Critical points of functionalized Lagrangians

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April 2013, 33(4): 1231-1246. doi: 10.3934/dcds.2013.33.1231

Critical points of functionalized Lagrangians

Keith Promislow ^1, and Hang Zhang ^2,

1.	Department of Mathematics, Michigan State University, East Lansing, MI, 48824
2.	Nico Holding LLC, 222 W. Adams Street, Chicago, IL, 60606, United States

Received September 2011 Revised October 2011 Published October 2012

Abstract
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We present a novel class of higher order energies motivated by the study of network formation in binary mixtures of functionalized polymers and solvent. For a broad class of Lagrangians, we introduce their functionalized form, which is a higher order energy balancing the square of the variational derivative against the original energy. We show that the functionalized energies have global minimizers over several natural spaces of admissible functions. The critical points of the functionalized Lagrangian contain those of the original Lagrangian, however we demonstrate that for a sufficient strength of the functionalization all the critical points of the original Lagrangian are saddle points of the functionalized Lagrangian, and the global minima is a new structure.

Keywords: Functionalized Lagrangian, variational approach, Cahn-Hilliard energy, network formation, minimization..

Mathematics Subject Classification: Primary: 49J45, 35Q74, 35Q5.

Citation: Keith Promislow, Hang Zhang. Critical points of functionalized Lagrangians. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1231-1246. doi: 10.3934/dcds.2013.33.1231

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, 7772-7783.
[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, 7772-7783.
[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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