Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T07:10:30.724Z Has data issue: false hasContentIssue false

On radial stationary solutions to a model of non-equilibrium growth

Published online by Cambridge University Press:  16 January 2013

CARLOS ESCUDERO
Affiliation:
Departamento de Matemáticas & ICMAT (CSIC-UAM-UC3M-UCM), Universidad Autónoma de Madrid, E-28049 Madrid, Spain email: cel@icmat.es
ROBERT HAKL
Affiliation:
Institute of Mathematics, AS CR, Žižkova 22, 616 62 Brno, Czech Republic
IRENEO PERAL
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, E-28049 Madrid, Spain
PEDRO J. TORRES
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, E-18071 Granada, Spain

Abstract

We present the formal geometric derivation of a non-equilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of non-equilibrium statistical mechanics.

Type
Papers
Copyright
Copyright © Cambridge University Press 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abdellaoui, B., Dall'Aglio, A. & Peral, I. (2006) Some remarks on elliptic problems with critical growth in the gradient. J. Differ. Equ. 222, 2162.Google Scholar
[2]Abdellaoui, B., Dall'Aglio, A. & Peral, I. (2008) Regularity and nonuniqueness results for parabolic problems arising in some physical models, having natural growth in the gradient. J. Math. Pure Appl. 90, 242269.CrossRefGoogle Scholar
[3]Ambrosetti, A. & Rabinowitz, P. H. (1973) Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349381.Google Scholar
[4]Andrews, B. (1999) Gauss curvature flow: The fate of the rolling stones. Invent. Math. 138, 151161.CrossRefGoogle Scholar
[5]Barabási, A.-L. & Stanley, H. E. (1995) Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, UK.Google Scholar
[6]Blömker, D., Flandoli, F. & Romito, M. (2009) Markovianity and ergodicity for a surface growth PDE. Ann. Probab. 37, 275313.CrossRefGoogle Scholar
[7]Blömker, D. & Romito, M. (2009) Regularity and blow-up in a surface growth model. Dyn. Partial Differ. Equ. 6, 227252.CrossRefGoogle Scholar
[8]Blömker, D. & Romito, M. (2012) Local existence and uniqueness in the largest critical space for a surface growth model, Nonlinear Differ. Equ. Appl. (NoDEA), 19, 365381CrossRefGoogle Scholar
[9]Chow, B. (1991) On Harnack's inequality and entropy for the Gaussian curvature flow. Comm. Pure Appl. Math. 44, 469483.Google Scholar
[10]Ekeland, I. (1974) On the variational principle. J. Math. Anal. Appl. 47, 324353.Google Scholar
[11]Escudero, C. (2008) Geometric principles of surface growth. Phys. Rev. Lett. 101, 196102.Google Scholar
[12]Escudero, C., Hakl, R., Peral, I. & Torres, P. J. preprint. Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth (2012). URL: http://www.ugr.es/~ecuadif/files/strong_singular_2part.pdfGoogle Scholar
[13]GarcíaAzorero, J. Azorero, J. & Peral, I. (1991) Multiplicity of solutions for elliptics problems with critical exponents or with a non-symmetric term. Trans. Am. Math. Soc. 323, 877895.Google Scholar
[14]Halpin-Healy, T. & Zhang, Y.-C. (1995) Kinetic roughening, stochastic growth, directed polymers & all that. Phys. Rep. 254, 215415.Google Scholar
[15]Haselwandter, C. A. & Vvedensky, D. D. (2007) Multiscale theory of fluctuating interfaces: Renormalization of atomistic models. Phys. Rev. Lett. 98, 046102.CrossRefGoogle ScholarPubMed
[16]Hornung, P. (2011) Euler–Lagrange equation and regularity for flat minimizers of the Willmore functional. Comm. Pure Appl. Math. 64, 367441.Google Scholar
[17]Kardar, M., Parisi, G. & Zhang, Y.-C. (1986) Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889892.Google Scholar
[18]Lai, Z.-W. & Das Sarma, S. (1991) Kinetic growth with surface relaxation: Continuum versus atomistic models. Phys. Rev. Lett. 66, 23482351.CrossRefGoogle ScholarPubMed
[19]Lengel, G., Phaneuf, R. J., Williams, E. D., Das Sarma, S., Beard, W. & Johnson, F. G. (1999) Nonuniversality in mound formation during semiconductor growth. Phys. Rev. B 60, R84698472.CrossRefGoogle Scholar
[20]Marsili, M., Maritan, A., Toigo, F. & Banavar, J. R. (1996) Stochastic growth equations and reparametrization invariance. Rev. Mod. Phys. 68, 963983.Google Scholar
[21]Sun, T., Guo, H. & Grant, M. (1989) Dynamics of driven interfaces with a conservation law. Phys. Rev. A 40, R67636766.Google Scholar
[22]Villain, J. (1991) Continuum models of crystal growth from atomic beams with and without desorption. J. Phys. I (France) 1, 1942.CrossRefGoogle Scholar
[23]Willmore, T. J. (1992) A survey on Willmore immersions. In: Dillen, F. & Verstraelen, L. (editors), Geometry and Topology of Submanifolds, IV (Proceedings of the Conference on Differential Geometry and Vision, Leuven, Belgium, 1991), World Scientific, Hackensack, NJ, pp. 1116.Google Scholar
[24]Wio, H. S. & Deza, R. R. (2007) Aspects of stochastic resonance in reaction-diffusion systems: The nonequilibrium-potential approach. Eur. Phys. J. Spec. Top. 146, 111126.Google Scholar