Modulated Energy Methods for
the Convergence of Allen-Cahn Equation
BICMR, Peking University
刘豫宁
Yuning LIU (NYU Shanghai)
上海纽约大学
yl67@nyu.edu
Table of contents
1
Phase field model for Mean Curvature Flow (MCF)
2
Phase transition for potentials of higher dimensional wells
3
Calibration and its application to minimal surface
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Mean Curvature flow
Consider the MCF {Σt }t>0 parametrized
by ϕt (s) with s being the local coordinate:
Σt
∂t ϕt (s) = κ(ϕt (s), t)ν(ϕt (s), t)
Let r = dΣ (x, t) be the signed distance
function (positive inside). Differentiating
the identity dΣ (ϕt (s) + rν(s, t), t) ≡ r leads to
∇dΣ = ν,
Ω−
t
Ω+
t
ν
∂t dΣ = −∂t ϕt (s) · ν
For fixed t, let π(x) be the projection of x on Σt
and {κi }16i6d−1 are the principal curvatures,
∆dΣ (x) =
d−1
X
i=1
d−1
d−1
X
X
−κi (π(x))
=−
κi − d Σ
κ2i + o(dΣ )
1 − κi (π(x))dΣ
i=1
i=1
Barrier formulation :
(∂t − ∆)dΣ = dΣ |A|2 + o(dΣ )
Example: let Σt = SR(t) be the sphere of radius R(t) centered at 0 with R(0) = R0 ,
dΣ = R(t) − |x|, ν = ∇dΣ = −x/|x|, div ν = ∆dΣ = −(d − 1)/|x|.
p
R02 − 2(d − 1)t.
If Σt evolves by MCF, then R(t) =
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Phase–field approximation
The phase–field models are widely adopted in the description of the evolution of
interfaces in continuum mechanics. They can be constructed to purposely
reproduce a given sharp interface model when the thickness of their diffused
interface, usually denoted by ε, trends to 0.
Other implicit representations: thresholding scheme, level–set method
Typical phase models: Allen–Cahn (L2 gradient flow) → MCF,. Cahn–Hilliard (H 1
gradient flow) → Hele–Shaw. 4-th order Allen–Cahn → Willmore flow.
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Ginzburg-Landau equation
Consider Ginzburg-Landau equation under diffusive scaling (x, t) → (εx, ε2 t):
∂t cε = ∆cε − ε−2 W 0 (cε )
(GL)
R ε
2
2
2
1
where W (c) = (c − 1) . It is the gradient flow of 2 |∇cε | + ε W (cε )dx. It satisfies
Z
Z
d
dµεt = − ε|∂t cε |2 dx
(1)
dt
where dµεt = 2ε |∇cε |2 + 1ε W (cε ) dx. Major challenge: ∇cε is not bounded in any
Lp space due to concentration.
ε→0
Modica-Mortola ’77, Bronsard-Kohn
’91: µεt −−−→ σHd−1 Σt under
R cε (x,t) p
Γ-convergence. ψε (x, t) = 0
2W (z) dz is bounded in BV class.
De Mottoni-Schatzman ’95: local, asymptotic expansion cε = θ( dεΣ ) + o(ε2 ).
Evans, Soner, Souganidis ’92: global convergence to viscosity sol. to
∂t u
∇u
= div |∇u|
: (∂t − ∆)dΣ > 0 in {x ∈ Ω : dΣ (x, t) > 0} as viscosity solu.
|∇u|
Ilmanen ’93, Chen ’96, Röger-Schätzle ’06: global, convergence to varifold
solution
BrakkeR’78: convergence of the
law:
localized energy
2
R
R
ε
0
0
d
1
1
φ(x)dµ
=
ε
−∆c
+
W
(c
)
∇c
·
∇φ
−
ε
∆c
φ.
ε
ε
ε − ε2 W (cε )
ε
2
t
dt
ε
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Asymptotical Analysis
Inner solution is the expansion of cε near the interface in a stretched variable
z = dΣ (x, t)/ε, which is introduced to relax the sharp transition of cε near the
interface. Outer solution determines the boundary condition of the inner solution at
z = ±∞. We use the Ansatz
cA (x, t) = c0 ( dεΣ , x, t) + εc1 + · · · near Σt
(inner solution)
and look for cA solving Allen-Cahn equations up to a tail:
∂t dΣ
∂z c0
ε
|
{z
+ ∂t c0 ≈ ∆dε Σ ∂z c0 +
} |
1 2
∂ c
ε2 z 0
=∂t cA
+ ∆c0 + · · · −
{z
1
W 0 (c0 )
ε2
1
=∆cA − 2 W 0 (cA )
ε
• O(ε−2 ) : we choose c0 = θ( dεΣ ), the optimal profile:
z
−θ00 (z) + W 0 (θ(z)) = 0, ∀z ∈ R,
⇒ θ(z) = tanh( √ )
θ(0) = 0, θ(±∞) = ±1.
2
• O(ε−1 ): (∂t − ∆)dΣ = 0 on Σt leads to MCF.
(∂t − ∆)c0 +
=
1
W 0 (c0 )
ε2
∂t dΣ − ∆dΣ dΣ 0
θ
dΣ
ε
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= (∂t dΣ − ∆dΣ ) θ0 ( dεΣ )
√
dΣ
≈ ε in L2 (Σt (δ))
ε
}
Brakke’s inequality
Let {Σt }t>0 be a family of (d − 1)-closed surface in Rd with normal velocity V , then
Z
Z
d
f (x) dHd−1 =
(V · ∇⊥ f − f V · H) dHd−1 .
(2)
dt Σt
Σt
Brakke’s lemma
Assume V be normal, then V = H is equivalent to the Brakke’s inequality:
Z
Z
d
f (x) dHd−1 6
(H · ∇f − f H · H) dHd−1
dt Σt
Σt
(3)
Z
Subtracting (2) from (3):
∇f · (H − V ) − f H · (H − V )
06
Σt
0
To blow-up at x0 ∈ Σt , we set w = H − V and replace f by λ2−d f ( x−x
).
λ
Z
0
0 6 λ1−d
(∇f )( x−x
) · w(x)dx + O(λ)
λ
Σt
Z
=
λ→0
Σt −x0
λ
Z
∇f (y) · w(x0 + λy)dy + O(λ) −−−→
(∇Σ f + n∂r f ) · w(x0 ).
Tx0 Σt
Since Tx0 Σt ∼
= Rd−1 and w is normal, this implies H = V . The book of Tonegawa’ 2019
gives a friendly introduction of Brakke flow.
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(4)
Phase-field approximation of Brakke’s inequality
R
Writing µt = Hd−1 Σt and µεt (A) , A ε|∇cε |2 + 1ε W (cε ) dx.
The Brakke’s inequality for MCF writes
Z
Z
Z
d
f dµt 6 H · ∇⊥ f dµt − |H|2 f dµt ,
dt
2
Z Z
Z 1
d
1
f dµεt = ε −∆cε + 2 W 0 (cε ) ∇cε · ∇f dx − ε ∆cε − 2 W 0 (cε ) f dx
dt
ε
ε
Based on Ilmanen ’93, Chen ’96, finally Röger-Schätzle ’06 proved the following
property under d = 2, 3: If µεt → µt weakly as Radon measure, then
the lower-semicontinuity property holds:
2
Z
Z 1 0
2
f |H| dµt 6 lim inf ε ∆cε − 2 W (cε ) f dx
ε→0
ε
for f conti. Here a localization is needed to insert f in the inequality.
µt is (d − 1)-integral, i.e. µt = θt Hd−1 Σt with Σt being a countably
(d − 1)-rectifiable, Hd−1 -measurable set and θ ∈ L1loc (Hd−1 Σt ).
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Modulated energy method: Barrier
Define dµεt =
ε
|∇cε |2
2
+ 1ε W (cε ) dx
ε
T=
|∇cε |2 +
2
and the energy stress tensor
1
W (cε ) Id − ε∇cε ⊗ ∇cε
ε
For any test function φ(x, t), one can derive
Z
Z
Z
d
φ(x, t) dµεt = ∂t φ dµεt + ε −∆cε + ε12 W 0 (cε ) ∇cε ·∇φ dx
dt
|
{z
}
div T
Z
2
− ε ∆cε − ε12 W 0 (cε ) φ dx
Motivated by Lin ’96, Jerrard-Soner ’98 and Ilmanen ’93, we choose φ = 12 dΣ (x, t)2
in Σt (δ) and constant outside Σt (2δ):
∂t φ − ∆φ + 1 . φ in Σt (2δ),
(5)
This function will cut-off the singularity at Σt and lead to
Z Z
Z d
ε
2
W (cε )
1
ε)
φ 2ε |∇cε |2 + W (c
dx
|∇c
|
−
dx .
.
φ
dµ
+
ε
ε
2
t
ε
2
ε
dt
|
|
{z
}
{z
}
dµε
t
dζtε
Modica ’85, Ilmanen ’93 use MP to show that the discrepancy dζtε preserves
negativity. This implies the strong convergence of cε in L2loc (Rd \Σt ).
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Table of contents
1
Phase field model for Mean Curvature Flow (MCF)
2
Phase transition for potentials of higher dimensional wells
3
Calibration and its application to minimal surface
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Keller–Rubinstein–Sternberg problem
We consider the vectorial (or matrix-valued) Ginzburg–Landau equation with
potentials of higher dimensional wells.
∂t Qε = ∆Qε − ε−2 W 0 (Qε )
Qε : Ω ⊂ Rd 7→ Rn
(GL)
where Argmin W (Q) = M , M+ t M− ⊂ Rn are two disjoint submanifolds.
Conjecture of Rubinstein–Sternberg–Keller ’89
For well-prepared initial data, the gradient of Qε will be concentrated on a moving
interface Σt governed by MCF and as ε ↓ 0,
Z X ±Z
1
ε
|∇Qε |2 + W (Qε ) dx ≈ σHd−1 (Σt ) + ε
σ
|∇Q± |2 dx
(6)
±
2
ε
Ω
Ω
±
t
±
Moreover Q± : Ω±
satisfy harmonic map heat flow.
t × (0, T ) 7→ M
1
Lin–Pan–Wang ’13: Establishing (6) in the steady case, M± hypersurfaces,
Σt = Σ unique smooth stable Minimum surface. Q± |∂Ω± form a minimal pair.
t
2
Fei–Wang–Zhang-Zhang ’18: dynamic case with M = CP2 t {05 } ⊂ R5 by
Hilbert expansion. Laux–L. ’21 by modulated energy.
3
Fei–Lin–Wang–Zhang ’ arxiv : dynamic case with M = O(n) by Hilbert
expansion.
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Isotropic–Nematic phase transition
Consider the matrix-valued GL equation
∂t Qε = ∆Qε −
1
W 0 (Qε )
ε2
where Qε : Ω → Q where
Q , {Q ∈ R3×3 : Q = QT , tr Q = 0},
2
W (Q) = a2 tr(Q2 ) − 3b tr(Q3 ) + 4c tr(Q2 )
is the Landau’s
expansion of the (molecular) free energy
and a, b, c > 0 are temperature dependent
constants. At the critical temperature
√
b2 = 27ac, s− = 0, s+ =
b+
b2 −24ac
.
4c
Argmin W (Q) = M := {s(n ⊗ n − 31 I3 ) : n ∈ S2 , s = s± }.
Z
where Q(x) =
f (x, p)(p ⊗ p − 31 I) dH2 (p)
S2
is the normalized second moment of the distribution f (x, ·) for fixed x ∈ Ω.
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Modulated energy method: Calibration
The quasi-distance is define by 1
Z 1
dW (Q) , inf
2W (γ(t))|γ 0 (t)|dt
γ
(7)
0
where γ is any curve satisfying γ(1) = Q and
Ω−
t
γ(0) ∈ M+ = {s+ (n ⊗ n − I3 /3) : n ∈ S2 }
ξ
Moreover, we define ψε (x, t) , dW
ε ◦ Qε (x, t)
W
and then dW
ε (Q) , (φε ∗Q d )(Q),
Ω
by an isotropic mollifier φε in Q. Motivated
Σ
by Fischer–Laux–Simon ’21 on the scalar GL
equation, we introduce
Z 1
ε
|∇Qε |2 + W (Qε ) + ε5 + (div ξ)ψε dx,
Eε [Qε |Σ](t) ,
2
ε
Ω
t
+
t
2
where ξ is an extension
pof the normal of Σt whose modulus decay in dΣ .
W
Note that |∂d (·)| = 2W (·) a.e. in Q. So (Q, 2W )/M is a (quotient) metric space
with distance function (7).
1
In the scalar case dW (c) =
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Rcp
2W (s) ds.
0
Modulated energy method: Calibration
√
Recall s− = 0, s+ =
b+
b2 −24ac
,
4c
Z Eε [Qε |Σ](t) ,
Ω
ψε (x, t) , dW
ε ◦ Qε (x, t) and the quasi-distance
ε
1
|∇Qε |2 + W (Qε ) + ε5 − ξ · ∇ψε dx
2
ε
Laux–L. ’ 21
Let Σt be a smooth MCF on [0, T ] and εkQin
ε kL∞ (Ω) + Eε [Qε |Σ](0) . ε,
then we have
Eε [Qε |Σ](t) . ε for any t ∈ [0, T ].
Moreover, up to the extraction of a subsequence ε = εk ,
k→∞
1
±
Qεk −
−−−
→ Q = s± n ⊗ n − 31 I3 weak-star in L∞
t (Hloc (Ωt ))
2
where n : Ω+
t 7→ S is a weak solution of harmonic heat flow with 0-Neumann
boundary condition on ∂Ω+
t = Σt .
By L.’ arxiv, one can derive, for some β > 0, a quantitative estimate
kψε − σ1Ω+ kL1 (Ω) . εβ
t
under an assumption on the initial data. This follows from a Gröwall inequality,
rather than the BV compactness of ψε .
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Calibration and discrepancy: scalar case
In the scalar case ψε (x, t) =
R cε (x,t) p
2W (z) dz, nε =
0
∇ψε
,
|∇ψε |
and ξ satisfies
T
∂t ξ + (H · ∇)ξ + (∇H) ξ = O(dΣ ).
We can write the modulated energy by:
ξ·∇ψε
Z
ε
|∇cε |2 +
Ω 2
Z
ε
=
|∇cε |2 +
2
Ω|
Eε [cε |Σ](t) =
By
z
}|
{
p
1
∇ψε
W (cε ) − ξ ·
|∇cε | 2W (cε )
ε
|∇ψε |
Z
p
1
(1 − ξ · nε ) |∇ψε |
W (cε ) − |∇cε | 2W (cε ) +
{z
}
ε
Ω|
{z
}
&min(d2
Σ ,1)
discrepancy2
ε|∇cε |2 = |∇ψε | +
√
ε|∇cε |
√
|
1 p
ε|∇cε | − √
2W (cε )
ε
{z
}
discrepancy
and the Cauchy–Schwarz inequality, one
Z can show
Eε [cε |Σ](t) &
ε|∇cε |2 min(d2Σ , 1)
Ω
One can show Eε [cε |Σ](t) satisfies a Grönwall inequality and thus with appropriate
initial data we have Eε [cε |Σ](t) . ε. This recovers the estimate of Lin ’96 when
co-dim = 1 without employing MP.
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Calibration and discrepancy: vectorial case
In the vectorial case, Eε [Qε |Σ](t) also enjoys a Grönwall inequality. Introducing an
orthogonal projection in the direction ∂dW
ε (Qε ):
∂dW
∂dW
ε (Qε )
ε (Qε )
ΠQε ∂xi Qε = ∂xi Qε :
, with x0 , t.
W
|∂dε (Qε )| |∂dW
ε (Qε )|
Z 1
1
By Grönwall’s inequality
|∇Qε |2 + 2 W (Qε ) (1 − ξ · nε )
2
ε
Ω
d Z
X
+
|∂xi Qε − ΠQε ∂xi Qε |2 . ε−1 Eε [Qε |Σ](t) . 1.
i=1
Ω
This leads to the modulated
estimate of ∇Qε without employing MP:
Z
|∇Qε |2 min(d2Σ , 1) . ε−1 Eε [Qε |Σ](t)
Ω
Define the phase-field analogy of the mean curvature vector by
∇Qε
, one can derive a differential inequality:
Hε = − ε∆Qε − ε−1 ∂F (Qε ) : |∇Q
ε|
Z 2
d
1
Eε [Qε |I] +
ε2 |∂t Qε |2 − |Hε |2 + Hε − ε|∇Qε |H
dx
dt
2ε
Z
2
1
+
ε∂t Qε − (div ξ)∂dW
ε (Qε ) dx . Eε [Qε |I].
2ε
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Modulated estimates
To derive the compactness of ∂t Qε , we use
Z
|∂t Qε + H · ∇Qε |2
R
Ω
|∇Qε |2 min(d2Σ , 1) . 1 and
ΩT
GL
6ε
−2
Z
ε2 |∂t Qε |2 − |Hε |2 + Hε − ε|∇Qε |H
2
.1
ΩT
Note that ∂t Qε + H · ∇Qε behaviors like ε−1 (∂t dΣ + H · ∇dΣ ).
Later we show [ΠQε ∂xi Qε , Qε ] = 0 for 0 6 i 6 d. So we have
k→∞
[∂i Qε , Qε ] = [∂i Qε − ΠQε ∂i Qε , Qε ] −
−−−
→ S̄i (x, t) weakly in L2t L2x (Ω)
By Chen’s formulation of GL,
[∂t Qε − ΠQε ∂t Qε , Qε ] = ∂xi [∂xi Qε − ΠQε ∂xi Qε , Qε ].
ε→0
Converge locally Qε −−−→ Q = s± (n ⊗ n − 31 I3 ) ∈ L2t L2loc (Ω±
t ) and n is a weak
solution of harmonic heat flow into S2 with 0 Neumann bdy condition on Σt :
Z TZ
Z TZ
∂t n ∧ n · ϕ +
(∂j n ∧ n) · ∂j ϕ = 0, ∀ϕ ∈ Cc1 (Ω)
0
Ω+
t
0
Ω+
t
Note that if n is regular enough, we have
∂t n − ∆n = |∇n|2 n in Ω+
t ,
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∂ν n|Σt = 0
Symmetry of the correctors
For any 0 6 i 6 d, we prove the commutator identity
[ΠQε ∂xi Qε , Qε ] = 0, ∀x ∈ Ω
W
Recall the orthogonal projection ΠQε ∂xi Qε k ∂dW
ε (Qε ). As dε (Q) is isotropic (i.e.
only depends on the eigenvalues of Q), there exists a smooth symmetric function
gε (λ1 , λ2 , λ3 ) such that
dW
ε (Q) = gε (λ1 (Q), λ2 (Q), λ3 (Q))
Let Q0 be a matrix having distinct eigenvalues, then λi (Q) and the eigenvectors
ni (Q) are real-analytic functions of Q near Q0 , and then by chain rule
3
3
X
X
∂gε
∂gε ∂λk
=
nk (Q) ⊗ nk (Q)
∂dW
ε (Q) =
∂λk ∂Q
∂λk
k=1
k=1
in a neighborhood of Q0 . We also have
3
X
Q=
λk (Q)nk (Q) ⊗ nk (Q)
k=1
∂dW
ε (Q), Q
So we have
= 0, holds in a neighborhood of Q0 having distinct
eigenvalues, and thus for a general Q by continuity of ∂dW
ε (Q).
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Boundary conditions on the free boundary
For any vector field ϕ ∈ Cc1 (Ω, R3 ), let Φ be the corresponding anti-symmetric
matrix-valued function. Apply the anti-symmetric product [·, Qε ] to the Qε equ:
Z TZ
Z TZ
[∂t Qε , Qε ] : Φ +
[∂j Qε , Qε ] : ∂j Φ = 0
0
Ω
0
Ω
We denote Σt (δ) the δ−neighborhood of Σt . Equivalently,
XZ T Z
[∂t Qε , Qε ] : Φ + [∂j Qε , Qε ] : ∂j Φ
Ω±
t \Σt (δ)
0
±
Z
T
Z
+
[∂t Qε , Qε ] : Φ + [∂j Qε , Qε ] : ∂j Φ = 0
0
Σt (δ)
The convergences limε→0 Qε are not up to the boundary Σt but ∇x,t Q ∈ L2t L2 (Ω+
t ):
Z TZ
Z TZ
[∂t Q, Q] : Φ + [∂j Q, Q] : ∂j Φ +
S̄0 : Φ + S̄j : ∂j Φ = 0
0
Ω+
t \Σt (δ)
0
Σt (δ)
for some S̄i ∈ L2x,t by modulated estimate. Take Q = s± (n ⊗ n − 31 I3 ) ∈ L2t L2x (Ω±
t ):
Z TZ
Z TZ
∂t n ∧ n · ϕ +
(∂j n ∧ n) · ∂j ϕ = 0, ∀ϕ ∈ Cc1 (Ω)
0
Ω+
t
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0
Ω+
t
Convergence estimate of ψε
We give a heuristic proof basic on the following result:
∂xi f + = ∂xi f 1{x∈Ω:f (x)>0} ,
∀f ∈ W 1,1 (Ω).
R
Recall ψε (x, t) = dW
(ψε − σ1Ω+ )+ dx leads to delta
ε ◦ Qε (x, t). Differentiating
mass ∇1Ω+ . To avoid it, we introduce the weighted energy
Z
gε (t) := (ψε − σ1Ω+ )+ ζ(dΣ ) dx
for a cut-off ζ and let Uε = {x ∈ Ω : ψε > σ1Ω+ }. Using (∂t + H · ∇) ζ(dΣ ) = O(dΣ ),
≈ε|∇Qε |
Z
z }| {
gε0 (t) =
∂t Qε + (H · ∇)Qε : ∂dW
ε (Qε ) ζ(dΣ )
Uε
Z
Z
−
H · ∇ψε ζ(dΣ ) + (ψε − σ1Ω+ )+ ∂t ζ(dΣ )
| {z }
Uε
|
{z
}
≈−H·∇ζ(dΣ )
R
= H·∇(ψε −σ1Ω+ )+ ζ(dΣ )
So we can derive a Grönwall inequality of gε (t) . ε. To get the desired estimate of
ψε , one simply employ standard trick to remove the weight ζ(dΣ ) at the price of a
weaker convergence rate ε1/2 .
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Table of contents
1
Phase field model for Mean Curvature Flow (MCF)
2
Phase transition for potentials of higher dimensional wells
3
Calibration and its application to minimal surface
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Minimal surface and its Euler–Lagrange equation
Suppose that u : Ω ⊂ R2 → R is a C 2 function and consider the graph
Graphu = {(x, y, u(x, y)) | (x, y) ∈ Ω}
Then the area is
Z
|(1, 0, ux ) × (0, 1, uy )| =
Area (Graphu ) =
Ω
Z p
1 + |∇u|2 ,
Ω
and the (upward pointing) unit normal is
ξ=
(1, 0, ux ) × (0, 1, uy )
(−ux , −uy , 1)
= p
|(1, 0, ux ) × (0, 1, uy )|
1 + |∇u|2
Therefore for the graphs Graphu+tη where η|∂Ω = 0 we get that
d
dt
t=0
Z
Area Graphu+tη =
Ω
h∇u, ∇ηi
p
=−
1 + |∇u|2
Z
η div
Ω
∇u
p
1 + |∇u|2
!
Therefore the graph of u is a critical point for the area functional if u satisfies the
divergence form equation
!
∇u
div p
=0
(8)
1 + |∇u|2
Yuning LIU (NYU Shanghai)
Calibration: divergence-free extension of the normal
Next we want to show that the graph of a function u on Ω satisfying the PDE is
area-minimizing amongst surfaces in the cylinder Ω × R ⊂ R3 : Actually let
Σ ⊂ Ω × R be any other surface with ∂Σ = ∂ Graphu and U be the domain enclosed
(−ux ,−uy ,1)
by Graphu and Σ. Observe that ξ = √
has a natural divergence-free
2
1+|∇u|
extension with variables (x, y, z), by Gauss’s theorem
Z
Z
Area (Graphu ) =
ξ · nu =
ξ · nΣ 6 Area(Σ)
Graphu
Σ
Let ω be the 2-form on Ω × R given by ω(X, Y ) = det(X, Y, ξ) for X, Y ∈ R3 .
|ω| 6 1 and given any orthogonal unit vectors X and Y at a point (x, y, z),
|ω(X, Y )| = 1 ⇐⇒ X, Y ⊂ T(x,y,u(x,y)) Graphu
ω=
dx ∧ dy − ux dy ∧ dz − uy dz ∧ dx
p
=⇒ dω = 0
1 + |∇u|2
Z
Z
Area (Graphu ) =
ω=
ω 6 Area(Σ)
Graphu
Σ
Note that a direct proof using convexity is available.
Yuning LIU (NYU Shanghai)
(Calibration)
Fundamental theorem of Calibration
Harvey–Lawson ’82
Let ω be a calibration, i.e., an m-form on Rn so that dω = 0 and
n
sup{hv, ω(x)i : x ∈ Rn , v = ∧m
j=1 vj is a tangent m-plane to R at x} = 1
Let S be a C 1 compact oriented submanifold a with/without boundary calibrated by
ω: hv, ω(x)i = 1 whenever v is a tangent plane to S at x.
Then S is homologically area-minimizing: i.e., if S 0 ∼ S, then area (S 0 ) > area (S).
a
S may allow singularities. Indeed the theorem holds in the category of
the rectifiable currents of geometric measure theory.
R
Proof. Since hv, ω(x)i
= 1 whenever v is a tangent plane to S at x, S ωR = areaR (S).
R
For any surface S 0 , S 0 ω 6 area (S 0 ). Since ω is closed, if S 0 ∼ S, then S 0 ω = S ω.
Assembling these three equalities and inequalities yields
Z
Z
area (S 0 ) >
ω=
ω = area (S)
S0
S
Such an argument was first used by Wirtinger ’36 to show that complex analytic
submanifold are always area-minimizing.
Yuning LIU (NYU Shanghai)
Partial regularity of minimal hyper-surfaces
Subsequent results of De Giorgi, Fleming, Almgren and Simons gave
more precise limitations to the singular set of area-minimizers: an
(n − 1)-dimensional area-minimizer in Rn is regular outside a singular
set whose dimension is at most n − 8 and is discrete when n = 8.
Simons cone is an example
showing that the partial regularity result is
optimal: the cone S = (x, y) ∈ R4 × R4 : |x| = |y| is minimal and has
a singular point in the origin. Bombieri–De Giorgi–Giusti ’69 proved
that this surface is indeed area-minimizing, thus complete the
regularity theory for area-minimizing hyper-surfaces2 .
De Philippis–Paolini ’09 gave a simplified proof by sub-calibration.
2
such a theory was employed by Schoen–Uhlenbeck ’82 in the study of
minimizing harmonic maps
Yuning LIU (NYU Shanghai)
Thank you for your attention !
A few related works and persectives:
1
Coupled with hydrodynamics. Abels–L. ’18, Abels–Fei ’21
2
Cahn–Hilliard to Hele-Shaw. Laux– ?
3
Phase–field to Willmore flow. L.–Fei’ 21
4
Triple junction dynamics. Fisher et al. ’20
5
Contact angle model. Abels–Moser ’20
6
Anisotropic models: Lin–Wang ’21 for Ericksen’s models (static),
L. ’arxiv for a simplified Landau-De Gennes dynamics.
Yuning LIU (NYU Shanghai)