Motion by curvature of planar networks
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 2, p. 235-324
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two-dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which are “essentially” non regular. As a first step, in this paper we study in detail the case of three curves in the plane meeting at a single triple junction and with the other ends fixed. We show some results about the existence, uniqueness and, in particular, the global regularity of the flow, following the line of analysis carried on in the last years for the evolution by mean curvature of smooth curves and hypersurfaces.
Classification:  53C44,  53A04,  35K55
@article{ASNSP_2004_5_3_2_235_0,
author = {Mantegazza, Carlo and Novaga, Matteo and Tortorelli, Vincenzo Maria},
title = {Motion by curvature of planar networks},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 3},
number = {2},
year = {2004},
pages = {235-324},
zbl = {1170.53313},
mrnumber = {2075985},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2004_5_3_2_235_0}
}

Mantegazza, Carlo; Novaga, Matteo; Tortorelli, Vincenzo Maria. Motion by curvature of planar networks. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 2, pp. 235-324. http://www.numdam.org/item/ASNSP_2004_5_3_2_235_0/

[1] D. Kinderlehrer - C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci. 11 (2001), 713-729. | MR 1833000 | Zbl 1036.74041

[2] J. Langer, A Compactness Theorem for Surfaces with ${L}_{p}$-Bounded Second Fundamental Form, Math. Ann. 270 (1985), 223-234. | MR 771980 | Zbl 0564.58010

[3] A. Lunardi, “Analytic semigroups and optimal regularity in parabolic problems,” Birkhäuser Verlag, Basel, 1995. | MR 1329547 | Zbl 0816.35001

[4] A. Lunardi - E. Sinestrari - W. Von Wahl, A semigroup approach to the time dependent parabolic initial-boundary value problem, Differential Integral Equations 5 (1992), 1275-1306. | MR 1184027 | Zbl 0758.34048

[5] L. Simon, “Lectures on Geometric Measure Theory”, Australian National University, Proc. Center Math. Anal., Vol. 3, Canberra, 1983. | MR 756417 | Zbl 0546.49019

[6] V. A. Solonnikov, “Boundary value problems of mathematical physics”. VIII, American Mathematical Society, Providence, R.I., 1975. | MR 369867

[7] H. M. Soner, Motion of a set by the curvature of its boundary, J. Differential Equations 101 (1993), 313-372. | MR 1204331 | Zbl 0769.35070

[8] A. Stone, A density function and the structure of singularities of the mean curvature flow, Calc. Var. Partial Differential Equations 2 (1994), 443-480. | MR 1383918 | Zbl 0833.35062

[9] A. Stone, A boundary regularity theorem for mean curvature flow, J. Differential Geom. 44 (1996), 371-434. | MR 1425580 | Zbl 0873.58066