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2015, 2015(special): 369-378. doi: 10.3934/proc.2015.0369

Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model

1. 

Instituto Matemático Interdisciplinar and Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040-Madrid, Spain

Received  September 2014 Revised  July 2015 Published  November 2015

This paper deals with several qualitative properties of solutions of some parabolic equations associated to the Monge--Ampère operator arising in suitable formulations of the Gauss curvature flow and the worn stone problems.
Citation: Gregorio Díaz, Jesús Ildefonso Díaz. Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model. Conference Publications, 2015, 2015 (special) : 369-378. doi: 10.3934/proc.2015.0369
References:
[1]

F. Bernis, J. Hulshof and J.L. Vázquez, A very singular solution for the dual porous media equation and the asymptotic behaviour of general solutions, J. reine und angew. Math., 435 (1993), 1-31.

[2]

B. Brandolini and J.I. Díaz, Qualitative properties of solutions of some parabolic Monge-Ampère type equations via perimeter symmetrization, work in progress.

[3]

C. Budd and V. Galaktionov, On self-similar blow-up in evolution equations of Monge-Ampère type, IMA J. Appl. Math., (2011), doi: 10.1093/imamat/hxr053.

[4]

D. Chopp, L.C. Evans and H. Ishii, Waiting time effects for Gauss curvature flows, Indiana Univ. Math. J., 48 (1999), 311-334.

[5]

B. Chow, Deforming convex hypersurfaces by the n-th root of the Gaussian curvature, J. Differential Geometry, 23 (1985), 117-138.

[6]

M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.

[7]

P. Daskalopoulos and K. Lee, Fully degenerate Monge-Ampère equations, J. Differential. Equations, 253 (2012), 1665-1691.

[8]

P. Daskalopoulos and O. Savin, $\mathcalC^{1,\alpha}$ regularirty of solutions to parabolic Monge-Ampère equations, Amer. Journal of Math, 134 (4) (2012), 1051-1087.

[9]

G. Díaz and J.I. Díaz, J.I, Finite extinction time for a class of non-linear parabolic equations, Comm. in Partial Equations, 4 (11) (1979), 1213-1231

[10]

G. Díaz and J.I. Díaz, Remarks on the Monge-Ampère equation, Contribuciones Matemáticas en homenaje a Juan Tarrés, UCM, Madrid (2012), 93-125.

[11]

G. Díaz and J.I. Díaz, On the free boundary associated to the stationary Monge-Ampére operator on the set of non strictly convex functions, Discrete Contin. Dyn. Syst., 35 (4) (2015), 1447-1468.

[12]

W.J. Fiery, Shapes of worn stones, Mathematika, 21 (1974), 1-11.

[13]

R. Hamilton, Worn stones with at sides; in a tribute to Ilya Bakelman, Discourses Math. Appl., 3 (1993), 69-78.

[14]

N. Igvida, Solutions auto-similaires pour une equation de Barenblatt, Rev. Mat. Apl., 17, (1991), 21-30.

show all references

References:
[1]

F. Bernis, J. Hulshof and J.L. Vázquez, A very singular solution for the dual porous media equation and the asymptotic behaviour of general solutions, J. reine und angew. Math., 435 (1993), 1-31.

[2]

B. Brandolini and J.I. Díaz, Qualitative properties of solutions of some parabolic Monge-Ampère type equations via perimeter symmetrization, work in progress.

[3]

C. Budd and V. Galaktionov, On self-similar blow-up in evolution equations of Monge-Ampère type, IMA J. Appl. Math., (2011), doi: 10.1093/imamat/hxr053.

[4]

D. Chopp, L.C. Evans and H. Ishii, Waiting time effects for Gauss curvature flows, Indiana Univ. Math. J., 48 (1999), 311-334.

[5]

B. Chow, Deforming convex hypersurfaces by the n-th root of the Gaussian curvature, J. Differential Geometry, 23 (1985), 117-138.

[6]

M.G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.

[7]

P. Daskalopoulos and K. Lee, Fully degenerate Monge-Ampère equations, J. Differential. Equations, 253 (2012), 1665-1691.

[8]

P. Daskalopoulos and O. Savin, $\mathcalC^{1,\alpha}$ regularirty of solutions to parabolic Monge-Ampère equations, Amer. Journal of Math, 134 (4) (2012), 1051-1087.

[9]

G. Díaz and J.I. Díaz, J.I, Finite extinction time for a class of non-linear parabolic equations, Comm. in Partial Equations, 4 (11) (1979), 1213-1231

[10]

G. Díaz and J.I. Díaz, Remarks on the Monge-Ampère equation, Contribuciones Matemáticas en homenaje a Juan Tarrés, UCM, Madrid (2012), 93-125.

[11]

G. Díaz and J.I. Díaz, On the free boundary associated to the stationary Monge-Ampére operator on the set of non strictly convex functions, Discrete Contin. Dyn. Syst., 35 (4) (2015), 1447-1468.

[12]

W.J. Fiery, Shapes of worn stones, Mathematika, 21 (1974), 1-11.

[13]

R. Hamilton, Worn stones with at sides; in a tribute to Ilya Bakelman, Discourses Math. Appl., 3 (1993), 69-78.

[14]

N. Igvida, Solutions auto-similaires pour une equation de Barenblatt, Rev. Mat. Apl., 17, (1991), 21-30.

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