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Evolution of networks with multiple junctions

Carlo Mantegazza, Matteo Novaga, Alessandra Pluda, Felix Schulze

Abstract

We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.

Evolution of networks with multiple junctions

Abstract

We consider the motion by curvature of a network of curves in the plane and we discuss existence, uniqueness, singularity formation and asymptotic behavior of the flow.

Paper Structure

This paper contains 54 sections, 117 theorems, 492 equations, 51 figures.

Table of Contents

  1. Introduction
  2. Notation, definitions and basic computations
  3. Networks
  4. Motion by curvature
  5. Basic computations
  6. Short time existence I
  7. Well--posedness in Sobolev spaces
  8. Well--posedness of the linearized system \ref{['linsys']} and of the special flow \ref{['problema-nogauge']}
  9. Existence and geometric uniqueness
  10. Extension to general regular networks
  11. Well--posedness in Hölder spaces
  12. Well--posedness of the linearized system \ref{['linsys']} and of the special flow \ref{['problema-nogauge']}
  13. Existence and geometric uniqueness
  14. Initial data with higher regularity
  15. Integral estimates
  16. ...and 39 more sections

Key Result

Lemma 2.17

If $\gamma$ is a curve moving by then the following commutation rule holds

Figures (51)

  • Figure 1.1: A planar network of curves in a convex domain.
  • Figure 2.1: An example of "violation" of Condition 3 in the definition of network.
  • Figure 2.2: A regular network.
  • Figure 2.4: Examples of networks.
  • Figure 3.1: A regular triod.
  • ...and 46 more figures

Theorems & Definitions (288)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 278 more