Energy Functionals for the Parabolic Monge-Ampere Equation
ENERGY FUNCTIONALS FOR THE PARABOLIC
MONGE-AMPÈRE EQUATION
ZUOLIANG HOU AND QI LI
1. Introduction
Because of its close connection with the Kähler-Ricci flow, the parabolic
complex Monge-Ampère equation on complex manifolds has been studied by
many authors. See, for instance, [Cao85, CT02, PS06]. On the other hand,
theories for complex Monge-Ampère equation on both bounded domains
and complex manifolds were developed in [BT76, Yau78, CKNS85, Ko l98].
In this paper, we are going to study the parabolic complex Monge-Ampère
equation over a bounded domain.
Let Ω ⊂ Cn be a bounded domain with smooth boundary ∂Ω. Denote
QT = Ω × (0, T ) with T > 0, B = Ω × {0}, Γ = ∂Ω × {0} and ΣT = ∂Ω ×
(0, T ). Let ∂pQT be the parabolic boundary of QT , i.e. ∂pQT = B∪Γ∪ΣT .
Consider the following boundary value problem:
− log det
= f(t, z, u) in QT ,
u = ϕ on ∂pQT .
where f ∈ C∞(R× Ω̄×R) and ϕ ∈ C∞(∂pQT ). We will always assume that
Then we will prove that
Theorem 1. Suppose there exists a spatial plurisubharmonic (psh) function
u ∈ C2(Q̄T ) such that
u t − log det
u αβ̄
≤ f(t, z, u) in QT ,
u ≤ ϕ on B and u = ϕ on ΣT ∩ Γ.
Then there exists a spatial psh solution u ∈ C∞(Q̄T ) of (1) with u ≥ u if
following compatibility condition is satisfied: ∀ z ∈ ∂Ω,
ϕt − log det
= f(0, z, ϕ(z)),
ϕtt −
log det(ϕαβ̄)
= ft(0, z, ϕ(z)) + fu(0, z, ϕ(z))ϕt .
Date: October 29, 2018.
http://arxiv.org/abs/0704.1822v1
2 ZUOLIANG HOU AND QI LI
Motivated by the energy functionals in the study of the Kähler-Ricci
flow, we introduce certain energy functionals to the complex Monge-Ampère
equation over a bounded domain. Given ϕ ∈ C∞(∂Ω), denote
(5) P(Ω, ϕ) =
u ∈ C2(Ω̄) | u is psh, and u = ϕ on ∂Ω
then define the F 0 functional by following variation formula:
(6) δF 0(u) =
δudet
We shall show that the F 0 functional is well-defined. Using this F 0 func-
tional and following the ideas of [PS06], we prove that
Theorem 2. Assume that both ϕ and f are independent of t, and
(7) fu ≤ 0 and fuu ≤ 0.
Then the solution u of (1) exists for T = +∞, and as t approaches +∞,
u(·, t) approaches the unique solution of the Dirichlet problem
= e−f(z,v) in QT ,
v = ϕ on ∂pQT ,
in C1,α(Ω̄) for any 0 < α < 1.
Remark : Similar energy functionals have been studied in [Bak83, Tso90,
Wan94, TW97, TW98] for the real Monge-Ampère equation and the real
Hessian equation with homogeneous boundary condition ϕ = 0, and the
convergence for the solution of the real Hessian equation was also proved in
[TW98]. Our construction of the energy functionals and the proof of the con-
vergence also work for these cases, and thus we also obtain an independent
proof of these results. Li [Li04] and Blocki [B lo05] studied the Dirichlet
problems for the complex k-Hessian equations over bounded complex do-
mains. Similar energy functional can also be constructed for the parabolic
complex k-Hessian equations and be used for the proof of the convergence.
2. A priori C2 estimate
By the work of Krylov [Kry83], Evans [Eva82], Caffarelli etc. [CKNS85]
and Guan [Gua98], it is well known that in order to prove the existence and
smoothness of (1), we only need to establish the a priori C2,1(Q̄T )1 estimate,
i.e. for solution u ∈ C4,1(Q̄T ) of (1) with
(9) u = u on ΣT ∪ Γ and u ≥ u in QT ,
(10) ‖u‖C2,1(QT ) ≤ M2,
where M2 only depends on QT , u, f and ‖u(·, 0)‖C2(Ω̄).
m,n(QT ) means m times and n times differentiable in space direction and time di-
rection respectively, same for Cm,n-norm.
ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 3
Proof of (10). Since u is spatial psh and u ≥ u, so
u ≤ u ≤ sup
(11) ‖u‖C0(QT ) ≤ M0.
Step 1. |ut| ≤ C1 in Q̄T .
Let G = ut(2M0−u)−1. If G attains its minimum on Q̄T at the parabolic
boundary, then ut ≥ −C1 where C1 depends on M0 and u t on Σ. Otherwise,
at the point where G attains the minimum,
Gt ≤ 0 i.e. utt + (2M0 − u)−1u2t ≤ 0,
Gα = 0 i.e. utα + (2M0 − u)−1utuα = 0,
Gβ̄ = 0 i.e. utβ̄ + (2M0 − u)
−1utuβ̄ = 0,
and the matrix Gαβ̄ is non-negative, i.e.
(13) utαβ̄ + (2M0 − u)
−1utuαβ̄ ≥ 0.
Hence
(14) 0 ≤ uαβ̄
utαβ̄ + (2M0 − u)
−1utuαβ̄
= uαβ̄utαβ̄ + n(2M0 − u)
−1ut,
where (uαβ̄) is the inverse matrix for (uαβ̄), i.e.
uαβ̄uγβ̄ = δ
Differentiating (1) in t, we get
(15) utt − uαβ̄utαβ̄ = ft + fu ut,
(2M0 − u)−1u2t ≤ −utt
= −uαβ̄utαβ̄ − ft − fu ut
≤ n(2M0 − u)−1ut − fu ut − ft,
hence
u2t − (n− (2M0 − u)fu)ut + ft(2M0 − u) ≤ 0.
Therefore at point p, we get
(16) ut ≥ −C1
where C1 depends on M0 and f .
Similarly, by considering the function ut(2M0 + u)
−1 we can show that
(17) ut ≤ C1.
Step 2. |∇u| ≤ M1
4 ZUOLIANG HOU AND QI LI
Extend u|Σ to a spatial harmonic function h, then
(18) u ≤ u ≤ h in QT and u = u = h on ΣT .
(19) |∇u|ΣT ≤ M1.
Let L be the linear differential operator defined by
(20) Lv =
− uαβ̄vαβ̄ − fuv.
L(∇u + eλ|z|2) = L(∇u) + Leλ|z|2
≤ ∇f − eλ|z|2
uαᾱ − fu).
Noticed that and both u and u̇ are bounded and
= eu̇−f ,
(22) 0 < c0 ≤ det
≤ c1,
where c0 and c1 depends on M0 and f . Therefore
uαᾱ ≥ nc−1/n1 .
Hence after taking λ large enough, we can get
L(∇u + eλ|z|2) ≤ 0,
(24) |∇u| ≤ sup
|∇u| + C2 ≤ M1.
Step 3. |∇2u| ≤ M2 on Σ.
At point (p, t) ∈ Σ, we choose coordinates z1, · · · , zn for Ω, such that
at z1 = · · · = zn = 0 at p and the positive xn axis is the interior normal
direction of ∂Ω at p. We set s1 = y1, s2 = x1, · · · , s2n−1 = yn, s2n = xn and
s′ = (s1, · · · , s2n−1). We also assume that near p, ∂Ω is represented as a
graph
(25) xn = ρ(s
j,k<2n
Bjksjsk + O(|s′|3).
Since (u− u)(s′, ρ(s′), t) = 0, we have for j, k < 2n,
(26) (u− u)sjsk(p, t) = −(u− u)xn(p, t)Bjk,
hence
(27) |usjsk(p, t)| ≤ C3,
where C3 depends on ∂Ω, u and M1.
ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 5
We will follow the construction of barrier function by Guan [Gua98] to
estimate |uxnsj |. For δ > 0, denote Qδ(p, t) =
Ω ∩Bδ(p)
× (0, t).
Lemma 3. Define the function
(28) d(z) = dist(z, ∂Ω)
(29) v = (u− u) + a(h− u) −Nd2.
Then for N sufficiently large and a, δ sufficiently small,
Lv ≥ ǫ(1 +
uαᾱ) in Qδ(p, t)
v ≥ 0 on ∂(Bδ(p) ∩ Ω) × (0, t)
v(z, 0) ≥ c3|z| for z ∈ Bδ(p) ∩ Ω
where ǫ depends on the uniform lower bound of he eigenvalues of {u αβ̄}.
Proof. See the proof of Lemma 2.1 in [Gua98]. �
For j < 2n, consider the operator
+ ρsj
Tj(u− u) = 0 on
∂Ω ∩Bδ(p)
× (0, t)
|Tj(u− u)| ≤ M1 on
Ω ∩ ∂Bδ(p)
× (0, t)
|Tj(u− u)(z, 0)| ≤ C4|z| for z ∈ Bδ(p)
So by Lemma 3 we may choose C5 independent of u, and A >> B >> 1 so
Av + B|z|2 − C5(uyn − u yn)
2 ± Tj(u− u)
≥ 0 in Qδ(p, t),
Av + B|z|2 − C5(uyn − u yn)
2 ± Tj(u− u) ≥ 0 on ∂pQδ(p, t).
Hence by the comparison principle,
Av + B|z|2 − C5(uyn − u yn)
2 ± Tj(u− u) ≥ 0 in Qδ(p, t),
and at (p, t)
(33) |uxnyj | ≤ M2.
To estimate |uxnxn |, we will follow the simplification in [Tru95]. For
(p, t) ∈ Σ, define
λ(p, t) = min{uξξ̄ | complex vector ξ ∈ Tp∂Ω, and |ξ| = 1}
Claim λ(p, t) ≥ c4 > 0 where c4 is independent of u.
Let us assume that λ(p, t) attains the minimum at (z0, t0) with ξ ∈ Tzo∂Ω.
We may assume that
λ(z0, t0) <
u ξξ̄(z0, t0).
6 ZUOLIANG HOU AND QI LI
Take a unitary frame e1, · · · , en around z0, such that e1(z0) = ξ, and Re en =
γ is the interior normal of ∂Ω along ∂Ω. Let r be the function which defines
Ω, then
(u− u )11̄(z, t) = −r11̄(z)(u − u )γ(z, t) z ∈ ∂Ω
Since u11̄(z0, t0) < u 11̄(z0, t0)/2, so
−r11̄(z0)(u− u )γ(z0, t0) ≤ −
u 11̄(z0, t0).
Hence
r11̄(z0)(u− u )γ(z0, t) ≥
u 11̄(z0, t) ≥ c5 > 0.
Since both ∇u and ∇u are bounded, we get
r11̄(z0) ≥ c6 > 0,
and for δ sufficiently small ( depends on r11̄ ) and z ∈ Bδ(z0) ∩ Ω,
r11̄(z) ≥
So by u11̄(z, t) ≥ u11̄(z0, t0), we get
u 11̄(z, t) − r11̄(z)(u− u )γ(z, t) ≥ u 11̄(z0, t0) − r11̄(z0)(u− u )γ(z0, t0).
Hence if we let
Ψ(z, t) =
r11̄(z)
r11̄(z0)(u− u )γ(z0, t0) + u 11̄(z, t) − u 11̄(z0, t0)
(u− u )γ(z, t) ≤ Ψ(z, t) on
∂Ω ∩Bδ(z0)
× (0, T )
(u− u )γ(z0, t0) = Ψ(z0, t0).
Now take the coordinate system z1, · · · , zn as before. Then
(u− u )xn(z, t) ≤
γn(z)
Ψ(z, t) on
∂Ω ∩Bδ(z0)
× (0, T )
(u− u )xn(z0, t0) =
γn(z0)
Ψ(z0, t0).
where γn depends on ∂Ω. After taking C6 independent of u and A >>
B >> 1, we get
Av + B|z|2 − C6(uyn − u yn)
Ψ(z, t)
γn(z)
− Tj(u− u)
≥ 0 in Qδ(p, t),
Av + B|z|2 − C6(uyn − u yn)
Ψ(z, t)
γn(z)
− Tj(u− u) ≥ 0 on ∂pQδ(p, t).
Av + B|z|2 − C6(uyn − u yn)
Ψ(z, t)
γn(z)
− Tj(u− u) ≥ 0 in Qδ(p, t),
|uxnxn(z0, t0)| ≤ C7.
ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 7
Therefore at (z0, t0), uαβ̄ is uniformly bounded, hence
u11̄(z0, t0) ≥ c4
with c4 independent of u. Finally, from the equation
detuαβ̄ = e
we get
|uxnxn | ≤ M2.
Step 4. |∇2u| ≤ M2 in Q.
By the concavity of log det, we have
L(∇2u + eλ|z|2) ≤ O(1) − eλ|z|2
uαᾱ − fu
So for λ large enough,
L(∇2u + eλ|z|2) ≤ 0,
(35) sup |∇2u| ≤ sup
|∇2u| + C8
with C8 depends on M0, Ω and f .
3. The Functionals I, J and F 0
Let us recall the definition of P(Ω, ϕ) in (5),
P(Ω, ϕ) =
u ∈ C2(Ω̄ | u is psh, and u = ϕ on ∂Ω
Fixing v ∈ P, for u ∈ P, define
(36) Iv(u) = −
(u− v)(
−1∂∂̄u)n.
Proposition 4. There is a unique and well defined functional Jv on P(Ω, ϕ),
such that
(37) δJv(u) = −
−1∂∂̄u)n − (
−1∂∂̄v)n
and Jv(v) = 0.
Proof. Notice that P is connected, so we can connect v to u ∈ P by a path
ut, 0 ≤ t ≤ 1 such that u0 = v and u1 = u. Define
(38) Jv(u) = −
−1∂∂̄ut)n − (
−1∂∂̄v)n
We need to show that the integral in (38) is independent of the choice of
path ut. Let δut = wt be a variation of the path. Then
w1 = w0 = 0 and wt = 0 on ∂Ω,
8 ZUOLIANG HOU AND QI LI
−1∂∂̄u)n − (
−1∂∂̄v)n
−1∂∂̄u)n − (
−1∂∂̄v)n
+ u̇ n
−1∂∂̄w(
−1∂∂̄u)n−1
Since w0 = w1 = 0, an integration by part with respect to t gives
−1∂∂̄u)n − (
−1∂∂̄v)n
−1∂∂̄u)n dt = −
−1nw∂∂̄u̇(
−1∂∂̄u)n−1 dt.
Notice that both w and u̇ vanish on ∂Ω, so an integration by part with
respect to z gives
−1nw∂∂̄u̇(
−1∂∂̄u)n−1 = −
−1n∂w ∧ ∂̄u̇(
−1∂∂̄u)n−1
−1nu̇∂∂̄w(
−1∂∂̄u)n−1.
(39) δ
−1∂∂̄u)n − (
−1∂∂̄v)n
dt = 0,
and the functional J is well defined. �
Using the J functional, we can define the F 0 functional as
(40) F 0v (u) = Jv(u) −
−1∂∂̄v)n.
Then by Proposition 4, we have
(41) δF 0v (u) = −
−1∂∂̄u)n.
Proposition 5. The basic properties of I, J and F 0 are following:
(1) For any u ∈ P(Ω, ϕ), Iv(u) ≥ Jv(u) ≥ 0.
(2) F 0 is convex on P(Ω, ϕ), i.e. ∀u0, u1 ∈ P,
(42) F 0
(u0 + u1
F 0(u0) + F
0(u1)
(3) F 0 satisfies the cocycle condition, i.e. ∀u1, u2, u3 ∈ P(Ω, ϕ),
(43) F 0u1(u2) + F
(u3) = F
(u3).
ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 9
Proof. Let w = (u− v) and ut = v + tw = (1 − t)v + tu, then
Iv(u) = −
−1∂∂̄u)n − (
−1∂∂̄v)n
−1∂∂̄ut)n dt
−1nw∂∂̄w(
−1∂∂̄ut)n−1
−1n∂w ∧ ∂̄w ∧ (
−1∂∂̄ut)n−1 ≥ 0,
Jv(u) = −
−1∂∂̄ut)n − (
−1∂∂̄v)n
−1∂∂̄us)n ds
−1nw∂∂̄w(
−1∂∂̄us)n−1 ds dt
(1 − s)
−1n∂w ∧ ∂̄w ∧ (
−1∂∂̄us)n−1 ds ≥ 0.
Compare (44) and (45), it is easy to see that
Iv(u) ≥ Jv(u) ≥ 0.
To prove (42), let ut = (1 − t)u0 + tu1, then
F 0(u1/2) − F 0(u0) = −
(u1 − u0) (
−1∂∂̄ut)n dt,
F 0(u1) − F 0(u1/2) = −
(u1 − u0) (
−1∂∂̄ut)n dt.
Since
(u1 − u0) (
−1∂∂̄ut)n dt−
(u1 − u0) (
−1∂∂̄ut)n dt.
(u1 − u0)
−1∂∂̄ut)n − (
−1∂∂̄ut+1/2)n
(ut+1/2 − ut)
−1∂∂̄ut)n − (
−1∂∂̄ut+1/2)n
dt ≥ 0.
F 0(u1) − F 0(u1/2) ≥ F 0(u1/2) − F 0(u0).
The cocycle condition is a simple consequence of the variation formula 41.
10 ZUOLIANG HOU AND QI LI
4. The Convergence
In this section, let us assume that both f and ϕ are independent of t. For
u ∈ P(Ω, ϕ), define
(46) F (u) = F 0(u) +
G(z, u)dV,
where dV is the volume element in Cn, and G(z, s) is the function given by
G(z, s) =
e−f(z,t) dt.
Then the variation of F is
(47) δF (u) = −
det(uαβ̄) − e−f(z,u)
Proof of Theorem 2. We will follow Phong and Sturm’s proof of the conver-
gence of the Kähler-Ricci flow in [PS06]. For any t > 0, the function u(·, t)
is in P(Ω, ϕ). So by (47)
F (u) = −
det(uαβ̄) − e
−f(z,u)
log det(uαβ̄) − (−f(z, u))
det(uαβ̄) − e
−f(z,u)
Thus F (u(·, t)) is monotonic decreasing as t approaches +∞. On the other
hand, u(·, t) is uniformly bounded in C2(Ω) by (10), so both F 0(u(·, t)) and
f(z, u(·, t)) are uniformly bounded, hence F (u) is bounded. Therefore
log det(uαβ̄) + f(z, u)
det(uαβ̄) − e
−f(z,u)
dt < ∞.
Observed that by the Mean Value Theorem, for x, y ∈ R,
(x + y)(ex − e−y) = (x + y)2eη ≥ emin(x,−y)(x− y)2,
where η is between x and −y. Thus
log det(uαβ̄) + f
det(uαβ̄) − e
log det(uαβ̄) + f
= C9|u̇|2
where C9 is independent of t. Hence
‖u̇‖2L2(Ω) dt ≤ ∞
(50) Y (t) =
|u̇(·, t)|2 det(uαβ̄) dV,
2üu̇ + u̇2uαβ̄ u̇αβ̄
det(uαβ̄) dV.
Differentiate (1) in t,
(51) ü− uαβ̄ u̇αβ̄ = fuu̇,
ENERGY FUNCTIONALS FOR THE PARABOLIC MONGE-AMPÈRE EQUATION 11
2u̇u̇αβ̄u
αβ̄ + u̇2
2fu + ü− fuu̇
det(uαβ̄) dV
2fu + ü− fuu̇
− 2u̇αu̇β̄u
det(uαβ̄) dV
From (51), we get
u − uαβ̄ üαβ̄ − fuü ≤ fuuu̇
Since fu ≤ 0 and fuu ≤ 0, so ü is bounded from above by the maximum
principle. Therefore
Ẏ ≤ C10
u̇2 det(uαβ̄) dV = C10Y,
(52) Y (t) ≤ Y (s)eC10(t−s) for t > s,
where C10 is independent of t. By (49), (52) and the uniform boundedness
of det(uαβ̄), we get
‖u(·, t)‖L2(Ω) = 0.
Since Ω is bounded, the L2 norm controls the L1 norm, hence
‖u(·, t)‖L1(Ω) = 0.
Notice that by the Mean Value Theorem,
|ex − 1| < e|x||x|
|eu̇ − 1| dV ≤ esup |u̇|
|u̇| dV
Hence eu̇ converges to 1 in L1(Ω) as t approaches +∞. Now u(·, t) is bounded
in C2(Ω), so u(·, t) converges to a unique function ũ, at least sequentially in
C1(Ω), hence f(z, u) → f(z, ũ) and
det(ũαβ̄) = lim
det(u(·, t)αβ̄) = lim
eu̇−f(z,u) = e−f(z,ũ),
i.e. ũ solves (8).
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Mathematics Department, Columbia University, New York, NY 10027
E-mail address: hou@math.columbia.edu
Mathematics Department, Columbia University, New York, NY 10027
E-mail address: liqi@math.columbia.edu
	1. Introduction
	2. A priori C2 estimate
	3. The Functionals I, J and F0
	4. The Convergence
	References
