list of open problems
{\L}ojasiewicz-Simon inequalities for minimal networks: stability and convergence
Full text linkWe investigate stability properties of the motion by curvature of planar
networks. We prove Lojasiewicz-Simon gradient inequalities for the length
functional of planar networks with triple junctions. In particular, such an
inequality holds for networks with junctions forming angles equal to
that are close in -norm to minimal networks, i.e., networks
whose edges also have vanishing curvature. The latter inequality bounds a
concave power of the difference between length of a minimal network
and length of a triple junctions network from above by the -norm
of the curvature of the edges of . We apply this result to prove the
stability of minimal networks in the sense that a motion by curvature starting
from a network sufficiently close in -norm to a minimal one exists for all
times and smoothly converges. We further rigorously construct an example of a
motion by curvature having uniformly bounded curvature that smoothly converges
to a degenerate network in infinite time
Minimizing properties of networks via global and local calibrations
Full text linkIn this note we prove that minimal networks enjoy minimizing properties for
the length functional. A minimal network is, roughly speaking, a subset of
composed of straight segments joining at triple junctions
forming angles equal to ; in particular such objects are just
critical points of the length functional a priori.
We show that a minimal network : i) minimizes mass among currents
with coefficients in a suitable group having the same boundary of ,
ii) identifies the interfaces of a partition of a neighborhood of
solving the minimal partition problem among partitions with same boundary
traces.
Consequences and sharpness of such results are discussed. The proofs reduce
to rather simple and direct arguments based on the exhibition of (global or
local) calibrations associated to the minimal network
Short‐time existence for the network flow
Full text linkThis paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated by Brakke using varifold methods. There is good reason to treat this problem by a direct PDE approach, but doing so requires one to deal with the singular nature of the PDE at the vertices of the network. This was handled in cases of increasing generality by Bronsard-Reitich, Mantegazza-Novaga-Tortorelli and eventually, in the most general case of irregular networks by Ilmanen-Neves-Schulze. Although the present paper proves a result similar to the one in Ilmanen et al., the method here provides substantially more detailed information about how an irregular network “resolves” into a regular one. Either approach relies on the existence of self-similar expanding solutions found in Mazzeo and Saez. As a precursor to the main theorem, we also prove an unexpected regularity result for the mixed Cauchy-Dirichlet boundary problem for the linear heat equation on a manifold with boundary
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