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Stationary solutions for the Cahn-Hilliard equation. (English) Zbl 0910.35049

The Cahn-Hilliard equation is an accepted macroscopic field-theoretical model of processes such as phase separation in a binary alloy. This equation is studied in a bounded domain without symmetry assumptions. It is assumed that the mean curvature of the boundary has a nondegenerate critical point. It is shown that there exists a spike-like stationary solution whose global maximum lies on the boundary. The derivation is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem.
Reviewer: I.Ginchev (Varna)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Agmon, S., Lectures on Elliptic Boundary Value Problems (1965), Von Nostrand: Von Nostrand Princeton · Zbl 0151.20203
[2] Alikakos, N.; Bates, P. W.; Chen, X., Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rat. Mech. Anal., Vol. 128, 165-205 (1994) · Zbl 0828.35105
[3] Alikakos, N.; Bates, P. W.; Fusco, G., Slow motion for the Cahn-Hilliard equation in one space dimension, J. Diff. Eqns., Vol. 90, 81-134 (1991) · Zbl 0753.35042
[4] Bates, P. W.; Fife, P. C., The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., Vol. 53, 990-1008 (1993) · Zbl 0788.35061
[5] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys., Vol. 28, 258-267 (1958) · Zbl 1431.35066
[6] X. ChenM. Kowalczyk; X. ChenM. Kowalczyk · Zbl 0863.35009
[7] E. N. Dancer; E. N. Dancer · Zbl 0945.35031
[8] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., Vol. 69, 397-408 (1986) · Zbl 0613.35076
[9] Gidas, B.; Ni, W.-M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Mathematical Analysis and Applications, Part A. Mathematical Analysis and Applications, Part A, Adv. Math. Suppl. Studies, Vol. 7A (1981), Academic Press: Academic Press New York), 369-402
[10] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer Berlin · Zbl 0691.35001
[11] Grinfeld, M.; Novick-Cohen, A., Counting stationary solutions of the Cahn-Hilliard equation by transversality arguments, (Proc. Roy. Soc. Edinburgh Sect. A, Vol. 125 (1995)), 351-370 · Zbl 0828.34007
[12] Helffer, B.; Sjöstrand, J., Multiple wells in the semi-classical limit I, Comm. PDE, Vol. 9, 337-408 (1984) · Zbl 0546.35053
[13] Kohn, R. V.; Sternberg, P., Local minimizers and singular perturbations, (Proc. Roy. Soc. Edinburgh Sect. A, Vol. 111 (1989)), 69-84 · Zbl 0676.49011
[14] Lin, C.-S.; Ni, W.-m; Takagi, I., Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eqns., Vol. 72, 1-27 (1988) · Zbl 0676.35030
[15] Lions, J. L.; Magenes, E., (Non-Homogeneous Boundary Value Problems and Applications, Vol I (1972), Springer-Verlag: Springer-Verlag New York/Heidelberg/Berlin) · Zbl 0223.35039
[16] Modica, L., The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., Vol. 107, 71-83 (1989)
[17] Ni, W.-M.; Pan, X.; Takagi, I., Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J., Vol. 67, 1-20 (1992) · Zbl 0785.35041
[18] Ni, W.-M.; Takagi, I., On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., Vol. 41, 819-851 (1991) · Zbl 0754.35042
[19] Ni, W.-M.; Takagi, I., Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., Vol. 70, 247-281 (1993) · Zbl 0796.35056
[20] Ni, W.-M.; Wei, J., On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., Vol. 48, 731-768 (1995) · Zbl 0838.35009
[21] Oh, Y. G., Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class \((V)_a\), Comm. PDE, Vol. 13, 12, 1499-1519 (1988) · Zbl 0702.35228
[22] Oh, Y. G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple-well potentials, Comm. Math. Phys., Vol. 131, 223-253 (1990) · Zbl 0753.35097
[23] Pego, R. L., Front migration in the nonlinear Cahn-Hilliard equation, (Proc. Roy. Soc. London A, Vol. 422 (1989)), 261-278 · Zbl 0701.35159
[24] Peletier, L. A.; Serrin, J., Uniqueness of positive solutions of semilinear equations in \(R^n\), Arch. Rational Mech. Anal., Vol. 81, 181-197 (1983) · Zbl 0516.35031
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