Minimal surfaces and entire solutions of the Allen-Cahn equation

116 Pages·2010·3.24 MB·English

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