Łojasiewicz-Simon inequalities for minimal networks: stability and convergence

Abstract

We investigate stability properties of the motion by curvature of planar networks. We prove Łojasiewicz-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 2 3 π that are close in H 2 -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 Γ * and length of a triple junctions network Γ from above by the L 2 -norm of the curvature of the edges of Γ. 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 H 2 -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.

On the uniqueness of nondegenerate blowups for the motion by curvature of networks

“…In view of [15,Theorem 5.14], this means that either the inferior limit of the length of at least one curve of the network is zero as t → T − , or the superior limit of the L 2 -norm of the curvature is +∞ as t → T − . We remind the reader that there are explicit example of singularity formations in the network flow, where one curves vanishes and two triple junctions coalesce, or entire regions disappears (see for instance [17,15,18]).…”

“…(3.19) 2. The expression for the first variation δG and the fact that it is Z ⋆ -valued follow from Proposition 3.5, while its analyticity follows as in [18,Lemma 3.11].…”

“…Hence, in order to perform a graph parametrization of networks close to Γ * we will need to allow for graphs having both a normal and a tangential component with respect to Γ * . The issue of performing such a graph parametrization and at the same time of achieving the linear functional analytic setup needed for deriving the Łojasiewicz-Simon inequality was investigated in [18] for the study of stability properties networks that are critical points of the length functional. We shall recall from [18] the tools needed here at the beginning of Section 3, and we refer to [18] for a more detailed account on the method.…”

“…Moreover δ 2 G 0 : V → Z ⋆ is a Fredholm operator of index zero.Proof. The proof follows by adapting the arguments in[18, Section 3.3].1. Analyticity of G easily follows as in[18, Lemma 3.11].…”

“…The proof follows by adapting the arguments in[18, Section 3.3].1. Analyticity of G easily follows as in[18, Lemma 3.11].…”