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Existence of equilibria for the Cahn-Hilliard equation via local minimizers of the perimeter. (English) Zbl 0863.35009
This paper is concerned with the Cahn-Hilliard equation: $u_t=\Delta(-\varepsilon^2\Delta u+ F'(u)),\quad x\in\Omega,\quad t>0,\tag{CH}$
${\partial\over\partial n} u={\partial\over\partial n} (-\varepsilon^2\Delta u+ F'(u))=0,\quad x\in\partial\Omega,\quad t>0,$ where $$\Omega$$ is a $$C^3$$ bounded domain in $$\mathbb{R}^2$$, $$\partial/\partial n$$ is the exterior normal derivative, $$\varepsilon$$ is a positive small parameter, and $$F\in C^2(\mathbb{R})$$ is a double, equal well potential (namely, $$F(-1)=F(1)=0$$, $$F(u)>0$$ for $$u\neq\pm1$$). The Cahn-Hilliard equation (CH) serves as a model for phase separation and coarsening phenomena in binary alloys at a temperature at which only two different concentration phases can coexist. The solution $$u^\varepsilon$$ of (CH) is approximately equal to 1 in a subregion $$\Omega^+$$ of $$\Omega$$ and is approximately equal to $$-1$$ in another subregion $$\Omega^-$$ of $$\Omega$$. These two subregions are separated by a thin region containing the interface $$\Gamma_\varepsilon\equiv\{u^\varepsilon=0\}$$. R. L. Pego [Proc. R. Soc. Lond., Ser. A 422, No. 1863, 261-278 (1989; Zbl 0701.35159)], by making use of the method of matched asymptotic expansions, derived formally the following law of motion, the Hele-Shaw problem, for the interface $$\Gamma=\lim_{\varepsilon\to0}\Gamma_\varepsilon$$ (assuming that $$\tau=\varepsilon t$$): $\Delta\mu=0,\quad x\in\Omega\backslash\Gamma,\quad \tau\geq 0,\quad \mu=\alpha_1\kappa,\quad x\in\Gamma,\quad \tau\geq 0,\tag{HS}$
${\partial\mu\over\partial n}=0,\quad x\in\partial\Omega,\quad \tau\geq 0,\quad V=\alpha_2\Biggl[{\partial\mu\over\partial n}\Biggr]_\Gamma,\quad\tau>0,$ where $$\mu$$ is an auxiliary function, $$\alpha_1$$ and $$\alpha_2$$ are positive constants, $$\kappa$$ is the mean curvature of $$\Gamma$$, $$V$$ is the velocity in the direction normal to $$\Gamma$$ and $$[\partial\mu/\partial n]_\Gamma$$ is the jump of the normal derivatives of $$\mu$$ across $$\Gamma$$. In the case when $$\Gamma\cap\partial\Omega\neq\emptyset$$, the above system is supplemented with the condition that $$\Gamma$$ meets $$\partial\Omega$$ orthogonally, which follows from the Neumann boundary condition in (CH). The connection between (HS) and (CH) was shown rigorously by N. D. Alikakos, P. W. Bates and the first author [Arch. Ration. Mech. Anal. 128, No. 2, 165-205 (1994; Zbl 0828.35105)]. Notice that in the two-dimensional case circles contained in $$\Omega$$ or circular arcs intersecting $$\partial\Omega$$ orthogonally are equilibria to (HS). In a series of papers of N. D. Alikakos and G. Fusco [see for example, Commun. Partial Differ. Equations 19, No. 9-10, 1397-1447 (1994; Zbl 0814.35042)], it was shown that the solutions of (CH) whose interfaces are close to circles contained in $$\Omega$$ evolve superslowly in such a way that $$\Gamma$$ moves towards the closest point on $$\partial\Omega$$.
The purpose of the present paper is to prove that there exist equilibria of (CH) with the property that the stationary interfaces $$\Gamma_\varepsilon$$ tend as $$\varepsilon\to0$$ to a circular arc $$\Gamma$$ intersecting $$\partial\Omega$$ orthogonally. The proof depends on the theory of functions of bounded variations and the result of R. V. Kohn and P. Sternberg [Proc. R. Soc. Edinb., Sect. A 111, No. 1/2, 69-84 (1989; Zbl 0676.49011)].

##### MSC:
 35B25 Singular perturbations in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
stationary interfaces; functions of bounded variations
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##### References:
 [1] Bates P.W., Indiana Univ.Math.J 41 pp 637– (1993) [2] DOI: 10.1080/03605309408821059 · Zbl 0814.35042 [3] 1994.Slow dynamics for the Cahn-Hilliard equation in higher space dimensions II: The motion of bubbles.Arch.Rational Mech.Anal., 19 to appear [4] DOI: 10.1063/1.1744102 [5] DOI: 10.1007/BF00695274 · Zbl 0780.35117 [6] DOI: 10.1512/iumj.1983.32.32003 · Zbl 0486.49024 [7] Quart.Appl.Math. 46 pp 301– (1988) [8] Proc.Roy.Soc,Edin pp 69– (1989) [9] Modica L., Arch.Rational Mech.Anal. 98 pp 123– (1986) [10] DOI: 10.1215/S0012-7094-93-07004-4 · Zbl 0796.35056 [11] Proc.Roy.Soc. 422 pp 261– (1989) · Zbl 0701.35159 [12] DOI: 10.1007/BF00253122 · Zbl 0647.49021
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