The anisotropic total variation flow. (English) Zbl 1109.35061

Summary: We prove existence and uniqueness of solutions of the anisotropic total variation flow when the initial data is an \(L^2\) function, and we give a characterization of such solutions that allows us to find explicit evolutions of sets in the presence of an anisotropy.


35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI


[1] Almgren, F.J., Taylor, J.E.: Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Diff. Geom. 42, 1-22 (1995) · Zbl 0867.58020
[2] Almgren, F.J., Taylor, J.E., Wang, L.-H.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31(2), 387-438 (1993) · Zbl 0783.35002 · doi:10.1137/0331020
[3] Alter, F.: Dualité et sous-differentielle pour une fonctionelle convexe homogéne positive. Unpublished
[4] Amar, M., Belletini, G.: A notion of total variation depending on a metric with discontinuous coefficients. Ann. Inst. Henri Poincaré 11, 91-133 (1994)
[5] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. 2000 · Zbl 0957.49001
[6] Ambrosio, L., Novaga, M., Paolini, E.: Some regularity results for minimal crystals. ESAIM Control Optim. Calc. Var. 8, 69-103 (2002) · Zbl 1066.49021 · doi:10.1051/cocv:2002018
[7] Andreu, F., Caselles, V., Díaz, J.I., Mazón, J.M.: Qualitative properties of the total variation flow. J. Funct. Analysis 188, 516-547 (2002) · Zbl 1042.35018 · doi:10.1006/jfan.2001.3829
[8] Andreu, F., Caselles, V., Mazón, J.M.: A parabolic quasilinear problem for linear growth functionals. Rev. Mat. Iberoamericana 18, 135-185 (2002) · Zbl 1010.35063
[9] Andreu, F., Caselles, V., Mazón, J.M.: Existence and uniqueness of solution for a parabolic quasilinear problem for linear growth functionals with L1 data. Math. Ann. 332, 139-206 (2002) · Zbl 1056.35085 · doi:10.1007/s002080100270
[10] Andreu, F., Caselles, V., Mazón, J.M.: Parabolic Quasilinear equations Minimizing Linear Growth Functionals. Progress in Math. vol 223, 2004. Birkhauser · Zbl 1053.35002
[11] Andreu, F., Caselles, V., Mazón, J.M., Moll, J.S.: The Total Variation Flow with Measure Initial Conditions. Communications in Contemporary Mathematics 6(3), 431-494 (2004) · Zbl 1082.35090 · doi:10.1142/S0219199704001367
[12] Andreu, F., Mazón, J.M., Moll, J.S.: The total variation flow with nonlinear boundary conditions. Asymptotic Analysis. To appear · Zbl 1082.35084
[13] Anzellotti, G.: Traces of bounded vector fields and the divergence theorem. Unpublished
[14] Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. di Matematica Pura ed Appl. IV 135, 293-318 (1983) · Zbl 0572.46023 · doi:10.1007/BF01781073
[15] Bellettini, G., Bouchitté, G., Fragalá, I.: BV functions with respect to a measure and relaxation of metric integral functions. J. Convex Anal. 6(2), 349-366 (2000) · Zbl 0959.49015
[16] Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in ?n. J. Diff. Equations 184, 475-525 (2002) · Zbl 1036.35099 · doi:10.1006/jdeq.2001.4150
[17] Bellettini, G., Caselles, V., Novaga, M.: Explicit solutions of the eigenvalue problem -div ( Du/ |Du |)= u. Preprint, 2003 · Zbl 1162.35379
[18] Bellettini, G., Novaga, M., Paolini, M.: Facet-breaking for three-dimensional crystal evolving by mean curvature. Interfaces and Free Boundaries 1, 39-55 (1999) · Zbl 0934.49023 · doi:10.4171/IFB/3
[19] Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem, part I: first variation and global L??regularity. Arch. Rational Mechanics and Anal. 157, 165-191 (2001) · Zbl 0976.58016 · doi:10.1007/s002050010127
[20] Bellettini, G., Novaga, M., Paolini, M.: On a crystalline variational problem, part II: BV-regularity and structure of minimizers on facets. Arch. Rational Mechanics and Anal. 157, 193-217 (2001) · Zbl 0976.58017 · doi:10.1007/s002050100126
[21] 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) · Zbl 0989.35127 · doi:10.4171/IFB/47
[22] Belloni, M., Kawohl, B.: The pseudo-p-Laplace eigenvalue problem and viscosity solutions as p??. ESAIM Control Optim. Calc. Var. 10, 28-52 (2004) · Zbl 1092.35074 · doi:10.1051/cocv:2003035
[23] Bénilan, Ph., Crandall, M.G., Pazy, A.: Evolution Equations Governed by Accretive Operators. Forthcoming
[24] Bouchitté, G., Dal Maso, G.: Integral representation and relaxation of convex local functionals on BV(?). Ann. Scuola Normale Superiore di Pisa IV 20, 483-533 (1993) · Zbl 0802.49008
[25] Bouchitté, G., Mascarenhas, L., Fonseca, I.: A global method for relaxation. Arch. Rational. Mech. Anal. 145, 51-98 (1998) · Zbl 0921.49004 · doi:10.1007/s002050050124
[26] Brezis, H.: Operateurs Maximaux Monotones. Amsterdam, 1973
[27] Caselles, V., Cambolle, A. Evolution of convex sets by the anisotropic mean curvature including the crystalline case. Preprint. 2004
[28] Chambolle, A. An algorithm for mean curvature motion. Interfaces and Free Boundaries 6(2), 195-218 (2004) · Zbl 1061.35147
[29] Demengel, F., Temam, R.: Convex function of a measure and applications. Indiana Univ. Math. J. 33(5), 673-709 (1984) · Zbl 0581.46036 · doi:10.1512/iumj.1984.33.33036
[30] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Math. 1992 · Zbl 0804.28001
[31] Gagliardo, E.: Caratterizzazione delle trace sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. mat. Padova 27, 284-305 (1957) · Zbl 0087.10902
[32] Giga, M.H., Giga, Y.: Evolving graphs by singular weighted curvature. Arch. Rational Mech. Anal. 141, 117-198 (1998) · Zbl 0896.35069 · doi:10.1007/s002050050075
[33] Giga, M.H., Giga, Y.: Generalized motion by nonlocal curvature in the plane. Arch. Rational Mech. Anal. 159, 295-333 (2001) · Zbl 1004.35075 · doi:10.1007/s002050100154
[34] Giga, M.H., Giga, Y., Kobayashi, R.: Very singular diffusion equations. Proc. of Taniguchi Conf. on Math. Advanced Studies in Pure Mathematics 31, 93-125 (2003) · Zbl 1002.35074
[35] Giga, Y., Paolini, M., Rybka, P.: On the motion by singular interfacial energy. Recent topics in mathematics moving toward science and engineering. Japan J. Indust. Appl. Math. 18(2), 231-248 (2001) · Zbl 0984.35090
[36] Leray, J., Lions, J.L.: Quelques résultats de visik sur le problèmes elliptiques non linéaires par le méthodes de Minty- Browder. Bul. Soc. Math. France 93, 97-107 (1965) · Zbl 0132.10502
[37] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969
[38] Taylor, J.E., Cahn, J.W., Hanwerker, A.: Geometric models of crystal growth. Acta Metall. 40, 1443-1474 (1992) · doi:10.1016/0956-7151(92)90090-2
[39] Ziemer, W.P.: Weakly Differentiable Functions. GTM 120. 1989 · Zbl 0692.46022
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