VISCOSITY SOLUTIONS FOR THE CRYSTALLINE MEAN CURVATURE FLOW WITH A NONUNIFORM DRIVING FORCE TERM

[AGu] S. Angenent and M. E. Gurtin, Multiphase thermomechanics with interfacial structure. II. Evolution of an isothermal interface, Arch. Rational Mech. Anal. 108 (1989), no. 4, 323–391, DOI 10.1007/BF01041068.

[A] G. Anzellotti, Pairings between measures and bounded functions and compen- sated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318 (1984), DOI 10.1007/BF01781073.

[AG] M. Arisawa and Y. Giga, Anisotropic curvature flow in a very thin domain, Indiana Univ. Math. J. 52 (2003), no. 2, 257–281, DOI 10.1512/iumj.2003.52.2099.

[B] G. Barles, A weak Bernstein method for fully nonlinear elliptic equations, Differential Integral Equations 4 (1991), no. 2, 241–262.

[BGN] G. Bellettini, R. Goglione, and M. Novaga, Approximation to driven motion by crys- talline curvature in two dimensions, Adv. Math. Sci. Appl. 10 (2000), no. 1, 467–493.

[BMN] A. Braides, A. Malusa, and M. Novaga, Crystalline evolutions with rapidly oscillating forcing terms (2017), available at https://arxiv.org/abs/1707.03342.

[CIL] 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. (N.S.) 27 (1992), no. 1, 1–67, DOI 10.1090/S0273-0979-1992-00266-5.

[CMNP] A. Chambolle, M. Morini, M. Novaga, and M. Ponsiglione, Existence and uniqueness for anisotropic and crystalline mean curvature flows, J. Amer. Math. Soc. 32 (2019), no. 3, 779–824, DOI 10.1090/jams/919.

[CMP] A. Chambolle, M. Morini, and M. Ponsiglione, Existence and uniqueness for a crys- talline mean curvature flow, Comm. Pure Appl. Math. 70 (2017), no. 6, 1084–1114, DOI 10.1002/cpa.21668.

[CN] A. Chambolle and M. Novaga, Existence and uniqueness for planar anisotropic and crystalline curvature flow, Variational methods for evolving objects (Y. Giga and Y. Tonegawa, eds.), Adv. Stud. Pure Math., vol. 67, Math. Soc. Japan, [Tokyo], 2015, pp. 87–113, DOI 10.2969/aspm/06710087.

[DZS] C. De Zan and P. Soravia, A comparison principle for the mean curvature flow equation with discontinuous coefficients, Int. J. Differ. Equ., posted on 2016, Art. ID 3627896, 6, DOI 10.1155/2016/3627896.

[G] Y. Giga, Surface evolution equations, Monographs in Mathematics, vol. 99, Birkhäuser Verlag, Basel, 2006. A level set approach.

[GG1] M.-H. Giga and Y. Giga, A subdifferential interpretation of crystalline motion under nonuniform driving force, Dynamical systems and differential equations, Vol. I (Spring- field, MO, 1996), 1998, pp. 276–287.

[GG2] , Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal. 141 (1998), no. 2, 117–198.

[GGN] M.-H. Giga, Y. Giga, and A. Nakayasu, On general existence results for one-dimensional singular diffusion equations with spatially inhomogeneous driving force, Geometric par- tial differential equations, CRM Series, vol. 15, Ed. Norm., Pisa, 2013, pp. 145–170.

[GGR] M.-H. Giga, Y. Giga, and P. Rybka, A comparison principle for singular diffusion equations with spatially inhomogeneous driving force for graphs, Arch. Ration. Mech. Anal. 211 (2014), no. 2, 419–453, DOI 10.1007/s00205-013-0676-y. Erratum: 212(2014), 707.

[GGoR1] Y. Giga, P. Górka, and P. Rybka, Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary, Discrete Contin. Dyn. Syst. 26 (2010), no. 2, 493–519, DOI 10.3934/dcds.2010.26.493.

[GGoR2] , A comparison principle for Hamilton-Jacobi equations with discontinu- ous Hamiltonians, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1777–1785, DOI 10.1090/S0002-9939-2010-10630-5.

[GGoR3] , Evolution of regular bent rectangles by the driven crystalline curvature flow in the plane with a non-uniform forcing term, Adv. Differential Equations 18 (2013), no. 3-4, 201–242.

[GOS] Y. Giga, M. Ohnuma, and M.-H. Sato, On the strong maximum principle and the large time behavior of generalized mean curvature flow with the Neumann boundary condition, J. Differential Equations 154 (1999), no. 1, 107–131, DOI 10.1006/jdeq.1998.3569.

[GP1] Y. Giga and N. Požár, A level set crystalline mean curvature flow of surfaces, Adv. Differential Equations 21 (2016), no. 7-8, 631–698.

[GP2] ――, Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to crystalline mean curvature flow, Comm. Pure Appl. Math. 71 (2018), no. 7, 1461–1491, DOI 10.1002/cpa.21752.

[GR1] Y. Giga and P. Rybka, Stability of facets of self-similar motion of a crystal, Adv. Differential Equations 10 (2005), no. 6, 601–634.

[GR2] ――, Stability of facets of crystals growing from vapor, Discrete Contin. Dyn. Syst. 14 (2006), no. 4, 689–706, DOI 10.3934/dcds.2006.14.689.

[GR3] ――, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal. 18 (2008), no. 1, 109–147, DOI 10.1007/s12220-007-9004-9.

[GR4] ――, Faceted crystal grown from solution - a Stefan type problem with a singular interfacial energy, Proceedings of the 4th JSAM-SIMAI Seminar on Industrial and Ap- plied Mathematics (H. Fujita and M. Nakamura, eds.), Gakuto International Series, Mathematical Sciences and Applications 28, Gakkotosho, Tokyo, 2008, pp. 31–41.

[GR5] ――, Facet bending driven by the planar crystalline curvature with a generic nonuniform forcing term, J. Differential Equations 246 (2009), no. 6, 2264–2303, DOI 10.1016/j.jde.2009.01.009.

[LSU] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian).

[L1] G. M. Lieberman, Second order parabolic differential equations, World Scientific Pub- lishing Co., Inc., River Edge, NJ, 1996.

[L2] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original].

[MR1] P. B. Mucha and P. Rybka, A new look at equilibria in Stefan-type problems in the plane, SIAM J. Math. Anal. 39 (2007/08), no. 4, 1120–1134, DOI 10.1137/060677124.

[MR2] ――, A caricature of a singular curvature flow in the plane, Nonlinearity 21 (2008), no. 10, 2281–2316, DOI 10.1088/0951-7715/21/10/005.

[T] J. E. Taylor, Constructions and conjectures in crystalline nondifferential geometry, Differential geometry, Proceedings of the Conference on Differential Geometry, Rio de Janeiro (B. Lawson and K. Tanenblat, eds.), Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 321–336, DOI 10.1111/j.1439- 0388.1991.tb00191.x.