1,720,969 research outputs found
Evolution of spoon-shaped networks
Full text linkWe consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain Ω. The two curves meet only at one point, forming angles of 120 degrees. The non-closed curve has a fixed end-point on ∂Ω. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon
On different notions of calibrations for minimal partitions and minimal networks in R2
Full text linkCalibrations are a possible tool to validate the minimality of a certain candidate. They have been introduced in the context of minimal surfaces and adapted to the case of the Steiner problem in several variants. Our goal is to compare the different notions of calibrations for the Steiner problem and for planar minimal partitions that are already present in the literature. The paper is then complemented with remarks on the convexification of the problem, on nonexistence of calibrations and on calibrations in families
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
Long time existence of solutions to an elastic flow of networks
Full text linkThe L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods
Lectures on curvature flow of networks
Full text linkWe present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities
Existence and uniqueness of the motion by curvature of regular networks
No full textWe prove existence and uniqueness of the motion by curvature of networks with triple junctions in Rd when the initial datum is of class Wp2-2/p and the unit tangent vectors to the concurring curves form angles of 120 degrees. Moreover, we investigate the regularisation effect due to the parabolic nature of the system. An application of the well-posedness is a new proof and a generalisation of the long-time behaviour result derived by Mantegazza et al. in 2004. Our study is motivated by an open question proposed in the 2016 survey from Mantegazza et al.: does there exist a unique solution of the motion by curvature of networks with initial datum being a regular network of class C 2? We give a positive answer
Variational models for the interaction of surfactants with curvature – existence and regularity of minimizers in the case of flexible curves
No full textExistence and regularity of minimizers for a geometric variational problem is shown. The variational integral models an energy contribution of the interface between two immiscible fluids in the presence of surfactants and includes a Helfrich type contribution, a Frank type contribution and a coupling term between the orientation of the surfactants and the curvature of the interface. Analytical results are proven in a one–dimensional situation for curves
The oriented mailing problem and its convex relaxation
Full text linkIn this note we introduce a new model for the mailing problem in branched transportation that takes into account the orientation of the moving particles. This gives an effective answer to Bernot et al. (2009, Problem 15.9). Moreover we define a convex relaxation in terms of rectifiable currents with group coefficients. We provide the problem with a notion of calibration. Using similar techniques we define a convex relaxation and a corresponding notion of calibration for a variant of the Steiner tree problem in which a connectedness constraint is assigned only among a certain partition of a given set of finitely many points
EVOLUTION OF NETWORKS WITH MULTIPLE JUNCTIONS
No full textWe consider the motion by curvature of a network of curves in theplane and we discuss existence, uniqueness, singularity formation, and asymptoticbehavior of the flow
Willmore flow of planar networks
Full text linkGeometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense
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