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

Evolution of spoon-shaped networks

Alessandra Pluda 1,

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.
Keywords: Curvature flow, grain boundaries., singularity formation, networks, Brakke spoon.
Mathematics Subject Classification: Primary: 53C44; Secondary: 53A04, 35K5.
Citation: Alessandra Pluda. Evolution of spoon-shaped networks. Networks & 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.   Google Scholar

[2]

K. A. Brakke, The Motion of a Surface by its Mean Curvature,, Princeton University Press, (1978).   Google Scholar

[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.  doi: 10.1007/BF00375607.  Google Scholar

[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.  doi: 10.1007/s00208-010-0558-7.  Google Scholar

[5]

K. S. Chou and X. P. Zhu, Shortening complete planar curves,, J. Diff. Geom., 50 (1998), 471.   Google Scholar

[6]

M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Diff. Geom., 23 (1986), 69.   Google Scholar

[7]

M. A. Grayson, The heat equation shrinks embedded plane curves to round points,, J. Diff. Geom., 26 (1987), 285.   Google Scholar

[8]

R. S. Hamilton, Four-manifolds with positive curvature operator,, J. Diff. Geom., 24 (1986), 153.   Google Scholar

[9]

R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane,, in Modern Methods in Complex Analysis (Princeton, 137 (1995), 201.  doi: 10.1016/1053-8127(94)00130-3.  Google Scholar

[10]

G. Huisken, Asymptotic behavior for singularities of the mean curvature flow,, J. Diff. Geom., 31 (1990), 285.   Google Scholar

[11]

G. Huisken, A distance comparison principle for evolving curves,, Asian J. Math., 2 (1998), 127.  doi: 10.4310/AJM.1998.v2.n1.a2.  Google Scholar

[12]

T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, preprint,, , ().   Google Scholar

[13]

D. Kinderlehrer and C. Liu, Evolution of grain boundaries,, Math. Models Methods Appl. Sci., 11 (2001), 713.  doi: 10.1142/S0218202501001069.  Google Scholar

[14]

A. Magni and C. Mantegazza, A note on Grayson's theorem,, Rend. Semin. Mat. Univ. Padova, 131 (2014), 263.  doi: 10.4171/RSMUP/131-16.  Google Scholar

[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.   Google Scholar

[16]

C. Mantegazza, M. Novaga, A. Pluda and F. Schulze, Evolution of network with multiple junctions,, Preprint, (2015).   Google Scholar

[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.   Google Scholar

[18]

L. Nirenberg, On elliptic partial differential equations,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 13 (1959), 115.   Google Scholar

show all references

References:
[1]

U. Abresch and J. Langer, The normalized curve shortening flow and homothetic solutions,, J. Diff. Geom., 23 (1986), 175.   Google Scholar

[2]

K. A. Brakke, The Motion of a Surface by its Mean Curvature,, Princeton University Press, (1978).   Google Scholar

[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.  doi: 10.1007/BF00375607.  Google Scholar

[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.  doi: 10.1007/s00208-010-0558-7.  Google Scholar

[5]

K. S. Chou and X. P. Zhu, Shortening complete planar curves,, J. Diff. Geom., 50 (1998), 471.   Google Scholar

[6]

M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves,, J. Diff. Geom., 23 (1986), 69.   Google Scholar

[7]

M. A. Grayson, The heat equation shrinks embedded plane curves to round points,, J. Diff. Geom., 26 (1987), 285.   Google Scholar

[8]

R. S. Hamilton, Four-manifolds with positive curvature operator,, J. Diff. Geom., 24 (1986), 153.   Google Scholar

[9]

R. S. Hamilton, Isoperimetric estimates for the curve shrinking flow in the plane,, in Modern Methods in Complex Analysis (Princeton, 137 (1995), 201.  doi: 10.1016/1053-8127(94)00130-3.  Google Scholar

[10]

G. Huisken, Asymptotic behavior for singularities of the mean curvature flow,, J. Diff. Geom., 31 (1990), 285.   Google Scholar

[11]

G. Huisken, A distance comparison principle for evolving curves,, Asian J. Math., 2 (1998), 127.  doi: 10.4310/AJM.1998.v2.n1.a2.  Google Scholar

[12]

T. Ilmanen, A. Neves and F. Schulze, On short time existence for the planar network flow, preprint,, , ().   Google Scholar

[13]

D. Kinderlehrer and C. Liu, Evolution of grain boundaries,, Math. Models Methods Appl. Sci., 11 (2001), 713.  doi: 10.1142/S0218202501001069.  Google Scholar

[14]

A. Magni and C. Mantegazza, A note on Grayson's theorem,, Rend. Semin. Mat. Univ. Padova, 131 (2014), 263.  doi: 10.4171/RSMUP/131-16.  Google Scholar

[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.   Google Scholar

[16]

C. Mantegazza, M. Novaga, A. Pluda and F. Schulze, Evolution of network with multiple junctions,, Preprint, (2015).   Google Scholar

[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.   Google Scholar

[18]

L. Nirenberg, On elliptic partial differential equations,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 13 (1959), 115.   Google Scholar

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