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Zbl 1147.35118
Le, Nam Q.
A Gamma-convergence approach to the Cahn-Hilliard equation. (English)
[J] Calc. Var. Partial Differ. Equ. 32, No. 4, 499-522 (2008). ISSN 0944-2669; ISSN 1432-0835/e

Summary: We study the asymptotic dynamics of the Cahn-Hilliard equation via the ``Gamma-convergence'' of gradient flows scheme initiated by Sandier and Serfaty. This gives rise to an $H^{1}$-version of a conjecture by De Giorgi, namely, the slope of the Allen-Cahn functional with respect to the $H^{- 1}$-structure Gamma-converges to a homogeneous Sobolev norm of the scalar mean curvature of the limiting interface. We confirm this conjecture in the case of constant multiplicity of the limiting interface. Finally, under suitable conditions for which the conjecture is true, we prove that the limiting dynamics for the Cahn-Hilliard equation is motion by Mullins-Sekerka law.

MSC 2000:: *35R35 Free boundary problems for PDE
35B40 Asymptotic behavior of solutions of PDE
80A22 Stefan problems, etc.
82C26 Dynamic and nonequilibrium phase transitions (general)
49J45 Optimal control problems inv. semicontinuity and convergence

Keywords: Cahn-Hilliard equation; Gamma-convergence; limiting dynamics

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Zentralblatt MATH Berlin [Germany]