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A Gamma-convergence approach to the Cahn-Hilliard equation. (English) Zbl 1147.35118
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:
 35R35 Free boundary problems for PDE 35B40 Asymptotic behavior of solutions of PDE 80A22 Stefan problems, phase changes, etc. 82C26 Dynamic and nonequilibrium phase transitions (general) 49J45 Optimal control problems involving semicontinuity and convergence; relaxation
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