The complex Monge-Amp\`ere equation on weakly pseudoconvex domains

TL;DR

This paper establishes boundary regularity results for solutions to the complex Monge-Ampère equation on weakly pseudoconvex domains, extending understanding of regularity under minimal assumptions on data and domain geometry.

Contribution

It proves weak Hölder regularity up to the boundary for solutions with data in L^p on domains satisfying the f-property, applicable to finite and infinite type pseudoconvex domains.

Findings

Boundary regularity of solutions established

Applicable to a broad class of pseudoconvex domains

Extends previous results to weaker data and domain conditions

Abstract

We show here a "weak" H\"older-regularity up to the boundary of the solution to the Dirichlet problem for the complex Monge-Amp\`{e}re equation with data in the $L^p$ space and the boundary of the domain satisfying an $f$-property. The $f$-property is a potential-theoretical condition which holds for all pseudoconvex domains of finite type and many examples of infinite type.

Equations

$$\{\begin{array}{ll}u\in PSH(\Omega )\cap L_{loc}^{\infty }(\Omega ),& \\ (d{d}^{c}u{)}^n=\psi dV& \text{in }\Omega ,\\ u=\phi & \text{on }b\Omega .\end{array}$$

$${\Lambda }^f(A)=\{u:\Vert u{\Vert }_{L^{\infty }(A)}+\underset{z,w\in A,z\ne w}{\sup }f(|z-w{|}^{-1})\cdot |u(z)-u(w)|<\infty \}$$

$$\Vert u{\Vert }_{{\Lambda }^f(A)}=\Vert u{\Vert }_{L^{\infty }(A)}+\underset{z,w\in A,z\ne w}{\sup }f(|z-w{|}^{-1})\cdot |u(z)-u(w)|.$$

$$\rho \in {\Lambda }^{g^2}(\bar{\Omega }),\Omega =\{\rho <0\}\text{and}i\partial \bar{\partial }\rho \ge \text{Id}.$$

$$\rho \in C^0(\bar{\Omega }),\Omega =\{\rho <0\}\text{and}i\partial \bar{\partial }\rho \ge \text{Id}.$$

$$v\le u(\Omega ,\phi ,\psi )\le w.$$

$$|u(z)-u(\zeta )|\lesssim [g(|z-\zeta {|}^{-1}){]}^{-\beta }z\in \bar{\Omega },\zeta \in b\Omega ,$$

$$|u(z)-u(z^{\prime })|\lesssim [g({\delta }^{-1}){]}^{-\beta },z,z^{\prime }\in \Omega \setminus {\Omega }_{\delta }\text{ and }|z-z^{\prime }|<\delta$$

$$u_{\frac{\delta}{2}}(z):=\underset{|\zeta|<{\frac{\delta}{2}}}{\sup}u(z+\zeta),\qquad\tilde{u}_{\frac{\delta}{2}}(z):=\frac{1}{\sigma_{2n-1}{\big{(}\frac{\delta}{2}}\big{)}^{2n-1}}\int_{b\mathbb{B}(z,{\frac{\delta}{2}})}u(\zeta)dS(\zeta),$$

$${\hat{u}}_{\frac{\delta }{2}}(z):=\frac{1}{{\sigma }_{2n}(\frac{\delta }{2}{)}^{2n}}{\int }_{\mathbb{B}(z,\frac{\delta }{2})}u(\zeta )dV(\zeta ),$$

$${\hat{u}}_{\frac{\delta }{2}}\le {\tilde{u}}_{\frac{\delta }{2}}\le u_{\frac{\delta }{2}}\text{in}{\Omega }_{\delta }.$$

$$\Vert {\tilde{u}}_{\frac{\delta }{2}}-u{\Vert }_{L^1({\Omega }_{\delta })}\lesssim {\delta }^{1-ϵ}.$$

$$\underset{\Omega }{\sup }(W-U)\le C\Vert (W-U{)}_{+}{\Vert }_{L^s(\Omega )}^{\eta },$$

$${\tilde{u}}_{\frac{\delta }{2}}\le u_{\frac{\delta }{2}}\le u+c[g({\delta }^{-1}){]}^{-\beta },\text{on bΩδ for suitable constant c}.$$

$$\begin{array}{rl}\underset{{\Omega }_{\delta }}{\sup }\left({\tilde{u}}_{\frac{\delta }{2}}-(u+c[g({\delta }^{-1}){]}^{-\beta })\right)& \underset{\text{Theorem reftse}}{\lesssim }\Vert {\left({\tilde{u}}_{\frac{\delta }{2}}-(u+c[g({\delta }^{-1}){]}^{-\beta })\right)}_{+}{\Vert }_{L^1({\Omega }_{\delta })}^{\eta }\\ & \lesssim \Vert {\tilde{u}}_{\frac{\delta }{2}}-u{\Vert }_{L^1({\Omega }_{\delta })}^{\eta }\underset{\text{Theorem reft3.1}}{\lesssim }{\delta }^{(1-ϵ)\eta },\end{array}$$

$$\underset{{\Omega }_{\delta }}{\sup }\left({\tilde{u}}_{\frac{\delta }{2}}-u\right)\lesssim {\delta }^{(1-ϵ)\eta }+[g({\delta }^{-1}){]}^{-\beta }\lesssim {\delta }^{\frac{\gamma }{2}}+[g({\delta }^{-1}){]}^{-\beta }\lesssim [g({\delta }^{-1}){]}^{-\beta },$$

$$\underset{{\Omega }_{\delta }}{\sup }(u_{\delta }-u)\lesssim [g({\delta }^{-1}){]}^{-\beta }\text{and}\underset{{\Omega }_{\delta }}{\sup }({\hat{u}}_{\delta }-u)\lesssim [g({\delta }^{-1}){]}^{-\beta }.$$

$$\underset{{\Omega }_{\delta }}{\sup }(u_{\frac{\delta }{2}}-u)\lesssim [g({\delta }^{-1}){]}^{-\beta }.$$

$$|u(z)-u(z^{\prime })|\lesssim [g(|z-z^{\prime }{|}^{-1}){]}^{-\beta }\text{for any z,z′∈Ωˉ}.$$

Full text

MSC: 32U05, 32U40, 53C55

The research of T. V. Khanh was supported by the Australian Research Council DE160100173.

1. Introduction

For a $C^{2}$, bounded, pseudoconvex domain $\Omega\subset\subset{\mathbb{C}}^{n}$, the Dirichlet problem for the Monge-Ampère equation consists in

[TABLE]

A great deal of work has been done when $\Omega$ is strongly pseudoconvex. On this domain, we can divide the literature by three kinds of data $\psi$.

• •

The Hölder data: Bedford-Taylor prove in [2] that $u\in C^{\frac{\alpha}{2}}(\bar{\Omega})$ if $\varphi\in C^{\alpha}(b\Omega)$, $\psi^{\frac{1}{n}}\in C^{\frac{\alpha}{2}}(\bar{\Omega})$ for $0<\alpha\leq 2$.

• •

The smooth data: Caffarelli, Kohn and Nirenberg prove in [4] that $u\in C^{\infty}(\bar{\Omega})$, for $\varphi\in C^{\infty}(b\Omega)$ and $\psi\in C^{\infty}(\bar{\Omega})$, in case $\psi>0$ in $\bar{\Omega}$ and $b\Omega$ is smooth.

• •

The $L^{p}$ data: Guedj, Kolodziej and Zeriahi prove in [6] that if $\psi\in L^{p}(\Omega)$ with $p>1$ and $\varphi\in C^{1,1}(b\Omega)$ then $u\in C^{\gamma}(\bar{\Omega})$ for any $\gamma<\gamma_{p}:=\frac{2}{qn+1}$ where $\frac{1}{q}+\frac{1}{p}=1$.

When $\Omega$ is no longer strongly pseudoconvex but has a certain “finite type”, there are some known results for this problem due to Blocki [3], Coman [5], and Li [11]. Recently, Ha and the second author gave a general related result to a Hölder data under the hypothesis that $\Omega$ satisfies an $f$-property (see Definition 2.1 below). The $f$-property is a consequence of the geometric “type” of the boundary. All pseudoconvex domains of finite type satisfy the $f$-property as well as many classes of domains of infinite type (see [7, 8, 9] for discussion on the $f$-property). Using the $f$-property, a “weak” Hölder regularity for the solution of the Dirichlet problem of complex Monge-Ampère equation is obtained in [7]. Coming back to the case of $\Omega$ of finite type, in a recent paper with Zampieri [1], we prove the Hölder regularity for $\psi\in L^{p}$ with $p>1$. The purpose of the present paper is to generalize the result in [1] to a pseudoconvex domain satisfying an $f$-property. For this purpose we recall the definition of a weak Hölder space in [7, 8]. Let $f$ be an increasing function such that $\underset{t\to+\infty}{\lim}f(t)=+\infty$, $f(t)\lesssim t$. For a subset $A$ of ${\mathbb{C}}^{n}$, define the $f$-Hölder space on $A$ by

[TABLE]

and set

[TABLE]

Note that the notion of the $f$-Hölder space includes the standard Hölder space $\Lambda_{\alpha}$ by taking $f(t)=t^{\alpha}$ (so that $f(|h|^{-1})=|h|^{-\alpha}$) with $0<\alpha\leq 1$. Here is our result

Theorem 1.1**.**

Let $\Omega\subset{\mathbb{C}}^{n}$ be a bounded, pseudoconvex domain admitting the $f$-property. Suppose that $\displaystyle\int_{1}^{\infty}\dfrac{da}{af(a)}<\infty$ and denote by $g(t):=\left(\displaystyle\int_{t}^{\infty}\dfrac{da}{af(a)}\right)^{-1}$ for $t\geq 1$. If $0<\alpha\leq 2$, $\varphi\in\Lambda^{t^{\alpha}}(b\Omega)$, and $\psi\geq 0$ on $\Omega$ with $\psi\in L^{p}$ with $p>1$, then the Dirichlet problem for the complex Monge-Ampère equation (1.1) has a unique plurisubharmonic solution $u\in\Lambda^{g^{\beta}}(\overline{\Omega})$. Here $\beta=\min(\alpha,\gamma)$, for any $\gamma<\gamma_{p}=\frac{2}{nq+1}$ where $\frac{1}{p}+\frac{1}{q}=1$.

The proof follows immediately from Theorem 2.2 and 2.5 below. Throughout the paper we use $\lesssim$ and $\gtrsim$ to denote an estimate up to a positive constant, and $\approx$ when both of them hold simultaneously. Finally, the indices $p$, $\alpha$, $\beta$, $\gamma$ and $\gamma_{p}$ only take ranges as in Theorem 1.1.

2. Hölder regularity of the solution

We start this section by defining the $f$-property as in [8, 9].

Definition 2.1**.**

For a smooth, monotonic, increasing function $f:[1,+\infty)\to[1,+\infty)$ with ${f(t)}{t^{-1/2}}$ decreasing, we say that $\Omega$ has the $f$-property if there exist a neighborhood $U$ of $b\Omega$ and a family of functions $\{\varphi_{\delta}\}$ such that

1. (i)

the functions $\varphi_{\delta}$ are plurisubharmonic, $C^{2}$ on $U$, and satisfy $-1\leq\varphi_{\delta}\leq 0$, 2. (ii)

$i\partial\bar{\partial}\varphi_{\delta}\gtrsim f(\delta^{-1})^{2}Id$ and $|D\varphi_{\delta}|\lesssim\delta^{-1}$ for any $z\in U\cap\{z\in\Omega:-\delta<r(z)<0\}$, where $r$ is a $C^{2}$-defining function of $\Omega$.

In [8], using the $f$-property, the second author constructed a family of plurisubharmonic peak functions with good estimates. This family of plurisubhamonic peak functions yields the existence of a defining function $\rho$ which is uniformly strictly-plurisubharmonic and weakly Hölder (see [7]).

Theorem 2.2** (Khanh [8] and Ha-Khanh[7]).**

Assume that $\Omega$ is a bounded, pseudoconvex domain admitting the $f$-property as in Theorem 1.1. Then there exists a uniformly strictly-plurisubharmonic defining function of $\Omega$ which belongs to the $g^{2}$-Hölder space of $\overline{\Omega}$, that means,

[TABLE]

The existence and uniqueness of the solution $u\in L^{\infty}(\Omega)$ to the equation (1.1) need a weaker condition, in particular, one only need $\rho\in C^{0}(\bar{\Omega})$ as shown by [10].

Theorem 2.3** (Kolodziej [10]).**

Let $\Omega$ be a bounded domain in ${\mathbb{C}}^{n}$. Assume that there exists a function $\rho$ such that

[TABLE]

Then for any $\varphi\in C^{0}(b\Omega)$, $\psi\in L^{p}(\Omega)$ there is a unique plurisubharmonic solution $u(\Omega,\varphi,\psi)\in C^{0}(\bar{\Omega})$.

To improve the smoothness of $u$, we increase the smoothness of $\rho$ and $\psi$.

Theorem 2.4** (Ha-Khanh [7]).**

Let $\rho$ satisfy (2.1). If $\varphi\in\Lambda^{t^{\alpha}}(b\Omega)$ and $\psi^{\frac{1}{n}}\in\Lambda^{g^{\alpha}}(\overline{\Omega})$, then the Dirichlet problem for the complex Monge-Ampère equation (1.1) has a unique plurisubharmonic solution $u(\Omega,\varphi,\psi)\in\Lambda^{g^{\alpha}}(\overline{\Omega})$.

Now we focus on lowering the smoothness of $\psi$ and prove the following theorem.

Theorem 2.5**.**

Let $\rho$ satisfy (2.1). If $\varphi\in\Lambda^{t^{\alpha}}(b\Omega)$ and $\psi\in L^{p}(\Omega)$, then the Dirichlet problem for the complex Monge-Ampère equation (1.1) has a unique plurisubharmonic solution $u(\Omega,\varphi,\psi)\in\Lambda^{g^{\beta}}(\overline{\Omega})$.

In order to prove this theorem, we need to construct a subsolution with $L^{p}$ data. Here, $v$ is a subsolution to (1.1) in the sense that $v$ is plurisubharmonic, $v|_{b\Omega}=\varphi$ and $(dd^{c}v)^{n}\geq\psi dV$ in $\Omega$.

Proposition 2.6**.**

Let $\rho$ satisfy (2.1).Then there is a subsolution $v\in\Lambda^{g^{\beta}}(\bar{\Omega})$ to (1.1) for $\varphi\in C^{\alpha}(b{\Omega})$ and $\psi\in L^{p}(\Omega)$.

Proof.

For a large ball $\mathbb{B}$ containing $\Omega$, we set \tilde{\psi}(z):=\begin{cases}\psi(z)\quad\text{if z\in\Omega},\\ 0\quad\text{ if z\in\mathbb{B}\setminus\Omega}.\end{cases} First, we apply Theorem 1 in [6] on $\mathbb{B}$ with $\tilde{\psi}\in L^{p}(\mathbb{B})$ and zero-valued boundary condition, it follows $u_{1}=u(\mathbb{B},0,\tilde{\psi})\in\Lambda^{t^{\gamma}}(\bar{\mathbb{B}})$. Second, we apply Theorem 2.4 on $\Omega$ twice: first for $u_{2}:=u(\Omega,-u_{1}|_{b\Omega},0)\in\Lambda^{g^{\gamma}}$ since $u_{1}|_{b\Omega}\in\Lambda^{t^{\gamma}}$ and second for $u_{3}:=u(\Omega,\varphi,0)\in\Lambda^{g^{\alpha}}$ by the hypothesis $\varphi\in\Lambda^{t^{\alpha}}$. Finally, taking the summation $v=u_{1}+u_{2}+u_{3}$, we have the conclusion.

∎

Proof of Theorem 2.5.

Keeping the notation of Theorem 2.3, let $u(\Omega,\varphi,\psi)\in C^{0}(\bar{\Omega})$ be the solution of (1.1). What follows is dedicated to showing that this $C^{0}$ plurisubharmonic solution $u(\Omega,\varphi,\psi)$ is in fact in $\Lambda^{g^{\beta}}(\bar{\Omega})$. By Theorem 2.4, $w:=u(\Omega,\varphi,0)\in\Lambda^{g^{\alpha}}(\bar{\Omega})$; let $v$ be as in Proposition 2.6, comparison principle yields at once

[TABLE]

By (2.2) and the $g^{\beta}$-Hölder regularity of $v$ and $w$ we get

[TABLE]

and therefore for $\delta$ suitably small

[TABLE]

where $\Omega_{\delta}:=\{z\in{\mathbb{C}}^{n}:\,r(z)<-\delta\}$ and $r$ is the $C^{2}$ defining function for $\Omega$ with $|\nabla r|=1$ on $b\Omega$. We have to prove that (2.3) also holds for $z,\,z^{\prime}\in\Omega_{\delta}$. For $z\in\bar{\Omega}_{\delta}$, we use the notation

[TABLE]

and

[TABLE]

where {\sigma_{2n-1}{\big{(}\frac{\delta}{2}}\big{)}^{2n-1}}=Vol(b\mathbb{B}(z,\frac{\delta}{2})) and {\sigma_{2n}{\big{(}\frac{\delta}{2}}\big{)}^{2n}}=Vol(\mathbb{B}(z,\frac{\delta}{2})). It is obvious that

[TABLE]

Furthermore, we have $L^{1}$ estimate of the difference between $u$ and $\tilde{u}_{\frac{\delta}{2}}$ and the stability estimate in the following theorems:

Theorem 2.7** (Baracco-Khanh-Pinton-Zampieri [1]).**

For any $0<\epsilon<1$, we have

[TABLE]

Theorem 2.8** (Guedj-Kolodziej-Zeriahi [6]).**

Fix $0\leq f\in L^{p}(\Omega),\,p>1$. Let $U,\,W$ be two bounded plurisubharmonic functions in $\Omega$ such that $(dd^{c}U)^{n}=fdV$ in $\Omega$ and let $U\geq W$ on $\partial\Omega$. Fix $s\geq 1$ and $0\leq\eta<\frac{s}{nq+s}$, $\frac{1}{p}+\frac{1}{q}=1$. Then there exists a uniform constant $C=C(\eta,\|f\|_{L^{p}(\Omega)})>0$ such that

[TABLE]

where $(W-U)_{+}:=\max(W-U,0)$.

By (2.3), we have

[TABLE]

Thus, we can apply Theorem 2.8 for $\Omega_{\delta}$ with $U:=u+c[g(\delta^{-1})]^{-\beta}$, $W:=\tilde{u}_{\frac{\delta}{2}}$ and $s:=1$; thus we get

[TABLE]

for any $\eta<\frac{1}{2}\gamma_{p}=\frac{1}{nq+1}$ where $\frac{1}{q}+\frac{1}{p}=1$. Taking $\gamma<\gamma_{p}$, $\beta=\min(\alpha,\gamma)$, $\epsilon=\frac{\gamma_{p}-\gamma}{\gamma_{p}+\gamma}>0$ and $\eta=\frac{1}{4}(\gamma_{p}+\gamma)<\frac{1}{2}\gamma_{p}$ so that $(1-\varepsilon)\eta=\frac{\gamma}{2}$, it follows

[TABLE]

where the last inequality of (2.7) follows by $g(\delta^{-1})\lesssim\delta^{-\frac{1}{2}}$ (by the conditions on $f$ in the $f$-property).

Similarly to [6, Lemma 4.2] by using the fact that $g(c\delta^{-1}))\approx g(\delta^{-1})$ for any constant $c>0$, one can state the equivalence between

[TABLE]

Using this equivalence together with the inequalities in (2.4), it follows that (2.7) is equivalent to

[TABLE]

From (2.3) and (2.8), it is easy to prove that

[TABLE]

∎