# American Institute of Mathematical Sciences

February 2017, 37(2): 663-683. doi: 10.3934/dcds.2017028

## Stability criteria for multiphase partitioning problems with volume constraints

 Department of Mathematics, University of Athens, Panepistemiopolis, 15784 Athens, Greece

* Corresponding author: A. Faliagas

Received  June 2015 Revised  February 2016 Published  November 2016

Fund Project: The first author was partially supported through the project PDEGE (Partial Differential Equations Motivated by Geometric Evolution), co-financed by the European Union European Social Fund (ESF) and national resources, in the framework of the program Aristeia of the Operational Program Education and Lifelong Learning of the National Strategic Reference Framework (NSRF)

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase 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 Sternberg-Zumbrun result on the instability of the disconnected phases in the more general setting of several phases.

Citation: N. Alikakos, A. Faliagas. Stability criteria for multiphase partitioning problems with volume constraints. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 663-683. doi: 10.3934/dcds.2017028
##### 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 and G. B. Tanoglu, Analysis of a corner layer problem in anisotropic interfaces, Discrete Cont. Dyn.-B, 6 (2006), 237-255. [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 and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008. doi: 10.1137/0153049. [5] P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations and time scales for coarsening, Physica D, 43 (1990), 335-348. doi: 10.1016/0167-2789(90)90141-B. [6] G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48 (1988), 506-518. doi: 10.1137/0148029. [7] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ Interscience Publishers, New York, 1953. [8] D. Depner and H. Garcke, Linearized stability analysis of surface diffusion for hypersurfaces with triple lines, Hokkaido Math. J, 42 (2013), 11-52. doi: 10.14492/hokmj/1362406637. [9] S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interface Free Bound., 1 (1999), 57-80. doi: 10.4171/IFB/4. [10] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces CBMS-NSF 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, Springer-Verlag, Berlin-New 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, Springer-Verlag, New York, 1991. [13] H. Garcke, K. Ito and Y. Kohsaka, Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions, SIAM J. Math. Anal., 36 (2005), 1031-1056. doi: 10.1137/S0036141003437939. [14] R. Ikota and E. Yanagida, Stability of stationary interfaces of binary-tree type, Calc. Var. PDE, 22 (2005), 375-389. doi: 10.1007/s00526-004-0281-x. [15] R. Ikota and E. Yanagida, A stability criterion for stationary curves to the curvature-driven motion with a triple junction, Differential Integral Equations, 16 (2003), 707-726. [16] W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), 900-904. doi: 10.1063/1.1722511. [17] J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. An., 89 (1985), 1-19. 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 two-phase systems, Mol. Biotechnol, 19 (2001), 269-277. [20] L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983. [21] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85. [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 least-energy configurations of immiscible fluids, J. Geom. Anal, 6 (1996), 151-161. doi: 10.1007/BF02921571. [25] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139171755.

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##### 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 and G. B. Tanoglu, Analysis of a corner layer problem in anisotropic interfaces, Discrete Cont. Dyn.-B, 6 (2006), 237-255. [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 and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53 (1993), 990-1008. doi: 10.1137/0153049. [5] P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations and time scales for coarsening, Physica D, 43 (1990), 335-348. doi: 10.1016/0167-2789(90)90141-B. [6] G. Caginalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48 (1988), 506-518. doi: 10.1137/0148029. [7] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. Ⅰ Interscience Publishers, New York, 1953. [8] D. Depner and H. Garcke, Linearized stability analysis of surface diffusion for hypersurfaces with triple lines, Hokkaido Math. J, 42 (2013), 11-52. doi: 10.14492/hokmj/1362406637. [9] S.-I. Ei, R. Ikota and M. Mimura, Segregating partition problem in competition-diffusion systems, Interface Free Bound., 1 (1999), 57-80. doi: 10.4171/IFB/4. [10] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces CBMS-NSF 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, Springer-Verlag, Berlin-New 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, Springer-Verlag, New York, 1991. [13] H. Garcke, K. Ito and Y. Kohsaka, Linearized stability analysis of stationary solutions for surface diffusion with boundary conditions, SIAM J. Math. Anal., 36 (2005), 1031-1056. doi: 10.1137/S0036141003437939. [14] R. Ikota and E. Yanagida, Stability of stationary interfaces of binary-tree type, Calc. Var. PDE, 22 (2005), 375-389. doi: 10.1007/s00526-004-0281-x. [15] R. Ikota and E. Yanagida, A stability criterion for stationary curves to the curvature-driven motion with a triple junction, Differential Integral Equations, 16 (2003), 707-726. [16] W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys., 27 (1956), 900-904. doi: 10.1063/1.1722511. [17] J. C. C. Nitsche, Stationary partitioning of convex bodies, Arch. Ration. Mech. An., 89 (1985), 1-19. 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 two-phase systems, Mol. Biotechnol, 19 (2001), 269-277. [20] L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983. [21] P. Sternberg and K. Zumbrun, A Poincaré inequality with applications to volume-constrained area-minimizing surfaces, J. Reine Angew. Math., 503 (1998), 63-85. [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 least-energy configurations of immiscible fluids, J. Geom. Anal, 6 (1996), 151-161. doi: 10.1007/BF02921571. [25] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9781139171755.
Three-phase partitioning of a set $\Omega$. Subsets painted with the same color contain material of the same phase. Interfaces ($M_{1},\cdots,M_{4}$) and subsets $(\Omega_{1},\cdots,\Omega_{5})$ are identified by successive indexing. Phases are enumerated in the same way: yellow is 1, green is 2, and cyan is 3. An alternative, more convenient for the calculations, identification scheme of subsets is shown in parentheses: the first index corresponds to a phase and the second enumerates the connected components of the subsets occupied by that phase; for example $\Omega_{12}(\equiv\Omega_{4})$ is the second connected component of phase 1. For connected phases (i.e. those occupying a single connected subset) we omit, for brevity, the second index; e.g. $\Omega_{31}\equiv\Omega_{3}$
Disconnected 3-phase partitioning. Differently shaded regions correspond to different phases. The notation is as in Figure 1. The phases are from left to right 1, 2, 3, and 1 (see first index of notation in parentheses). $N_{i}$ is the unit normal field of interface $M_{i}$ (in the indicated orientation)
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