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A note on non lower semicontinuous perimeter functionals on partitions
Evolution of spoon-shaped networks
1. | Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, Pisa, 56127, Italy |
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,, , ().
|
[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. |
show all references
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,, , ().
|
[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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