Abstract
We introduce the notions of viscosity super- and subsolutions suitable for singular diffusion equations of non-divergence type with a general spatially inhomogeneous driving term. In particular, the viscosity super- and subsolutions support facets and allow a possible facet bending. We prove a comparison principle by a modified doubling variables technique. Finally, we present examples of viscosity solutions. Our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph-like curve.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Angenent S.B., Gurtin M.E.: Multiphase thermomechanics with interfacial structure 2 Evolution of an isothermal interface. Arch. Ration. Mech. Anal. 108, 323–391 (1989)
Bellettini G., Caselles V., Chambolle A., Novaga M.: Crystalline mean curvature flow of convex sets. Arch. Ration. Mech. Anal. 179, 109–152 (2006)
Bellettini G., Goglione R., Novaga M.: Approximation to driven motion by crystalline curvature in two dimensions. Adv. Math. Sci. Appl., 10, 467–493 (2000)
Bellettini G., Novaga M.: Approximation and comparison for non-smooth anisotropic motion by mean curvature in R N. Math. Mod. Meth. Appl. Sc. 10, 1–10 (2000)
Bellettini G., Novaga M., Paolini M.: Facet-breaking for three-dimensional crystals evolving by mean curvature.. Interfaces Free Bound. 1, 39–55 (1999)
Bellettini G., Novaga M., Paolini M.: Characterization of facet breaking for nonsmooth mean curvature flow in the convex case.. Interfaces and Free Boundaries 3, 415–446 (2001)
Bellettini G., Novaga M., Paolini M.: On a crystalline variational problem, part I: first variation and global L ∞-regularity. Arch. Ration. Mech. Anal. 157, 165–191 (2001)
Bellettini G., Novaga M., Paolini M.: On a crystalline variational problem, part II: BV regularity and structure of minimizers on facets. Arch. Ration. Mech. Anal. 157, 193–217 (2001)
Chambolle, A., Novaga, M.: Existence and uniqueness for planar anisotropic and crystalline curvature flow. Advanced Studies in Pure Math. (to appear)
Chen Y.-G., Giga Y., Goto S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33, 749–786 (1991)
Crandall M., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)
Elliott C.M., Gardiner A., Schätzle R.: Crystalline curvature flow of a graph in a variational setting. Adv. Math. Sci. Appl. 8, 425–460 (1998)
Evans L.C., Spruck J.: Motion of level sets by mean curvature, I. J. Differential Geom. 33, 635–681 (1991)
Fukui, T., Giga, Y.: Motion of a graph by nonsmooth weighted curvature. In: Lakshmikantham, V. (ed.) World Congress of Nonlinear Analysts ’92, pp. 47–56. Walter de Gruyter, Berlin, 1996
Giga, Y.: Singular diffusivity—facets, shocks and more. In: Hill, J.M., Moore, R. (eds.) Applied Math Entering the 21st Century. ICIAM 2003 Sydney, pp. 121–138. SIAM, Philadelphia, 2004
Giga, Y.: Surface Evolution Equations—A Level Set Approach. Birkhäuser, Basel, 2006
Giga M.-H., Giga Y.: Evolving graphs by singular weighted curvature. Arch. Ration. Mech. Anal. 141, 117–198 (1998)
Giga, M.-H., Giga, Y.: A subdifferential interpretation of crystalline motion under nonuniform driving force. In: Chen, W.-X., Hu, S.-C. (eds.) In: Proceedings of the International Conference in Dynamical Systems and Differential Equations, Springfield Missouri, 1996, Dynamical Systems and Differential Equations, vol. 1. Southwest Missouri University, Missouri, pp. 276–287, 1998
Giga, M.-H., Giga, Y.: Remarks on convergence of evolving graphs by nonlocal curvature. In: Amann, H., Bandle, C., Chipot, M., Conrad, F., Shafrir, L. (eds.) Progress in Partial Differential Equations, vol. 1. Pitman Research Notes in Mathematics Series, vol. 383, pp. 99–116. Longman, Essex, 1998
Giga M.-H., Giga Y.: Stability for evolving graphs by nonlocal weighted curvature. Commun. Partial Differ. Equ. 24, 109–184 (1999)
Giga M.-H., Giga Y.: Generalized motion by nonlocal curvature in the plane. Arch. Ration. Mech. Anal. 159, 295–333 (2001)
Giga, M.-H., Giga, Y.: A PDE approach for motion of phase-boundaries by a singular interfacical energy. In: Funaki, F., Osada, H. (eds.) Stochastic Analysis on Large Scale Interacting Systems, Advanced Studies in Pure Math., vol. 39, pp. 212–232. Mathematical Society of Japan, 2004
Giga M.-H., Giga Y.: Very singular diffusion equations: second and fourth order problems. Japan J. Ind. Appl. Math. 27, 323–345 (2010)
Giga, M.-H., Giga, Y., Kobayashi, R.: Very singular diffusion equations. In: Maruyama, M., Sunada, T. (eds.) Taniguchi Conference on Mathematics, Nara ’98. Adv. Studies in Pure Math., vol. 31, pp. 93–125. Mathematical Society of Japan, 2001
Giga, M.-H., Giga, Y., Nakayasu, A.: On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force. In: Chambolle, A., et al. (eds.) Proceedings of ERC Workshop on “Geometric Partial Differential Equations”, CRM Series 15, pp. 143–170. Ennio De Giorgi Mathematical Research Center, Pisa, 2013
Giga Y., Goto S., Ishii H., Sato M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40, 443–470 (1991)
Giga Y., Górka P., Rybka P.: Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete Contin. Dyn. Syst. 26, 493–519 (2010)
Giga Y., Gurtin M.-E., Matias J.: On the dynamics of crystalline motions. Jpn J. Ind. Appl. Math. 15, 7–50 (1998)
Giga Y., Rybka P.: Facet bending in the driven crystalline curvature flow in the plane. J. Geom. Anal. 18, 109–147 (2008)
Giga Y., Rybka P.: Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term. J. Differ. Equ. 246, 2264–2303 (2009)
Hörmander, L.: Notions of Convexity. Birkhäuser, Boston, 1994
Kielak K., Mucha P.B., Rybka P.: Almost classical solutions to the total variation flow. J. Evol. Equ. 13, 21–49 (2013)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. Academic Press, New York, 1980
Mucha P.B., Rybka P.: A caricature of a singular curvature flow in the plane.. Nonlinearity 21, 2281–2316 (2008)
Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. Elsevier, Amsterdam, 1987
Roosen, A.R.: Crystalline curvature and flat flow in a diffusion field. Unpublished note, 1994
Schwartz, L.: Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg. Hermann, Paris, 1966
Taylor, J.: Constructions and conjectures in crystalline nondifferential geometry. In: Lawson, B., Tanenblat, K. (eds.) Differential geometry, Proceedings of the Conference on Differential Geometry, Rio de Janeiro, Pitman Monographs in Pure and Applied Math., vol. 52, pp. 321–336. Pitman, London, 1991
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. Otto
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Giga, MH., Giga, Y. & Rybka, P. A Comparison Principle for Singular Diffusion Equations with Spatially Inhomogeneous Driving Force for Graphs. Arch Rational Mech Anal 211, 419–453 (2014). https://doi.org/10.1007/s00205-013-0676-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-013-0676-y