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By C. Bardos, J. M. Lasry, M. Schatzman

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Going back to a global minimizer u of J, this implies that the blow-down sets εk {u = 0} converge uniformly to a minimal surface. It is a difficult problem to understand how well the original level set {u = 0} can be approximated by a minimal surface. For example in the case when ∂E = {xn = 0} it is not clear wether or not {u = 0} stays at a bounded distance from ∂E. Also, given a nontrivial global minimal surface ∂E it is not evident if there exists at least one minimizer for which {u = 0} stays at a bounded distance from ∂E.

Consider a minimizers u of J which is bounded by −1 and 1. The classical double-well potential W to have in mind is W (s) = 14 (1 − s2 )2 . In order for our ideas to become more transparent, later we will consider the discontinuous potential W (s) = χ(−1,1) (s), where χ(−1,1) represents the characteristic function of the interval (−1, 1). The behavior of u in large domains is given by the behavior of the rescaled functions uε in B1 , x uε (x) := u( ). ε If u minimizes J in the ball B 1ε then uε minimizes the rescaled energy Jε in B1 ˆ ε 1 |∇v|2 + W (v) dx.

We say that a vector ν ∈ RN \ {0} is orthogonal to F at a point y ∈ Ω ∩ F if B(y + ν, |ν|) ⊂ (Ω \ F ) ∪ {y}. In this case we write ν ⊥ F at y. We let F := { y ∈ Ω ∩ F : ∃ ν ⊥ F at y }. 4. We have A(y) ν, ν = 0 ∀y ∈ F , ∀ ν ⊥ F at y. Proof. Every point y ∈ F ∗ is a maximum point of u. 3 one straghtforwardly obtains the classical strong Maximum Principle for Elliptic Operators. 5. If L is ellpitic in Ω, then F ≡ Ω. Proof. By contradiction, assume F ≡ Ω. Then F ∗ = ∅. Thus, there exists at least one point y ∈ F ∗ and a vector ν ⊥ F at y.

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