Motion by curvature of planar networks
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 2, pp. 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
Mantegazza, Carlo 1; Novaga, Matteo 2; Tortorelli, Vincenzo Maria 2

1 Scuola Normale Superiore, 56126 Pisa, Italy
2 Dipartimento di Matematica Università degli Studi 56127 Pisa, Italy novagadm.unipi.it
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Mantegazza, Carlo; Novaga, Matteo; Tortorelli, Vincenzo Maria. Motion by curvature of planar networks. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

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