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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. Google Scholar

[2]

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

[3]

C. Budd and V. Galaktionov, On self-similar blow-up in evolution equations of Monge-Ampère type,, IMA J. Appl. Math., (2011). Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[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 (1979), 1213. Google Scholar

[10]

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

[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 (2015), 1447. Google Scholar

[12]

W.J. Fiery, Shapes of worn stones,, Mathematika, 21 (1974), 1. Google Scholar

[13]

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

[14]

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

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. Google Scholar

[2]

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

[3]

C. Budd and V. Galaktionov, On self-similar blow-up in evolution equations of Monge-Ampère type,, IMA J. Appl. Math., (2011). Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[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 (1979), 1213. Google Scholar

[10]

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

[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 (2015), 1447. Google Scholar

[12]

W.J. Fiery, Shapes of worn stones,, Mathematika, 21 (1974), 1. Google Scholar

[13]

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

[14]

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

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