Weak solutions of the equation

of complex Monge-Ampère type

by

Sławomir Kołodziej

Abstract. We prove some existence results for the equation of the complex Monge-Ampère type in strictly pseudoconvex domains and on Kähler manifolds.

0. Introduction.

In this paper we extend the results on the existence of bounded (resp. continuous) solutions of the complex Monge-Ampère equation

(0.1)  (dd^{c}u)^{n} = dμ

(with given positive measure dμ, a plurisubharmonic solution u, and the wedge product on the left defined as in [BT1]) to the case of more general equations of Monge-Ampère type

(0.2)  (dd^{c}u)^{n} = F (u, ·)dμ,

We shall assume throughout that F (t, z) ≥ 0, that it is nondecreasing and continuous in the first variable and measurable in the other one.

One can study the Dirichlet problem for the equation (0.1) in a strictly pseudoconvex domain in ℂ^{n} imposing a boundary condition

lim_{z→x} u(z) = φ(x) for x ∈ ∂Ω.

with given φ ∈ C(∂Ω) (see [BT1], [Ce], [CKNS], [K1], [K2], [K3]). By [K1] this problem has a unique bounded solution, provided a bounded subsolution exists. We shall prove (Theorem 1.1 below) that the same conclusion holds for (0.2) when F is bounded. Thus we generalize the results of Bedford-Taylor [BT2] and Cegrell [Ce]. We refer to [CKNS] for the study of the classical solutions of the equation. Furtheremore, let the measure dμ be represented as f dλ with dλ denoting the Lebesgue measure and f ≥ 0 belonging to the Orlicz space

L^{χ}(Ω) = {g ∈ L^{1}(Ω) : ∫_{Ω} χ(g) dλ < ∞}.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. primary 32F07, secondary 32F05.

Key words and phrases. plurisubharmonic function, complex Monge-Ampère operator.

Partially supported by KBN Grant No. PO3A 003 13

If

(0.3)  χ(t) = |t|(log(1 + |t|))^{n}h(log(1 + |t|)),

where h : ℝ_{+} → (1, ∞) is an increasing function satisfying

∫_{1}^{∞}(yh^{1/n}(y))^{−1} dy < ∞,

then the Dirichlet problem for (0.1) has a unique continuous solution (see [K2], [K3]). Corollary 1.2 says that it is still true for (0.2) with dμ = dλ and F (t, z) ≤ f(z) ∈ L^{χ}(Ω).

In the next section we study the complex Monge-Ampère equation on a compact Kähler manifold M with the fundamental form ω :

(0.2’)  (ω + dd^{c}u)^{n} = F (u, ·)ω^{n}

By the Stokes theorem, if u satisfies (0,2’) then

∫_{M} F (u, ·)ω^{n} = ∫_{M} ω^{n}.

For F positive and smooth, satisfying the normalizing condition, the equation has been solved by Aubin [A1], [A2] and Yau [Y]. It is particulary interesting for F (t, z) = exp(ct + f(z)) when the solution serves to produce Kähler-Einstein metrics (see [A1], [A2], [A3], [S], [Y]). Using a result from [K3], where the case of F not depending on t was treated, we show in Theorem 2.1 that for F only nonnegative and F ∈ L^{χ}(M), χ as above, one can find a continuous solution of (0.2’). This result, applied to M = ℙ^{n}, leads to solving (0.2) in the family of entire plurisubharmonic functions of minimal growth.

1. The equation of Monge-Ampère type in a strictly pseudoconvex domain.

Theorem 1.1. Let Ω be a strictly pseudoconvex domain and φ ∈ C(∂Ω). Suppose there exists v ∈ P SH ∩ L^{∞}(Ω) such that (dd^{c}v)^{n} = dμ and lim_{z→x} v(z) = φ(x) for x ∈ ∂Ω. Furtheremore assume that F : ℝ × Ω → ℝ is a bounded nonnegative function which is nondecreasing and continuous in the first variable and dμ measurable in the second one. Then there exists a unique bounded plurisubharmonic solution of the Dirichlet problem

(1.1) u ∈ P SH ∩ L^{∞}(Ω), (dd^{c}u)^{n} = F (u, ·)dμ,

lim_{z→x} u(z) = φ(x) for x ∈ ∂Ω.

Proof. Without loss of generality we assume F ≤ 1. As in [Ce] we shall use Schauder’s fixed point theorem. Consider L^{2}(Ω, dλ) equipped with the weak topology. Let h be the maximal plurisubharmonic function in Ω with the boundary data equal to φ. Then the set

A = {u ∈ P SH(Ω) : v ≤ u ≤ h}

is convex and compact in L^{2}(Ω, dλ). We define the mapping

F : A → A

taking for F(u) the solution of

(1.2)  \begin{matrix}w ∈ A, \\ (dd^{c}w)^{n} = F (u, ·)dμ. \end{matrix}

This solution exists by [K1]. We need to show that F is contiuous. Let u_{j} → u in A. Set w = F(u) and w_{j} = F(u_{j}). By Hartogs’ lemma u = (lim sup u_{j})^{∗}. Consider auxiliary functions ũ_{k} = (sup_{j≥k} u_{j})^{∗}, w̃_{k} = F(ũ_{k}). Since (dd^{c}w̃_{k})^{n} is decreasing it follows from the comparison principle [BT3] that the sequence w̃_{k} is increasing to some w̃ ∈ A. The convergence theorem [BT3] gives (dd^{c}w̃)^{n} = lim_{k}(dd^{c}w̃_{k})^{n}. We also have

(dd^{c}w_{k})^{n} = F (u_{k}, ·)dμ ≤ F (ũ_{k}, ·)dμ = (dd^{c}w̃_{k})^{n}.

Hence, using the comparison principle, w̃_{k} ≤ w_{k}.

Furtheremore, ũ_{k} ↓ u, from the Lebesgue monotone convergence theorem one infers

(dd^{c}w̃)^{n} = lim_{k→∞} F (ũ_{k}, ·)dμ = F (u, ·)dμ.

Since the solution to the Dirichlet problem (1.2) is unique we thus get

w = w̃ = lim ↑ w̃_{k} ≤ lim inf w_{k}.

It remains to prove that

lim sup w_{k} ≤ w.

For this consider the sequence ŵ_{k} of functions in A solving

(dd^{c}ŵ_{k})^{n} = F (û_{k}, ·)dμ,

where û_{k} = inf_{k≤j} u_{j}. Then, by the comparison principle w_{k} ≤ ŵ_{k} and ŵ_{k} decreases to ŵ ∈ A. Since û_{k} ↑ u we obtain, applying the convergence theorem

(dd^{c}ŵ)^{n} = lim ↑ (dd^{c}ŵ_{k})^{n} = F (u, ·)dμ.

This implies w = ŵ = lim ↑ ŵ_{k} ≥ lim sup w_{k}. Thus we have proved the continuity of the mapping F. The Schauder theorem now says that F has a fixed point which gives the existence part of the statement. Uniqueness follows from the comparison principle and the fact that F (·, z) is nondecreasing.

Remark. The above theorem remains true for hyperconvex domains provided there exists a maximal plurisubharmonic function with boundary data equal to φ.

Remark. The existence part of the theorem still holds when we drop the hypothesis that F (·, z) be monotone, but uniqueness is then lost (see [Ce]).

Corollary 1.2. Let Ω, φ and F be as in the above theorem except that instead of assuming that F is bounded we now suppose

F (t, ·) ≤ ψ ∈ L^{χ}(Ω),

with χ given by (0.3). Then there exists a continuous solution to

u ∈ P SH(Ω) ∩ C(Ω̄), (dd^{c}u)^{n} = F (u, ·) dλ,

lim_{z→x} u(z) = φ(x) for x ∈ ∂Ω,

where dλ denotes the Lebesgue measure.

Proof. For the subsolution required in Theorem 1.1 we take v ∈ P SH(Ω) ∩ C(Ω̄) solving

(dd^{c}v)^{n} = ψ dλ,

with given boundary data. The existence of such v and continuity of u has been proved in [K2] and [K3].

The equation of Monge-Ampère type on Kähler manifolds.

Our next result is concerned with the equation (0.2’).

Theorem 2.1. Let (M, ω) be a compact Kähler manifold with the fundamental form ω. Assume F : ℝ × M → ℝ, 0 ≤ F (t, z) ≤ ψ(z) ∈ L^{χ}(M) (with χ as in (0.3)) is a function such that F (·, z) is continuous and nondecreasing, F (t, ·) is ω^{n} measurable and

(2.1)  lim_{t→−∞} ∫_{M} F (t, z)ω^{n} ≤ ∫_{M} ω^{n} ≤ lim_{t→∞} ∫_{M} F (t, z)ω^{n}.

Then the equation

Mu := (dd^{c}u + ω)^{n} = F (u, ·)ω^{n}

has a continuous solution such that dd^{c}u + ω ≥ 0.

We shall need the following lemma.

Lemma 2.2. Let (M, ω) and F be as in Theorem 2.1 except that we replace the condition (2.1) by slightly stronger one

(2.1’)  lim_{t→−∞} ∫_{M} F (t, z)ω^{n} < ∫_{M} ω^{n} < lim_{t→∞} ∫_{M} F (t, z)ω^{n}.

Then there exists a sequence F_{k} ∈ C^{∞}(ℝ×M) with the following properties: F_{k} > 0,

\frac{∂}{∂t}F_{k}(t, z) ≥ 0,

sup_{t} sup_{k} ∫_{M} χ(F_{k}(t, ·))ω^{n} < c < ∞,

(2.1’) is fulfilled with F_{k} in place of F, and for any bounded function u in M:

lim_{k→∞} F_{k}(u(z), z) = F (u(z), z)

almost everywhere in M.

Proof. Using (2.1’) we fix R > 0 such that

(2.2)  ∫_{M} F (−R, z)ω^{n} + R^{−1} < ∫_{M} ω^{n} < ∫_{M} F (R, z)ω^{n} − R^{−1}.

For any positive integer k > R we shall construct F_{k}(t, z) as follows. First, note that the assumptions on F (·, z) allow us to find for given z ∈ M a positive integer N such that

(2.3)  F (t_{j+1}, z) − F (t_{j}, z) < 2^{−k−4}

where

t_{j} := −k + j2^{−N}, j = 0, 1, ..., k2^{N+1}.

We fix N such that (2.3) holds true for all z ∉ E with

(2.4)  ∫_{E} ω^{n} < 2^{−k−2}.

In the next step we apply Luzin’s theorem to choose g_{j} ∈ C(M), g_{j} ≥ 0, ∫_{M} χ(g_{j})ω^{n} < c satisfying

(2.5)  ∫_{E′_{j}} ω^{n} < 2^{−k−j−3}, E′_{j} := {g_{j} ≠ F (t_{j}, ·)}.

Set h_{j} = min_{j≤s} g_{j}. Then {h_{j} ≠ F (t_{j}, ·)} ⊂ E′ := ∪_{0}^{N}E′_{s}. Since h_{j} ≤ h_{j+1} one can approximate h_{j} by smooth positive f_{j} such that f_{j} ≤ f_{j+1}, ∫_{M} χ(f_{j})ω^{n} < c and

(2.6)  {|f_{j} − F (t_{j}, ·)| > 2^{−k−3}} ⊂ E′.

Fix ψ_{j} smooth in a neighbourhood of the interval [t_{j−1}, t_{j}] with 0 ≤ ψ_{j} ≤ 1, ψ_{j} = 1 close to t_{j} and ψ_{j} = 0 close to t_{j−1}. Set

F_{k}(t, z) = ψ_{j}(t)f_{j+1}(z) + (1 − ψ_{j}(t))f_{j}(z), t ∈ [t_{j−1}, t_{j}].

By our choice of ψ_{j} those functions are smooth and the choice of f_{j} guarantees that F_{k} are nondecreasing in t, satisfy the inequalities

(2.2’)  ∫_{M} F_{k}(−R, z)ω^{n} < ∫_{M} ω^{n} < ∫_{M} F_{k}(R, z)ω^{n},

and the condition

sup_{t} sup_{k} ∫_{M} χ(F_{k}(t, ·))ω^{n} < c < ∞.

Note that for z ∉ E′, t ∈ [t_{j−1}, t_{j}] we have (see (2.6))

(2.7)  F_{k}(t, z) ≥ f_{j}(z) ≥ F (t_{j}, z) − 2^{−k−3} ≥ F (t, z) − 2^{−k−3}.

If moreover z ∉ E then (see (2.3) and (2.6))

(2.8)  \begin{aligned}F (t, z) & ≥ F (t_{j−1}, z) > F (t_{j+1}, z) − 2^{−k−3} \\ & ≥ f_{j+1}(z) − 2^{−k−2} ≥ F_{k}(t, z) − 2^{−k−2}. \end{aligned}

By (2.4) and (2.5) we have

∫_{E∪E′} ω^{n} < 2^{−k−1},

and writing E = E(k), E′ = E′(k) to indicate the dependece of those sets on k

∑_{j≤k} ∫_{E(k)∪E′(k)} ω^{n} < 2^{−j}.

Since, by (2.7) and (2.8), for any z ∉ ∪_{j≤k}[E(k) ∪ E′(k)] and any t we have lim_{k→∞} F_{k}(t, z) = F (t, z) the last part of the statement follows.

Proof of Theorem 2.1. First we prove the statement under extra hypothesis (2.1). We fix a sequence F_{k} as in the above lemma. Yau’s theorem [Y, Theorem 4] provides smooth u_{k} satisfying dd^{c}u_{k} + ω > 0 and

Mu_{k} = F_{k}(u_{k}, ·).

Let us define some auxiliary functions

u_{jk} = max_{j≤l≤k} u_{l}, v_{j} = ( lim_{k→∞} u_{jk})^{∗}, u = (lim sup u_{k})^{∗}.

One can apply [K3, section 2.3] to conclude that

(2.9)  sup_{M} u_{k} − inf_{M} u_{k} < c_{0},

with c_{0} depending only on c from the assumptions. By Stokes’ theorem

∫_{M} F (u_{k}, ·)ω^{n} = ∫_{M} ω^{n}.

So, in view of (2.2’) we obtain

inf_{M} u_{k} < R, sup_{M} u_{k} > −R, k > R.

Those inequalities and (2.9) imply that the sequence u_{k} is uniformly bounded. Passing to a subsequence we assume

u_{k} → u a.e.

Applying [BT1, Proposition 2.8] one gets

Mu_{jk} ≥ min_{j≤l≤k} F_{l}(u_{l}, ·).

Now, for fixed ϵ > 0 we find an integer j_{0} and a set E with

u_{j}(z) ≥ u(z) − ϵ, j ≥ j_{0}, z ∉ E,

and ∫_{E} ω^{n} < ϵ. Then for j, z as above

Mu_{jk}(z) ≥ inf_{j≤l≤k} F_{l}(u(z) − ϵ, z).

Letting k to ∞ and using the convergnce theorem [BT3] we obtain

Mv_{j}(z) ≥ inf_{j≤l} F_{l}(u(z) − ϵ, z), z ∉ E.

Since v_{j} decrease to u one can apply the convergence theorem and Lemma 2.2 to get

Mu(z) ≥ F (u(z) − ϵ, z) a.e. in M \ E.

This is true for any ϵ > 0 and F (·, z) is continuous, so

(2.10)  Mu(z) ≥ F (u(z), z) a.e.

Since by the lemma and the argument above F_{k}(u_{k}(z), z) → F (u(z), z) almost everywhere and, on the other hand,

∫_{M} F_{k}(u_{k}, ·)ω^{n} = ∫_{M} ω^{n}

for any k, we conclude that the integrals over M of both sides of inequality (2.10) are equal. So the functions are equal. To get the general case, note that if we had equalities in (2.1’) then F would be independent of t and the equation would reduce to the one solved in [K3]. If we have one strict inequality we can find a monotone sequence c_{j} → 0 such that for any j

lim_{t→−∞} ∫_{M}(F (t, z) + c_{j})ω^{n} < ∫_{M} ω^{n} < lim_{t→∞} ∫_{M}(F (t, z) + c_{j})ω^{n}.

Then by the preceding part of the proof, there exist u_{j} satisfying

Mu_{j} = F (u_{j}(z), z) + c_{j}.

The function u = (lim sup u_{k})^{∗} is the desired solution as can be seen by repeating the above reasoning.

Theorem 2.1 when applied for M = ℙ^{n}, equipped with the Fubini-Study metric, allow us to solve the Monge-Ampère type equation in the class of entire plurisubharmonic functions of minimal growth usually denoted by L_{+} .

L_{+} = {u ∈ P SH(ℂ^{n}) : |u(z) − log(1 + |z|^{2})| < const.}.

The function

v_{0}(z) = \frac{1}{2} log(1 + |z|^{2}) ∈ L_{+}

is a potential of the Fubini-Study metric in ℙ^{n} restricted to ℂ^{n} (which is embedded in the standard way). We have

∫ (dd^{c}v_{0})^{n} = ∫ ω^{n} = ∫ \frac{n!}{(1 + |z|^{2})^{n+1}} dλ = (2π)^{n}.

Let F : ℝ × ℂ^{n} → ℝ, 0 ≤ F (t, z) be continuous and nondecreasing in t, measurable in z, and such that for some t_{0}

∫_{ℂ^{n}} F (t_{0}, z) dλ = (2π)^{n}.

Suppose also that

F (t, z) ≤ f(z)(1 + |z|^{2})^{−n−1},

with f ∈ L^{χ}(ω^{n}).

Corollary 2.3. For F introduced above the following equation

\begin{matrix}u ∈ L_{+} \\ (dd^{c}u)^{n} = F (u − v_{0}, ·) dλ \end{matrix}

has a solution.

The uniqueness of those solutions has been shown in [BT4] in the case when F is independent either of t or z.

References

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postal address: Jagiellonian University, Institute of Mathematics Reymonta 4, 30-059 Kraków, Poland e-mail: kolodzie@ im.uj.edu.pl