From Constant Mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions
نویسنده
چکیده
∂νu = 0, on ∂Ω, where ν denotes a unit vector field normal to ∂Ω and where ε λ ∈ R corresponds to the Lagrange multiplier associated to the constraint V (u) = c0 |Ω|. One can also ignore the volume constraint, in which case a critical point would satisfy equation (1.1) with λ = 0. Since classical methods of the calculus of variation apply, there is no difficulty in finding minimizers of Eε. The real issue is the study of the asymptotic behavior of the minimizers (or more generally of the critical points) of Eε as the parameter ε tends to 0. There has been a number of important work on this question over the last two decades and the basic result can be described as follows : Assume that (εk)k>0 tends to 0 and let (uk)k>0 be a sequence of minimizers of Eεk under the constraint V (u) = c0 |Ω|. Then, up to a subsequence, one can assume that (|uk|)k>0 converges a.e. to the constant function 1. In the definition of the energy Eε, the role of the term
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