# American Institute of Mathematical Sciences

September  2016, 11(3): 509-526. doi: 10.3934/nhm.2016007

## Evolution of spoon-shaped networks

 1 Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy

Received  March 2015 Revised  September 2015 Published  August 2016

We consider a regular embedded network composed by two curves, one of them closed, in a convex and smooth domain $\Omega$. The two curves meet only at one point, forming angles of $120$ degrees. The non-closed curve has a fixed end--point on $\partial\Omega$. We study the evolution by curvature of this network. We show that the maximal time of existence is finite and 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.
Citation: Alessandra Pluda. Evolution of spoon-shaped networks. Networks and Heterogeneous Media, 2016, 11 (3) : 509-526. doi: 10.3934/nhm.2016007
##### References:
 [1] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Diff. Geom., 23 (1986), 175-196. [2] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, NJ, 1978. [3] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), 355-379. doi: 10.1007/BF00375607. [4] X. Chen and J. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311. doi: 10.1007/s00208-010-0558-7. [5] K. S. Chou and X. P. Zhu, Shortening complete planar curves, J. Diff. Geom., 50 (1998), 471-504. [6] M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. [7] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom., 26 (1987), 285-314. [8] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diff. Geom., 24 (1986), 153-179. [9] R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Princeton University Press, Princeton, NJ, 137 (1995), 201-222. doi: 10.1016/1053-8127(94)00130-3. [10] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31 (1990), 285-299. [11] G. Huisken, A distance comparison principle for evolving curves, Asian J. Math., 2 (1998), 127-133. doi: 10.4310/AJM.1998.v2.n1.a2. [12] T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, preprint, arXiv:1407.4756. [13] D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), 713-729. doi: 10.1142/S0218202501001069. [14] A. Magni and C. Mantegazza, A note on Grayson's theorem, Rend. Semin. Mat. Univ. Padova, 131 (2014), 263-279. doi: 10.4171/RSMUP/131-16. [15] A. Magni, C. Mantegazza and M. Novaga, Motion by curvature of planar networks II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (2016), 117-144. [16] C. Mantegazza, M. Novaga, A. Pluda and F. Schulze, Evolution of network with multiple junctions, Preprint, 2015. [17] C. Mantegazza, M. Novaga and V. M. Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3 (2004), 235-324. [18] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 13 (1959), 115-162.

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##### References:
 [1] U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions, J. Diff. Geom., 23 (1986), 175-196. [2] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Princeton University Press, Princeton, NJ, 1978. [3] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), 355-379. doi: 10.1007/BF00375607. [4] X. Chen and J. Guo, Motion by curvature of planar curves with end points moving freely on a line, Math. Ann., 350 (2011), 277-311. doi: 10.1007/s00208-010-0558-7. [5] K. S. Chou and X. P. Zhu, Shortening complete planar curves, J. Diff. Geom., 50 (1998), 471-504. [6] M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. [7] M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom., 26 (1987), 285-314. [8] R. S. Hamilton, Four-manifolds with positive curvature operator, J. Diff. Geom., 24 (1986), 153-179. [9] R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Princeton University Press, Princeton, NJ, 137 (1995), 201-222. doi: 10.1016/1053-8127(94)00130-3. [10] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom., 31 (1990), 285-299. [11] G. Huisken, A distance comparison principle for evolving curves, Asian J. Math., 2 (1998), 127-133. doi: 10.4310/AJM.1998.v2.n1.a2. [12] T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, preprint, arXiv:1407.4756. [13] D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Math. Models Methods Appl. Sci., 11 (2001), 713-729. doi: 10.1142/S0218202501001069. [14] A. Magni and C. Mantegazza, A note on Grayson's theorem, Rend. Semin. Mat. Univ. Padova, 131 (2014), 263-279. doi: 10.4171/RSMUP/131-16. [15] A. Magni, C. Mantegazza and M. Novaga, Motion by curvature of planar networks II, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (2016), 117-144. [16] C. Mantegazza, M. Novaga, A. Pluda and F. Schulze, Evolution of network with multiple junctions, Preprint, 2015. [17] C. Mantegazza, M. Novaga and V. M. Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3 (2004), 235-324. [18] L. Nirenberg, On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 13 (1959), 115-162.
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