W, W, W(r) = (1/4)(r 2 1), W (r) = r 3 r

2018 : msjmeeting-2018sep-09i004 ( ), Cahn Hilliard..,.,. 1.,,,. Dirichlet( ), Neumann( ), Robin(, ), 1, 2, 3., dynamic( ).,.,, system( ) transmission problem( ). 2. Cahn Hilliard 0 < T < +, R d (d = 2, 3) :=. Cahn Hilliard( ) [7] 2 (, Milanville[43] ): u µ = 0 in Q := (0, T ), (1) t µ = u + W (u) f in Q, (2) W, W, W(r) = (1/4)(r 2 1), W (r) = r 3 r., u := u(t, x) 1, u = 1 A, u = 1 B, u = 0 Supported by JSPS KAKENHI Grant Number 17K05321. e-mail: fukao@kyokyo-u.ac.jp

., ν ν, µ := µ(t, x) Neumann ν µ = 0 on Σ := (0, T ) Σ u, : u(t)dx = u(0)dx for all t [0, T ]. u Neumann., { } 1 E CH (z) := 2 z 2 dx + W(z) dx, d dt E ( ) CH u(t) + µ(t) 2 dx = 0 for all t [0, T ].., Gal[28] Goldstein Miranville Schimperna[31], Cahn Hilliard : u t + νµ µ = 0 on Σ, (3) µ = ν u u + W (u) f on Σ. (4) (3) (1), (2) (1) (4) Cahn Hilliard., (1) (3) u(t)dx + u(t)d = u(0)dx + u(0)d for all t [0, T ].,. (1) (4) { } { } 1 1 E GMS (z) := 2 z 2 dx + W(z) dx + 2 z 2 d + W (z) d, d dt E ( ) GMS u(t) + µ(t) 2 dx + µ(t) 2 d = 0 for all t [0, T ]..

3., (1) (4),., (GMS). u, µ : Q R u, µ : Σ R, : u µ = 0 t a.e. in Q, (5) µ = u + ξ + π(u) f, ξ β(u) a.e. in Q, (6) u = u, µ = µ, u t + νµ µ = 0 a.e. on Σ, (7) µ = ν u u + ξ + π (u ) f, ξ β (u ) a.e. on Σ, (8) u(0) = u 0 a.e. in, u (0) = u 0 a.e. on, (9), u u, Laplace Beltrami (, Grigor yan[32] )., W := β + π, β ˆβ W, π ˆπ W, W = ˆβ + ˆπ, β, β R 2 R,. β(r) = r 3, π(r) = r, D(β) = R, W W(r) = 1 4 (r2 1) 2 ; β(r) = ln((1 + r)/(1 r)), π(r) = 2cr, D(β) = ( 1, 1), W log W(r) = ( (1 + r) ln(1 + r) + (1 r) ln(1 r) ) cr 2,, c > 0 ( ) ; β(r) = I [ 1,1] (r), π(r) = r, D(β) = [ 1, 1], W W(r) = I [ 1,1] (r) 1 2 r2,, I [ 1,1] [ 1, 1], I [ 1,1] (r) = { 0 if r [ 1, 1], + otherwise., I [ 1,1] I [ 1,1] (, Barbu[6] )., : H := L 2 (), V := H 1 (), W := H 2 () H := L 2 (), V := H 1 (), W := H 2 (), H := H H, V := { z := (z, z ) V V : z = z a.e. on }, W := (W W ) V., : H 0 := {z H : m(z) = 0},, m : H R m(z) := 1 + { zdx + } z d for all z H

, m 0 := m(u 0 )., u 0 := (u 0, u 0 )., V 0 := V H 0., V 0, Poincaré Wirtinger z V 0 := a(z, z) 1/2 (z V 0 )., a(, ) : V V R a(z, z) := z zdx + z z d for all z, z V,., V 0 V 0 F F z, z V 0,V 0 := a(z, z) for all z, z V 0 F : V 0 V 0, C P > 0 C P z 2 V F z, z V 0,V 0 = z 2 V 0 for all z V 0., V 0 H 0 V 0 (Colli F[11, Appendix] )., Kenmochi Niezgódka Paw low[38] (, [37, 39] )., : v := (v, v ) := (u m 0, u m 0 ), v 0 := (v 0, v 0 ) := (u 0 m 0, u 0 m 0 )., : β(z) := (β(z), β (z )), π(z) := (π(z), π (z ))., : (A1) β, β : R 2 R, ˆβ, ˆβ : R [0, + ] ˆβ(0) = ˆβ (0) = 0 β = ˆβ, β = ˆβ ; (A2) π, π : R R Lipshitz ; (A3) D(β ) D(β), ϱ, c 0 > 0 β (r) ϱ β (r) + c0 for all r D(β ),, β (r) := {r β(r) : r = min s β(r) s }; (A4) m 0 intd(β ) ˆβ(u 0 ) L 1 (), ˆβ(u 0 ) L 1 () ; (A5) f := (f, f ) W 1,1 (0, T ; H) f L 2 (0, T ; V ), u 0 V. β β = β. β = β (A3)., (A3), β (A3) β (Calatroni Colli[8])., :

2.1.[11, Theorem 2.2] (A1) (A5), (v, µ, ξ) ( µ(t), z ) v H 1 (0, T ; V 0) L (0, T ; V 0 ) L 2 (0, T ; W ), µ L 2 (0, T ; V ), ξ L 2 (0, T ; H), ξ β(v + m 0 ) a.e. in Q, ξ β (v + m 0 ) a.e. on Σ, v (t), z V 0,V 0 + a( µ(t), z ) = 0 for all z V 0, H = a( v(t), z ) + ( ξ(t) + π(v(t) + m 0 1) f(t), z ) H for all z V, for a.a. t (0, T ), v(0) = v 0 in H 0. : (A5) f H 1 (0, T ; H), f(0) V, u 0 V (H 3 () H 3 ()), β (u 0 ) V. 2.2.[11, Theorem 4.2] (A1) (A4), (A5), (u, µ, ξ) u W 1, (0, T ; V ) H 1 (0, T ; V ) L (0, T ; W ), µ L (0, T ; V ) L 2 (0, T ; W ), ξ L (0, T ; H), ξ β(u) a.e. in Q, ξ β (u ) a.e. on Σ, (5) (9)., : F 1 v (t) + ϕ ( v(t) ) = P ( ξ(t) π ( v(t) + m 0 1 ) + f(t) ) in H 0, ξ(t) β ( v(t) + m 0 1 ) in H, for a.a. t (0, T ), v(0) = v 0 in H 0., ϕ : H 0 [0, + ] 1 z 2 dx + 1 z 2 d if z V 0, ϕ(z) := 2 2 + if z H 0 \ V 0., P : H H 0 P z := z m(z)1, 1 := (1, 1)., z D( ϕ) = W V 0, ϕ(z) = ( z, ν z z ) (Colli F[11, Appendix], [13] )., µ = ϕ(v)+ξ+π(v +m 0 1) f, V µ, µ m(µ) Poincaré Wirtinger Kenmochi Niezgódka Paw low[38] 1.

GMS, [11, Theorem 2.1]., f F Yamazaki[25], Kajiwara[35]., Cherfils Petcu Pierre[10], Cherfils Petcu[9] Furihata Matsuo[27] F Yoshikawa Wada[26]. Liu Wu [42]. 4. GMS u, ξ : Q R, u, ξ : Σ R u ξ = g, ξ β(u) in Q, (10) t u ξ = ξ, ξ β(u ), t + νξ ξ = g on Σ, (11) u(0) = u 0 in, u (0) = u 0 on. (12) β(= β ) : R 2 R (10) (12). k s r r < 0, β(r) := 0 0 r L, β(r) := r q 1 r (q > 0), β(r) := I [0,1] (r) (13) k l (r L) r > L,, (10) (12) Stefan (k l, k s, L ), (q > 1), fast diffusion (0 < q < 1), Hele-Shaw. Stefan Aiki[1, 2, 3] 1990, Laplace Beltrami, Laplace Beltrami,., Igbida, [5, 34]. Hele-Shaw Igbida[33]., GMS : ε (0, 1] u ε t µ ε = 0 in Q, (14) µ ε = ε u ε + ξ ε + π ε (u ε ) f, ξ ε β(u ε ) in Q, (15) u,ε u,ε = (u ε ), µ,ε = (µ ε ), + ν µ ε µ,ε = 0 a.e. on Σ, (16) t µ,ε = ε ν u ε ε u,ε + ξ,ε + π ε (u,ε ) f, ξ,ε β(u,ε ) a.e. on Σ, (17) u ε (0) = u 0,ε a.e. in, u,ε (0) = u 0,ε a.e. on, (18), π ε : R R, β (13) W := ˆβ + ˆπ ε (14) (18) Cahn Hilliard., β(r) = r 3, π ε (r) = εr., :

(A6) g L 2 (0, T ; H 0 ), u 0 H; (A7) ε (0, 1] π ε : R R Lipschitz., c 1 > 0 σ : [0, 1] [0, 1] σ(0) = 0, σ(1) = 1 πε (0) + π ε L (R) c 1 σ(ε)., f L 2 (0, T ; V 0 ) ϕ(f) = g in H 0 a.e. in (0, T ), v 0,ε V 0 v 0,ε + ε ϕ(v 0,ε ) = v 0 := u 0 m 0 1 in H 0, C > 0 v 0,ε 2 H 0 C, ε v 0,ε 2 V 0 C, ˆβ(v 0,ε + m 0 )dx C, ˆβ(v 0,ε + m 0 )d C for all ε (0, 1], v 0,ε v 0 in H 0, u 0,ε := v 0,ε + m 0 1 u 0 in H as ε 0. (10) (12) (14) (18) : 3.1.[21, Theorem 2.1] [22, Theorem 2.1] (A1), (A4), (A6), (A7), (u, ξ) u H 1 (0, T ; V ) L (0, T ; H), ξ L 2 (0, T ; V ), ξ β(u) a.e. in Q, ξ β(u ) a.e. on Σ, u (t), z V,V + u (t), z = V,V + g(t)zdx + ξ(t) zdx + g (t)z d for all z V, for a.a. t (0, T ), u(0) = u 0 in H. ξ (t) z d [21, Theorem 2.2]. Cahn Hilliard., Neumann Colli-F[12] u ε u 2 C([0,T ];V ) + T 0 ( ξε (s) ξ(s), u ε (s) u(s) ) H ds Const.ε 1 2, Robin F Motoda[24] Neumann., [12], Cahn Hilliard β., H 1 () ˆβ, H 1 (),. Neumann, Kenmochi[36], Akagi[4], log Kubo Lu[40]., Robin

Damlamian Kenmochi[15],. Cahn Hilliard, Cahn Hilliard. Damlamian[14] H 1 () [15] 1 (L 1 Barbu[6] ). F Kurima Yokota[23]. 5., Σ Cahn Hilliard : u, µ : Σ R u t µ = ν µ on Σ, (19) µ = u + W (u ) f + ν u on Σ, (20), ν Q µ, u : µ = 0, τ u t u + W (u) = f in Q, (21), τ > 0 0,., Cahn Hilliard,., (quasi-static)., (19) (21). u (t)d = u 0 d for all t [0, T ]. (19) (21) u = u, µ = µ GMS [13].,., Lions[41], Escher[16], Fila Quittner[20], Fila Ishige Kawakami[17, 18, 19], Gal Meyries[29], Cahn Hilliard Garcke Kampmann Rätz Röger[30]. [1] T. Aiki, Two-phase Stefan problems with dynamic boundary conditions, Adv. Math. Sci. Appl., 2 (1993), 253 270. [2] T. Aiki, Multi-dimensional Stefan problems with dynamic boundary conditions, Appl. Anal., 56 (1995), 71 94. [3] T. Aiki, Periodic stability of solutions to some degenerate parabolic equations with dynamic boundary conditions, J. Math. Soc. Japan, 48 (1996), 37 59. [4] G. Akagi, Energy solutions of the Cauchy Neumann problem for porous medium equations, pp.1 10 in Discrete and Continuous Dynamical Systems, supplement 2009, American Institute of Mathematical Sciences, 2009.

[5] F. Andreu, J. M. Mazón, J. Toledo and N. Igbida, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces Free Bound., 8 (2006), 447 479. [6] V. Barbu, Nonlinear differential equations of monotone types in Banach spaces, Springer, London 2010. [7] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 2 (1958), 258 267. [8] L. Calatroni and P. Colli, Global solution to the Allen Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12 27. [9] L. Cherfils and M. Petcu, A numerical analysis of the Cahn Hilliard equation with non-permeable walls, Numer. Math., 128 (2014), 518 549. [10] L. Cherfils, M. Petcu and M. Pierre, A numerical analysis on the Cahn Hilliard equation dynamic boundary conditions, Discrete Contin. Dyn. Syst., 27 (2010), 1511 1533. [11] P. Colli and T. Fukao, Equation and dynamic boundary condition of Cahn Hilliard type with singular potentials, Nonlinear Anal., 127 (2015), 413 433. [12] P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn Hilliard systems, J. Differential Equations, 260 (2016), 6930 6959. [13] P. Colli and T. Fukao, Cahn Hilliard equation on the boundary with bulk condition of Allen Cahn type, preprint arxiv:1803.05314v4 [math.ap] (2018), pp. 1 30, to appear in Adv. Nonlinear Anal. [14] A. Damlamian, Some results on the multi-phase Stefan problem, Comm. Partial Differential Equations, 2 (1977), 1017 1044. [15] A. Damlamian and N. Kenmochi, Evolution equations generated by subdifferentials in the dual space of (H 1 ()), Discrete Contin. Dyn. Syst., 5 (1999), 269 278. [16] J. Escher, Smooth solutions of nonlinear elliptic systems with dynamic boundary conditions, pp. 173 183 in Evolution equations, control theory, and biomathematics, Lecture Notes in Pure and Appl. Math., 155, Dekker, New York, 1994. [17] M. Fila, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure. Appl. Anal., 11 (2012), 1285 1301. [18] M. Fila, K. Ishige and T. Kawakami, Existence of positive solutions of a semilinear elliptic equation with a dynamical boundary condition, Calc. Var. Partial Differential Equations, 54 (2015), 2059 2078. [19] M. Fila, K. Ishige and T. Kawakami, The large diffusion limit for the heat equation with a dynamical boundary condition, preprint arxiv:1806.06308 [math.ap] (2018), pp. 1 19. [20] M. Fila and P. Quittner, Global solutions of the Laplace equation with a nonlinear dynamical boundary condition, Math. Methods Appl. Sci., 20 (1997), 1325 1333. [21] T. Fukao, Convergence of Cahn Hilliard systems to the Stefan problem with dynamic boundary conditions, Asymptot. Anal., 99 (2016), 1 21. [22] T. Fukao, Cahn Hilliard approach to some degenerate parabolic equations with dynamic boundary conditions, pp. 282 291 in: L. Bociu, J-A. Désidéri and A. Habbal (Eds.), System Modeling and Optimization, Springer, Switzerland, 2016. [23] T. Fukao, S. Kurima and T. Yokota, Nonlinear diffusion equations as asymptotic limits of Cahn Hilliard systems on unbounded domains via Cauchy s criterion, Math. Methods Appl. Sci., 41 (2018), 2590 2601. [24] T. Fukao and T. Motoda, Nonlinear diffusion equations with Robin boundary conditions as asymptotic limits of Cahn Hilliard systems, J. Elliptic Parabol. Equ., 4 (2018), 271

291. [25] T. Fukao and N. Yamazaki, A boundary control problem for the equation and dynamic boundary condition of Cahn Hilliard type, pp.255 280 in Solvability, Regularity, Optimal Control of Boundary Value Problems for PDEs, Springer INdAM Series, Vol.22, Springer, Cham, 2017. [26] T. Fukao, S. Yoshikawa and S. Wada, Structure-preserving finite difference schemes for the Cahn Hilliard equation with dynamic boundary conditions in the one-dimensional case, Commun. Pure Appl. Anal., 16 (2017), 1 20. [27] D. Furihata and T. Matsuo, Discrete variational derivative method. A structurepreserving numerical method for partial differential equations, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, 2011. [28] C. Gal, A Cahn Hilliard model in bounded domains with permeable walls, Math. Methods Appl. Sci., 29 (2006), 2009 2036. [29] C. Gal and M. Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc. Lond. Math. Soc. (3), 108 (2014), 1351 1380. [30] H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-cahn Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci., 26 (2016), 1149 1189. [31] G. R. Goldstein, A. Miranville and G. Schimperna, A Cahn Hilliard model in a domain with non-permeable walls, Physica D, 240 (2011), 754 766. [32] A. Grigor yan, Heat kernel and analysis on manifolds, American Mathematical Society, International Press, Boston, 2009. [33] N. Igbida, Hele-Shaw type problems with dynamical boundary conditions, J. Math. Anal. Appl., 335 (2007), 1061 1078. [34] N. Igbida and M. Kirane, A degenerate diffusion problem with dynamical boundary conditions, Math. Ann. 323 (2002), 377 396. [35] N. Kajiwara, Maximal L p regularity of a Cahn Hilliard equation in bounded domains with permeable and non-permeable walls, preprint 2017. [36] N. Kenmochi, Neumann problems for a class of nonlinear degenerate parabolic equations, Differential Integral Equations, 2 (1990), 253 273. [37] N. Kenmochi and M. Niezgódka, Viscosity approach to modelling non-isothermal diffusive phase separation, Japan J. Indust. Appl. Math., 13 (1996), 135 169. [38] N. Kenmochi, M. Niezgódka and I. Paw low, Subdifferential operator approach to the Cahn Hilliard equation with constraint, J. Differential Equations, 117 (1995), 320 354. [39] M. Kubo, The Cahn Hilliard equation with time-dependent constraint, Nonlinear Anal., 75 (2012), 5672 5685. [40] M. Kubo and Q. Lu, Nonlinear degenerate parabolic equations with Neumann boundary condition, J. Math. Anal. Appl., 307 (2005), 232 244. [41] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villas, Paris, 1968. [42] C. Liu and H. Wu, An energetic variational approach for the Cahn Hilliard equation with dynamic boundary conditions: derivation and analysis, preprint arxiv:1710.08318 [math.ap] (2017), pp. 1 68. [43] A. Miranville, The Cahn Hilliard equation and some of its variants, AIMS Mathematics., 2 (2017), 479 544.