Abstract:We consider the evolution by curvature of a general embedded network with two triple junctions. We classify the possible singularities and we discuss the long time existence of the evolution.MSC: 53C44 (primary); 53A04, 35K55 (secondary)

Geometric 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.Mathematics Subject Classification (2010): 35K52, 53C44 (primary); 35K61, 35K41 (secondary). *

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We consider a regular embedded network composed by two curves, one of them closed, in a convex domain Ω. The two curves meet only in one point, forming angle 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 existence time 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.Mathematics Subject classification (2010). 53C44 (primary); 53A04, 35K55 (secondary).with loops. We call Brakke spoon a spoon-shaped network composed by a half-line which meets a closed curve shrinking in a self-similar way during the evolution. The main result is the following: Theorem 1.1. Given a spoon-shaped network Γ t evolving by curvature in a strictly convex and open set Ω ∈ R 2 , there exists a finite time T such that either the inferior limit of the length of the non-closed curve goes to zero as t → T , or there exists a point x 0 ∈ Ω such that for a subsequence of rescaled time t j the associate rescaled network tends to a Brakke spoon as j → ∞.This result can be seen as an analogue of Grayson's theorem for a closed curve, which says that an embedded evolving curve in R 2 becomes eventually convex without developing singularities and then shrinks to a point (see [6,7]). A new proof of this theorem is given in [14]. In Proposition 5.4, we obtain, up to subsequence, as possible limits, sets that satisfy the equation k + x|ν = 0 for all x, where k is the curvature and ν is the unit normal vector. Satisfying the previous equation, the sets shrink homothetically. The existence and uniqueness of self-similar shrinking solution of a problem similar to ours is proved in [4]: Chen and Guo consider an equivalent system that describes the motion by curvature, but they focus on the evolution of a curve symmetric about the x−axis, and consider the part of the curve in the upper half-plane, which forms fixed angles with the axis. In Proposition 5.4, to gain existence of a unique limit set, without self-intersections, with multiplicity one, with curvature not constantly zero, we will apply [4, Theorem 3], as our asymptotic solution is one of the Abresch-Langer curves, symmetric and convex.

We 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 $${\mathbb {R}}^2$$ R 2 . In the last section of the paper we also discuss a list of open problems.

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We prove existence and uniqueness of the motion by curvature of networks in R n when the initial datum is of class W 2− 2 /p p , with triple junction where the unit tangent vectors to the concurring curves form angles of 120 degrees. Moreover we investigated the regularization effect due to the parabolic nature of the system. An application of this wellposedness result is a new proof of [25, Theorem 3.18] where the possible behaviours of the solutions at the maximal time of existence are described. Our study is motivated by an open question proposed in [24]: does there exist a unique solution of the motion by curvature of networks with initial datum a regular network of class C 2 ? We give a positive answer.