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 & 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. doi: 10.1017/S0956792500002369. Google Scholar

[2]

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

[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. doi: 10.3792/pjaa.65.207. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

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. doi: 10.1017/S0956792500002369. Google Scholar

[2]

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

[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. doi: 10.3792/pjaa.65.207. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

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