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A nonlocal bistable reactiondiffusion equation with a gap
Stability criteria for multiphase partitioning problems with volume constraints
Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece 
We study the stability of partitions involving two or more phases in convex domains under the assumption of at most twophase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign of the principal eigenvalue of the Jacobi operator. We thus derive certain stability criteria, and in particular we recapture the SternbergZumbrun result on the instability of the disconnected phases in the more general setting of several phases.
References:
[1] 
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003. 
[2] 
N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco, G. B. Tanoglu, Analysis of a corner layer problem in anisotropic interfaces, Discrete Cont. Dyn.B, 6 (2006), 237255. 
[3] 
F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. , 4 (1976), ⅷ+199pp. 
[4] 
P. W. Bates, P. C. Fife, The dynamics of nucleation for the CahnHilliard equation, SIAM J. Appl. Math., 53 (1993), 9901008. doi: 10.1137/0153049. 
[5] 
P. W. Bates, P. C. Fife, Spectral comparison principles for the CahnHilliard and phasefield equations and time scales for coarsening, Physica D, 43 (1990), 335348. doi: 10.1016/01672789(90)90141B. 
[6] 
G. Caginalp, P. C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48 (1988), 506518. doi: 10.1137/0148029. 
[7] 
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ Interscience Publishers, New York, 1953. 
[8] 
D. Depner, H. Garcke, Linearized stability analysis of surface diffusion for hypersurfaces with triple lines, Hokkaido Math. J, 42 (2013), 1152. doi: 10.14492/hokmj/1362406637. 
[9] 
S.I. Ei, R. Ikota, M. Mimura, Segregating partition problem in competitiondiffusion systems, Interface Free Bound., 1 (1999), 5780. doi: 10.4171/IFB/4. 
[10] 
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces CBMSNSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. 
[11] 
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, SpringerVerlag, BerlinNew York, 1979. 
[12] 
P. C. Fife, A. Liñán and F. Williams (Editors), Dynamical Issues in Combustion Theory, Proceedings of the Workshop held in Minneapolis, Minnesota, November 1989 The IMA Volumes in Mathematics and its Applications, 35, SpringerVerlag, New York, 1991. 
[13] 
H. Garcke, K. Ito, Y. Kohsaka, Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions, SIAM J. Math. Anal., 36 (2005), 10311056. doi: 10.1137/S0036141003437939. 
[14] 
R. Ikota, E. Yanagida, Stability of stationary interfaces of binarytree type, Calc. Var. PDE, 22 (2005), 375389. doi: 10.1007/s005260040281x. 
[15] 
R. Ikota, E. Yanagida, A stability criterion for stationary curves to the curvaturedriven motion with a triple junction, Differential Integral Equations, 16 (2003), 707726. 
[16] 
W. W. Mullins, Twodimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), 900904. doi: 10.1063/1.1722511. 
[17] 
J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. An., 89 (1985), 119. doi: 10.1007/BF00281743. 
[18] 
J. C. C. Nitsche, Corrigendum to: Stationary partitioning of convex bodies Arch. Ration. Mech. An. 95 (1986), p389. 
[19] 
H.K. Rajni, Aqueous twophase systems, Mol. Biotechnol, 19 (2001), 269277. 
[20] 
L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983. 
[21] 
P. Sternberg, K. Zumbrun, A Poincaré inequality with applications to volumeconstrained areaminimizing surfaces, J. Reine Angew. Math., 503 (1998), 6385. 
[22] 
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces Princeton University Press, Princeton, 1971. 
[23] 
M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2008. 
[24] 
B. White, Existence of leastenergy configurations of immiscible fluids, J. Geom. Anal, 6 (1996), 151161. doi: 10.1007/BF02921571. 
[25]  J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139171755. 
show all references
References:
[1] 
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003. 
[2] 
N. D. Alikakos, P. W. Bates, J. W. Cahn, P. C. Fife, G. Fusco, G. B. Tanoglu, Analysis of a corner layer problem in anisotropic interfaces, Discrete Cont. Dyn.B, 6 (2006), 237255. 
[3] 
F. J. Jr. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. , 4 (1976), ⅷ+199pp. 
[4] 
P. W. Bates, P. C. Fife, The dynamics of nucleation for the CahnHilliard equation, SIAM J. Appl. Math., 53 (1993), 9901008. doi: 10.1137/0153049. 
[5] 
P. W. Bates, P. C. Fife, Spectral comparison principles for the CahnHilliard and phasefield equations and time scales for coarsening, Physica D, 43 (1990), 335348. doi: 10.1016/01672789(90)90141B. 
[6] 
G. Caginalp, P. C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48 (1988), 506518. doi: 10.1137/0148029. 
[7] 
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ Interscience Publishers, New York, 1953. 
[8] 
D. Depner, H. Garcke, Linearized stability analysis of surface diffusion for hypersurfaces with triple lines, Hokkaido Math. J, 42 (2013), 1152. doi: 10.14492/hokmj/1362406637. 
[9] 
S.I. Ei, R. Ikota, M. Mimura, Segregating partition problem in competitiondiffusion systems, Interface Free Bound., 1 (1999), 5780. doi: 10.4171/IFB/4. 
[10] 
P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces CBMSNSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1988. 
[11] 
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, SpringerVerlag, BerlinNew York, 1979. 
[12] 
P. C. Fife, A. Liñán and F. Williams (Editors), Dynamical Issues in Combustion Theory, Proceedings of the Workshop held in Minneapolis, Minnesota, November 1989 The IMA Volumes in Mathematics and its Applications, 35, SpringerVerlag, New York, 1991. 
[13] 
H. Garcke, K. Ito, Y. Kohsaka, Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions, SIAM J. Math. Anal., 36 (2005), 10311056. doi: 10.1137/S0036141003437939. 
[14] 
R. Ikota, E. Yanagida, Stability of stationary interfaces of binarytree type, Calc. Var. PDE, 22 (2005), 375389. doi: 10.1007/s005260040281x. 
[15] 
R. Ikota, E. Yanagida, A stability criterion for stationary curves to the curvaturedriven motion with a triple junction, Differential Integral Equations, 16 (2003), 707726. 
[16] 
W. W. Mullins, Twodimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), 900904. doi: 10.1063/1.1722511. 
[17] 
J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. An., 89 (1985), 119. doi: 10.1007/BF00281743. 
[18] 
J. C. C. Nitsche, Corrigendum to: Stationary partitioning of convex bodies Arch. Ration. Mech. An. 95 (1986), p389. 
[19] 
H.K. Rajni, Aqueous twophase systems, Mol. Biotechnol, 19 (2001), 269277. 
[20] 
L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983. 
[21] 
P. Sternberg, K. Zumbrun, A Poincaré inequality with applications to volumeconstrained areaminimizing surfaces, J. Reine Angew. Math., 503 (1998), 6385. 
[22] 
E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces Princeton University Press, Princeton, 1971. 
[23] 
M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2008. 
[24] 
B. White, Existence of leastenergy configurations of immiscible fluids, J. Geom. Anal, 6 (1996), 151161. doi: 10.1007/BF02921571. 
[25]  J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139171755. 
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