American Institute of Mathematical Sciences

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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

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 & Continuous Dynamical Systems, 2013, 33 (4) : 1231-1246. doi: 10.3934/dcds.2013.33.1231
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show all references

References:
[1]

J. Chem. Phys., 28 (1958), 258-267. Google Scholar

[2]

J. Theor. Biol., 26 (1970), 61-81. doi: 10.1016/S0022-5193(70)80032-7.  Google Scholar

[3]

Nano Letters, 9 (2009), 2807-2812. doi: 10.1021/nl803174p.  Google Scholar

[4]

Nonlinearity, 18 (2005), 1249-1267.  Google Scholar

[5]

American Mathematical Society, Providence, R. I., 2000.  Google Scholar

[6]

Physica D, 240 (2011), 675-693. doi: 10.1016/j.physd.2010.11.016.  Google Scholar

[7]

Phys. Rev. Lett., 65 (1990), 1116-1119. doi: 10.1103/PhysRevLett.65.1116.  Google Scholar

[8]

Zeitshcrift fur naturforschung C, 28 (1973), 693-703. Google Scholar

[9]

J. Membrane Science, 13 (1983), 307-326. doi: 10.1016/S0376-7388(00)81563-X.  Google Scholar

[10]

Solar Energy Materials & Solar cells, 93 (2009), 2003-2007. Google Scholar

[11]

SIAM Journal Math. Anal., 37 (2005), 712-736. doi: 10.1016/j.ijpharm.2004.11.015.  Google Scholar

[12]

Arch. Rational. Mech. Anal., 98 (1987), 123-142.  Google Scholar

[13]

Accounts of Chemical Research, 42 (2009), 1700-1708. doi: 10.1016/j.ssc.2008.12.019.  Google Scholar

[14]

SIAM Math. Analysis, 70 (2009), 369-409.  Google Scholar

[15]

Math. Z., 254 (2006), 675-714.  Google Scholar

[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.   Google Scholar

[17]

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). Google Scholar

[18]

Arch. Rational Mech. Anal., 101 (1988), 209-260.  Google Scholar

[19]

Springer-Verlag, Berlin, 1990.  Google Scholar

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