Modica, Luciano Gradient theory of phase transitions with boundary contact energy. (English) Zbl 0642.49009 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 487-512 (1987). The author studies the asymptotic behavior, as \(\epsilon\) goes to zero, of solutions of the variational problems for the van der Waals-Cahn- Hilliard theory of phase transitions in a fluid. The internal free energy, per unit volume, is assumed to be given by \(\epsilon^ 2\) times the square of the gradient of the density \(\delta\) plus a function of \(\delta\), and the contact energy with the container walls, per unit surface area, is given by \(\epsilon\) times a function of the density \(\delta\). The result is the asymptotic behavior, as \(\epsilon\) goes to zero, of solutions by looking for a variational problem solved by the limit solution. This limit problem exists and agrees with the so-called liquid-drop problem. Reviewer: M.Codegone Cited in 1 ReviewCited in 41 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 49Q05 Minimal surfaces and optimization 76T99 Multiphase and multicomponent flows 80A17 Thermodynamics of continua Keywords:van der Waals-Cahn-Hilliard theory; phase transitions; asymptotic behavior; limit solution; liquid-drop problem PDFBibTeX XMLCite \textit{L. Modica}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 487--512 (1987; Zbl 0642.49009) Full Text: DOI Numdam EuDML References: [1] Anzellotti, G.; Giaquinta, M., Funzioni BV e tracce, Rend. Sem. Mat. Univ. Padova, Vol. 60, 1-22 (1978) · Zbl 0432.46031 [2] Cahn, J. W., Critical Point Wetting, J. Chem. Phys., Vol. 66, 3667-3672 (1977) [3] Cahn, J. W.; Heady, R. B., Experimental Test of Classical Nucleation Theory in a Liquid-Liquid Miscibility Gap System, J. Chem. Phys., Vol. 58, 896-910 (1973) [4] Dal Maso, G.; Modica, L., Nonlinear Stochastic Homogenization, Ann. Mat. Pura Appl., Vol. 144, 4, 347-389 (1986) · Zbl 0607.49010 [5] Giusti, E., The Equilibrium Configuration of Liquid Drops, J. Reine Angew. Math., Vol. 331, 53-63 (1981) · Zbl 0438.76078 [6] Giusti, E., Minimal Surfaces and Functions of Bounded Variation (1984), Birkhäuser Verlag: Birkhäuser Verlag Basel, Boston, Stuttgart · Zbl 0545.49018 [7] Gonzalez, E.; Massari, U.; Tamanini, I., On the Regularity of Boundaries of Sets Minimizing Perimeter with a Volume Constraint, Indiana Univ. Math. J., Vol. 32, 25-37 (1983) · Zbl 0486.49024 [8] Gurtin, M. E., Some Results and Conjectures in the Gradient Theory of Phase Transitions, No. 156 (1985), Institute for Mathematics and Its Applications, University of Minnesota, Preprint [10] Modica, L., Gradient Theory of Phase Transitions and Minimal Interface Criterion, Arch. Rat. Mech. Anal., Vol. 98, 123-142 (1987) · Zbl 0616.76004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.