Minimal surfaces and entire solutions of the Allen-Cahn equation Manuel del Pino DIM and CMM Universidad de Chile in collaboration with Michal Kowalczyk (Universidad de Chile) and Juncheng Wei (Chinese University of Hong-Kong). The Allen-Cahn Equation (AC) ∆u + u − u3 = 0 in RN Euler-Lagrange equation for the energy functional (cid:90) (cid:90) 1 1 J(u) = |∇u|2+ (1−u2)2 2 4 1 F(u) = (1−u2)2, 4 F ‘‘double-well potential’’: F(u) > 0, u (cid:54)= ±1; F(+1) = 0 = F(−1). ε-version: Ω bounded domain in RN, ε > 0 small. (AC) ε2∆u + u − u3 = 0 in Ω ε ε (cid:90) 1 (cid:90) J (u) = |∇u|2+ (1−u2)2 ε 2 4ε Ω Ω Critical points: continuous realization of phase. Allen-Cahn: Gradient theory of phase transitions. ε (cid:90) 1 (cid:90) J (u) = |∇u|2+ (1−u2)2. ε 2 4ε Ω Ω u = ±1 global minimizers: two phases of a material. For Λ ⊂ Ω, the function (cid:26) u = +1 in Λ u := χ −χ = Γ Λ Ω\Λ u = −1 in Ω\Λ minimizes second integral in J . The inclusion of ε the gradient term prevents the interface from being too wild: The interface: Γ = ∂Λ∩Ω. Nature selects it approximately locally minimal. Critical points of ε (cid:90) 1 (cid:90) J (u) = |∇u|2+ (1−u2)2. ε 2 4ε Ω Ω in H1(Ω), correspond to solutions of ε2∆u+u−u3 = 0 in Ω, ∂ u = 0 on ∂Ω. ν Modica and Mortola (1977): u family of local ε minimizers with J (u ) ≤ C. Then, up to subsequences: ε ε 2√ u → χ −χ in L1, J (v ) → 2HN−1(Γ) as ε → 0, ε Λ Ω\Λ ε ε 3 Perimeter HN−1(Γ) is minimal: Γ = ∂Λ∩Ω is a (generalized) minimal surface. This result was the motivation of the theory of Γ-convergence in the calculus of variations. u ≈ χ − χ ε Λ Ω\Λ [u = 0] ≈ Γ ε Γ is a minimal surface if and only if H = 0, H = Γ Γ mean curvature of Γ ⇐⇒ Γ is stationary for surface area. Formal asymptotic behavior of u ε Assume that Γ is a smooth hypersurface and let ν designate a choice of its unit normal. Local coordinates near Γ: x = y +zν(y), y ∈ Γ, |z| < δ Laplacian in these coordinates: ∆ = ∂ + ∆ − H (y)∂ x zz Γz Γz z Γz := {y +zν(y) / y ∈ Γ}. ∆ is the Laplace-Beltrami operator on Γz acting on Γz functions of y, and H (y) its mean curvature at the Γz point y +zν(y). Let k ,...,k denote the principal curvatures of Γ. 1 N Then N (cid:88) ki H = Γz 1−zk i i=1 For later reference, we expand N (cid:88) H (y) = H (y)+z2|A (y)|2+z3 k3+··· Γz Γ Γ i i=1 where N N (cid:88) (cid:88) H = k , |A |2 = k2 . Γ i Γ i i=1 i=1 (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) mean curvature norm second fundamental form