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Preface
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 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.
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Second edition, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[11] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin-New York, 1979. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[19] |
H.-K. Rajni, Aqueous two-phase systems, Mol. Biotechnol, 19 (2001), 269-277. Google Scholar |
[20] |
L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983. Google Scholar |
[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. Google Scholar |
[23] |
M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2008. Google Scholar |
[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.![]() |
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[11] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics 28, Springer-Verlag, Berlin-New York, 1979. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[19] |
H.-K. Rajni, Aqueous two-phase systems, Mol. Biotechnol, 19 (2001), 269-277. Google Scholar |
[20] |
L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 1983. Google Scholar |
[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. Google Scholar |
[23] |
M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2008. Google Scholar |
[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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