Abstract
The pioneering work of Ennio De Giorgi and Herbert Federer in the 1950’s on the structure of sets of finite perimeter laid a mathematical foundation for much of modern geometric measure theory. This contribution to continuum thermodynamics is formulated in this tradition. We here wish to examine possible interpretations of Gibbs’s phase rules for equilibrium in fluid systems of several phases in which surface energies are a dominant component of the relevant free energy. We initially consider a simple model problem in some detail and then briefly discuss more elaborate situations.
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References
W.Allard, On the first variation of a varifold ,Ann. of Math. 95 (1972), 417–491.
F.Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints ,Mem. Amer. Math. Soc. No. 165 (1976).
F.Almgren, Deformations and multiple valued functions ,Geometric Measure Theory and the Calculus of Variations, Proceedings of Symposia in Pure Mathematics 44 (1986), 29–130.
F.Almgren, Optimal isoperimetric inequalities ,Indiana Univ. Math. J. 13 (1986), 451–547.
F.Almgren, Spherical symmetrization ,Integral Functionals in the Calculus of Variations, Editrice Tecnico Scientifica, Pisa (to appear).
H.Federer, Geometric Measure Theory ,Springer-Verlag, New-York (1969).
H.Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension ,Bull. Amer. Math. Soc. 76 (1970), 767–771
E.Giusti, Minimal Surfaces and Functions of Bounded Variation ,Monographs in Mathematics 80, Birkhäuser, Boston (1984).
M.Gurtin, W.Williams, W.Ziemer, Geometric measure theory and the axioms of continuum thermodynamics ,Arch. Rational Mech. Anal. 92 (1986), 1–22.
J.Taylor, Boundary regularity for solutions of various capillary and free boundary problems ,Comm. Partial Differential Equations 2 (1977), 323–358.
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Dedicated to Ennio De Giorgi on his sixtieth birthday
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© 1989 Birkhauser Boston
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Almgren, F.J., Gurtin, M.E. (1989). A Mathematical Contribution to Gibbs’s Analysis of Fluid Phases in Equilibrium. In: Colombini, F., Marino, A., Modica, L., Spagnolo, S. (eds) Partial Differential Equations and the Calculus of Variations. Progress in Nonlinear Differential Equations and Their Applications, vol 1. Birkhäuser Boston. https://doi.org/10.1007/978-1-4615-9828-2_2
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DOI: https://doi.org/10.1007/978-1-4615-9828-2_2
Publisher Name: Birkhäuser Boston
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