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AIMS Mathematics

2017, Volume 2, Issue 2: 207-214. doi: 10.3934/Math.2017.2.207
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Surface tension, higher order phase field equations, dimensional analysis and Clairaut’s equation

  • Mathematics Department University of Pittsburgh, Pittsburgh, PA 15260, USA
  • Received: 11 October 2016 Accepted: 28 October 2016 Published: 28 March 2017

  • A higher order phase field free energy leads to higher order differential equations. The surface tension involves L2 norms of higher order derivatives. An analysis of dimensionless variables shows that the surface tension satisfies a Clairaut's equation in terms of the coeffcients of the higher order phase field equations. The Clairaut's equation can be solved by characteristics on a suitable surface in the $\mathbb{R}^N$ space of coeffcients. This perspective may also be regarded as interpreting dimensional analysis through Clairaut's equation. The surface tension is shown to be a homogeneous function of monomials of the coeffcients.

    Citation: Gunduz Caginalp. Surface tension, higher order phase field equations, dimensional analysis and Clairaut’s equation[J]. AIMS Mathematics, 2017, 2(2): 207-214. doi: 10.3934/Math.2017.2.207

    Related Papers:

  • A higher order phase field free energy leads to higher order differential equations. The surface tension involves L2 norms of higher order derivatives. An analysis of dimensionless variables shows that the surface tension satisfies a Clairaut's equation in terms of the coeffcients of the higher order phase field equations. The Clairaut's equation can be solved by characteristics on a suitable surface in the $\mathbb{R}^N$ space of coeffcients. This perspective may also be regarded as interpreting dimensional analysis through Clairaut's equation. The surface tension is shown to be a homogeneous function of monomials of the coeffcients.


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    [7] G. Caginalp and P. Fife, Higher order phase field models and detailed anisotropy, Physical Review B., 34 (1986), 4940-4943.
    [8] L. C. Evans, Partial Differential Equations, American Math Soc. , Providence, RI, 2010.
    [9] N. Goldenfeld, Lectures on phase transitions and renormalization group, Perseus Books, 1992.
    [10] A. Miranville, Higher-order anisotropic Caginalp phase-field systems Mediterr. J. Math., 13 (2016), 4519-4535.
    [11] M. Niezgodka and P. Strzelecki, Free Boundary Problems, Theory and Applications Proceedings of the Zakopane Conference, 1995.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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