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Critical points of functionalized Lagrangians
1. | Department of Mathematics, Michigan State University, East Lansing, MI, 48824 |
2. | Nico Holding LLC, 222 W. Adams Street, Chicago, IL, 60606, United States |
References:
[1] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy, J. Chem. Phys., 28 (1958), 258-267. |
[2] |
P. Canham, Minimum energy of bending as a possible explanation of biconcave shape of human red blood cell, J. Theor. Biol., 26 (1970), 61-81.
doi: 10.1016/S0022-5193(70)80032-7. |
[3] |
E. Crossland, M. Kamperman, M. Nedelcu, C. Ducati, U. Wiesner, D. Smilgies, G. Toombes, M. Hillmyer, S. Ludwigs, U. Steiner and H. Snaith, A Bicontinuous double gyroid hybrid solar cell, Nano Letters, 9 (2009), 2807-2812.
doi: 10.1021/nl803174p. |
[4] |
Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18 (2005), 1249-1267. |
[5] |
L. C. Evans, "Partial Differential Equations," American Mathematical Society, Providence, R. I., 2000. |
[6] |
N. Gavish, G. Hayrapetyan, K. Promislow and L. Yang, Curvature driven flow of bi-layer interfaces, Physica D, 240 (2011), 675-693.
doi: 10.1016/j.physd.2010.11.016. |
[7] |
G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphillic systems, Phys. Rev. Lett., 65 (1990), 1116-1119.
doi: 10.1103/PhysRevLett.65.1116. |
[8] |
W. Helfrich, Elastic properties of lipid bilayers - theory and possible experiments, Zeitshcrift fur naturforschung C, 28 (1973), 693-703. |
[9] |
William Hsu and Timothy Gierke, Ion transport and clustering in Nafion perfluorinated membranes, J. Membrane Science, 13 (1983), 307-326.
doi: 10.1016/S0376-7388(00)81563-X. |
[10] |
Kun-Mu Lee, Chih-Yu Hsu, Wei-Hao Chiu, Meng-Chin Tsui, Yung-Liang Tung, Song-Yeu Tsai and Kuo-Chuan Ho, Dye-sensitized solar cells with mirco-porous TiO$_2$ electrode and gel polymer electrolytes prepared by in situ cross-link reaction, Solar Energy Materials & Solar cells, 93 (2009), 2003-2007. |
[11] |
Roger Moser, A higher order asymptotic problem related to phase transition, SIAM Journal Math. Anal., 37 (2005), 712-736.
doi: 10.1016/j.ijpharm.2004.11.015. |
[12] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational. Mech. Anal., 98 (1987), 123-142. |
[13] |
J. Peet, A. Heeger and G. Bazan, "Plastic'' Solar cells: Self-assembly of bulk hetrojunction nanomaterials by spontaneous phase separation, Accounts of Chemical Research, 42 (2009), 1700-1708.
doi: 10.1016/j.ssc.2008.12.019. |
[14] |
K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM Math. Analysis, 70 (2009), 369-409. |
[15] |
Matthias Röger and Reiner Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714. |
[16] |
L. Rubatat, G. Gebel and O. Diat, Fibriallar structure of Nafion: Matching Fourier and real space studies of corresponding films and solutions,, Macromolecules, 2004 (): 7772.
|
[17] |
U. Schwarz and G. Gompper, Bicontinuous surfaces in self-assembled amphiphilic systems, in "Morphology of Condensed Matter: Physics and Geometry of spatially complex systems" (eds. K. R. Mecke and D. Stoyan), 107-151, Springer Lecture Notes in Physics, 600 (2002). |
[18] |
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. |
[19] |
M. Struwe, "Variational Methods," Springer-Verlag, Berlin, 1990. |
show all references
References:
[1] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial energy, J. Chem. Phys., 28 (1958), 258-267. |
[2] |
P. Canham, Minimum energy of bending as a possible explanation of biconcave shape of human red blood cell, J. Theor. Biol., 26 (1970), 61-81.
doi: 10.1016/S0022-5193(70)80032-7. |
[3] |
E. Crossland, M. Kamperman, M. Nedelcu, C. Ducati, U. Wiesner, D. Smilgies, G. Toombes, M. Hillmyer, S. Ludwigs, U. Steiner and H. Snaith, A Bicontinuous double gyroid hybrid solar cell, Nano Letters, 9 (2009), 2807-2812.
doi: 10.1021/nl803174p. |
[4] |
Qiang Du, Chun Liu, Rolf Ryham and Xiaoqiang Wang, A phase field formulation of the Willmore problem, Nonlinearity, 18 (2005), 1249-1267. |
[5] |
L. C. Evans, "Partial Differential Equations," American Mathematical Society, Providence, R. I., 2000. |
[6] |
N. Gavish, G. Hayrapetyan, K. Promislow and L. Yang, Curvature driven flow of bi-layer interfaces, Physica D, 240 (2011), 675-693.
doi: 10.1016/j.physd.2010.11.016. |
[7] |
G. Gompper and M. Schick, Correlation between structural and interfacial properties of amphillic systems, Phys. Rev. Lett., 65 (1990), 1116-1119.
doi: 10.1103/PhysRevLett.65.1116. |
[8] |
W. Helfrich, Elastic properties of lipid bilayers - theory and possible experiments, Zeitshcrift fur naturforschung C, 28 (1973), 693-703. |
[9] |
William Hsu and Timothy Gierke, Ion transport and clustering in Nafion perfluorinated membranes, J. Membrane Science, 13 (1983), 307-326.
doi: 10.1016/S0376-7388(00)81563-X. |
[10] |
Kun-Mu Lee, Chih-Yu Hsu, Wei-Hao Chiu, Meng-Chin Tsui, Yung-Liang Tung, Song-Yeu Tsai and Kuo-Chuan Ho, Dye-sensitized solar cells with mirco-porous TiO$_2$ electrode and gel polymer electrolytes prepared by in situ cross-link reaction, Solar Energy Materials & Solar cells, 93 (2009), 2003-2007. |
[11] |
Roger Moser, A higher order asymptotic problem related to phase transition, SIAM Journal Math. Anal., 37 (2005), 712-736.
doi: 10.1016/j.ijpharm.2004.11.015. |
[12] |
L. Modica, The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational. Mech. Anal., 98 (1987), 123-142. |
[13] |
J. Peet, A. Heeger and G. Bazan, "Plastic'' Solar cells: Self-assembly of bulk hetrojunction nanomaterials by spontaneous phase separation, Accounts of Chemical Research, 42 (2009), 1700-1708.
doi: 10.1016/j.ssc.2008.12.019. |
[14] |
K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM Math. Analysis, 70 (2009), 369-409. |
[15] |
Matthias Röger and Reiner Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714. |
[16] |
L. Rubatat, G. Gebel and O. Diat, Fibriallar structure of Nafion: Matching Fourier and real space studies of corresponding films and solutions,, Macromolecules, 2004 (): 7772.
|
[17] |
U. Schwarz and G. Gompper, Bicontinuous surfaces in self-assembled amphiphilic systems, in "Morphology of Condensed Matter: Physics and Geometry of spatially complex systems" (eds. K. R. Mecke and D. Stoyan), 107-151, Springer Lecture Notes in Physics, 600 (2002). |
[18] |
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. |
[19] |
M. Struwe, "Variational Methods," Springer-Verlag, Berlin, 1990. |
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