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On certain convex compactifications for relaxation in evolution problems
Stability analysis for phase field systems associated with crystallinetype energies
1.  Department of Mathematics, Department of Mathematics Faculty of Education, Chiba University, 133 Yayoichō, Inage, Chiba, 2638522, Japan 
References:
[1] 
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Science Publications, 2000. 
[2] 
M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91133. 
[3] 
F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321360. 
[4] 
F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347403. doi: 10.1006/jfan.2000.3698. 
[5] 
V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff International Publishing, 1976. 
[6] 
G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Rat. Mech. Anal., 179 (2006), 109152. doi: 10.1007/s0020500503870. 
[7] 
H. Brézis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espace de Hilbert," NorthHolland, Amsterdam, 1973. 
[8] 
V. Caselles, A. Chambolle, S. Moll and M. Novaga, A characterization of convex calibrable sets in $\mathbbR^N$ with respect to anisotropic norms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 803832. doi: 10.1016/j.anihpc.2008.04.003. 
[9] 
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions", Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992. 
[10] 
Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal., 18 (2008), 109147. doi: 10.1007/s1222000790049. 
[11] 
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser, 1984. 
[12] 
T. Ishiwata, Motion of nonconvex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233253. doi: 10.1007/BF03167521. 
[13] 
N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions, in "Phase Transitions and Hysteresis (Montecatini Terme, 1993)," Lecture Notes in Math., 1584, Springer, Berlin, (1994), 3986. 
[14] 
N. Kenmochi, Y. Mizuta and T. Nagai, Projections onto convex sets, convex functions and their subdifferentials, Bull. Fac. Edu., Chiba Univ., 29 (1980), 1122. 
[15] 
J. L. Lions and E. Magenes, "NonHomogeneous Boundary Value Problems and Applications Vol. I," SpringerVerlag, 1972. 
[16] 
J. S. Moll, The anisotropic total variation flow, Math. Annalen., 332 (2005), 177218. doi: 10.1007/s0020800406240. 
[17] 
M. Novaga and E. Paolini, Stability of crystalline evolutions, Math. Mod. Meth. Appl. Sci., 15 (2005), 117. doi: 10.1142/S0218202505000571. 
[18] 
A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883890. doi: 10.1090/S000299041942078116. 
[19] 
K. Shirakawa, Stability for steadystate solutions of a nonisothermal AllenCahn equation generated by a total variation energy, in "Nonlinear Partial Differential Equations and Their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, (2004), 289304. 
[20] 
K. Shirakawa, Largetime behavior for a phase field system associated with total variation energy, Adv. Math. Sci. Appl., 15 (2005), 127. 
[21] 
K. Shirakawa, Stability for steadystate patterns in phase field dynamics associated with total variation energies, Discrete Contin. Dyn. Syst., 15 (2006), 12151236. doi: 10.3934/dcds.2006.15.1215. 
[22] 
K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions," Series on Advances in Mathematics for Applied Sciences, 71, World Scientific Publishing, (2006), 269288. 
[23] 
K. Shirakawa, Stability analysis for two dimensional AllenCahn equations associated with crystalline type energies, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., 697707. 
[24] 
K. Shirakawa and M. Kimura, Stability analysis for AllenCahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257282. 
[25] 
A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser, Boston, 1996. 
show all references
References:
[1] 
L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Science Publications, 2000. 
[2] 
M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91133. 
[3] 
F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321360. 
[4] 
F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347403. doi: 10.1006/jfan.2000.3698. 
[5] 
V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff International Publishing, 1976. 
[6] 
G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Rat. Mech. Anal., 179 (2006), 109152. doi: 10.1007/s0020500503870. 
[7] 
H. Brézis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espace de Hilbert," NorthHolland, Amsterdam, 1973. 
[8] 
V. Caselles, A. Chambolle, S. Moll and M. Novaga, A characterization of convex calibrable sets in $\mathbbR^N$ with respect to anisotropic norms, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 803832. doi: 10.1016/j.anihpc.2008.04.003. 
[9] 
L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions", Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992. 
[10] 
Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal., 18 (2008), 109147. doi: 10.1007/s1222000790049. 
[11] 
E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser, 1984. 
[12] 
T. Ishiwata, Motion of nonconvex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233253. doi: 10.1007/BF03167521. 
[13] 
N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions, in "Phase Transitions and Hysteresis (Montecatini Terme, 1993)," Lecture Notes in Math., 1584, Springer, Berlin, (1994), 3986. 
[14] 
N. Kenmochi, Y. Mizuta and T. Nagai, Projections onto convex sets, convex functions and their subdifferentials, Bull. Fac. Edu., Chiba Univ., 29 (1980), 1122. 
[15] 
J. L. Lions and E. Magenes, "NonHomogeneous Boundary Value Problems and Applications Vol. I," SpringerVerlag, 1972. 
[16] 
J. S. Moll, The anisotropic total variation flow, Math. Annalen., 332 (2005), 177218. doi: 10.1007/s0020800406240. 
[17] 
M. Novaga and E. Paolini, Stability of crystalline evolutions, Math. Mod. Meth. Appl. Sci., 15 (2005), 117. doi: 10.1142/S0218202505000571. 
[18] 
A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883890. doi: 10.1090/S000299041942078116. 
[19] 
K. Shirakawa, Stability for steadystate solutions of a nonisothermal AllenCahn equation generated by a total variation energy, in "Nonlinear Partial Differential Equations and Their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, (2004), 289304. 
[20] 
K. Shirakawa, Largetime behavior for a phase field system associated with total variation energy, Adv. Math. Sci. Appl., 15 (2005), 127. 
[21] 
K. Shirakawa, Stability for steadystate patterns in phase field dynamics associated with total variation energies, Discrete Contin. Dyn. Syst., 15 (2006), 12151236. doi: 10.3934/dcds.2006.15.1215. 
[22] 
K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions," Series on Advances in Mathematics for Applied Sciences, 71, World Scientific Publishing, (2006), 269288. 
[23] 
K. Shirakawa, Stability analysis for two dimensional AllenCahn equations associated with crystalline type energies, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., 697707. 
[24] 
K. Shirakawa and M. Kimura, Stability analysis for AllenCahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257282. 
[25] 
A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser, Boston, 1996. 
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