September  2013, 2(3): 517-530. doi: 10.3934/eect.2013.2.517

On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions

1. 

Dipartimento di Scienze di Base e Applicate, per l'Ingegneria-Sezione di Matematica, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy

Received  December 2012 Revised  April 2013 Published  July 2013

In this paper we consider a geometric motion associated with the minimization of a functional which is the sum of a kinetic part of $p$-Laplacian type, a double well potential $\psi$ and a curvature term. In the case $p=2$, such a functional arises in connection with the image segmentation problem in computer vision theory. By means of matched asymptotic expansions, we show that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear $p$-order equation. The singular limit depends on a complex way on the mean and Gaussian curvatures and the surface Laplacian of the mean curvature of the evolving front.
Citation: Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
References:
[1]

J. W. Cahn, C. M. Elliott and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: Motion by minus the Laplacian of the mean curvature, European J. Appl. Math., 7 (1996), 287-301. doi: 10.1017/S0956792500002369.

[2]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.

[3]

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 207-210. doi: 10.3792/pjaa.65.207.

[4]

E. De Giorgi, Convergence problems for functionals and operators, Prooceedings of the International Meeting on Recent Methods in Nonlinear Analysis, (1979), 131-188.

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1977.

[6]

P. Loreti and R. March, Propagation of fronts in a nonlinear fourth order equation, European J. Appl. Math., 11 (2000), 203-213. doi: 10.1017/S0956792599004131.

[7]

B. Lou, Singular limit of a $p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383. doi: 10.1023/A:1021083015108.

[8]

R. March and M. Dozio, A variational method for the recovery of smooth boundaries, Image and Vision Computing, 15 (1997), 705-712. Available from: http://dx.doi.org/10.1016/S0262-8856(97)00002-4. doi: 10.1016/S0262-8856(97)00002-4.

[9]

L. Modica and S. Mortola, Un esempio di $\Gamma ^-$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.

[10]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.

[11]

C. Pocci, Singular limit of a nonlinear fourth order equation with spatially inhomogeneous terms, submitted.

[12]

B. Sciunzi and E. Valdinoci, Mean curvature properties for $p$-Laplace phase transitions, J. Eur. Math. Soc. (JEMS), 7 (2005), 319-359. doi: 10.4171/JEMS/31.

show all references

References:
[1]

J. W. Cahn, C. M. Elliott and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: Motion by minus the Laplacian of the mean curvature, European J. Appl. Math., 7 (1996), 287-301. doi: 10.1017/S0956792500002369.

[2]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.

[3]

Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 207-210. doi: 10.3792/pjaa.65.207.

[4]

E. De Giorgi, Convergence problems for functionals and operators, Prooceedings of the International Meeting on Recent Methods in Nonlinear Analysis, (1979), 131-188.

[5]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1977.

[6]

P. Loreti and R. March, Propagation of fronts in a nonlinear fourth order equation, European J. Appl. Math., 11 (2000), 203-213. doi: 10.1017/S0956792599004131.

[7]

B. Lou, Singular limit of a $p$-Laplacian reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 110 (2003), 377-383. doi: 10.1023/A:1021083015108.

[8]

R. March and M. Dozio, A variational method for the recovery of smooth boundaries, Image and Vision Computing, 15 (1997), 705-712. Available from: http://dx.doi.org/10.1016/S0262-8856(97)00002-4. doi: 10.1016/S0262-8856(97)00002-4.

[9]

L. Modica and S. Mortola, Un esempio di $\Gamma ^-$-convergenza, Boll. Un. Mat. Ital. B (5), 14 (1977), 285-299.

[10]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.

[11]

C. Pocci, Singular limit of a nonlinear fourth order equation with spatially inhomogeneous terms, submitted.

[12]

B. Sciunzi and E. Valdinoci, Mean curvature properties for $p$-Laplace phase transitions, J. Eur. Math. Soc. (JEMS), 7 (2005), 319-359. doi: 10.4171/JEMS/31.

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