$C^{2,\alpha }$-estimate for Monge-Ampère equations with Hölder-continuous right hand side

Abstract

We present a somewhat new proof to the $C^{2,\alpha }$ a priori estimate for the uniform elliptic Monge-Ampère equations, in both the real and complex settings. Our estimates do not need to differentiate the equation, and only depends on the $C^{\alpha }$-norm of the right-hand side of the equation.

Appendix: A new proof of the a priori version of Cafferelli’s estimate for real Monge-Ampère equations

The proof of Theorem 1.1 is exactly parallel to the proof in Sect. 2. Namely, to translate the “complex” proof in Sect. 2 to the real case of Theorem 1.1, we only need to

• change the $``\sqrt{-1}\partial {\bar{\partial }}''$ in Sect. 2 to “${\nabla }^{2}$” (Hessian);

• change the $\omega _{k{\bar{l}}}$ to $g_{kl}$, $\phi _{k{\bar{l}}}$ to $u_{kl}$;

• change the complex coordinates “$z_{1}\ldots z_{n}$” in Sect. 2 to real coordinates “$x_{1}\ldots x_{n}$”;

• change the words “plurisubharmonic” to “convex”;

• change the equation from (2) to (1).

By translating as above, Lemma 2.1 corresponds to Lemma 2.10, Theorem 2.3 corresponds to Theorem 2.11, Proposition 2.5 corresponds to Proposition 2.13. One thing worth mentioning is, while the proof of Lemma 2.1 requires the Griffith–Harris trick [16] and Hormander’s results [19], Lemma 2.10 can be proved in one line.

Lemma 2.10

Suppose $\frac{1}{\Lambda }<r<\Lambda$ for some $\Lambda >0$, then there exists a constant $C_{\Lambda }$ depending on $\Lambda$ and n with the following properties.

Suppose g is a matrix-valued function over $B_{r}$, such that $g=\nabla ^{2}u$ for some function $u\in C^{2,\alpha }(B_{r})$. Then there exists a function $v\in C^{2,\alpha }(B(\frac{r}{2}))$ such that $g=\nabla ^{2}v$ and

$$\begin{aligned} |v|_{0, B_{\frac{r}{2}}}\le Cr^2|g|_{0,B_r}, \end{aligned}$$

where C is constant depending on n. Consequently,

$$\begin{aligned} |v|_{2,\alpha ,B_{\frac{r}{4}}}\le C_{\Lambda }|g|_{\alpha ,B_{r}}. \end{aligned}$$

Proof of Lemma. 2.10

The proof can not be easier. Just take v as u minus its linearization i.e

$$\begin{aligned} v=u-u(0)-x\cdot \nabla u(0), \end{aligned}$$

(23)

then

$$\begin{aligned} \nabla ^{2}v=g,\ v(0)=0,\ (\nabla v)(0)=0. \end{aligned}$$

Thus the estimate of $|v|_{L^{\infty }(B(\frac{r}{2}))}$ follows by applying the mean value theorem to $\nabla v$ and then to v. $\square$

Theorem 2.11

(Calabi [5]) (Pogorelov [23]) Suppose g is a symmetric-matrix-valued function defined over $\mathbb R^{n}$. Suppose g admits a $C^{2,\alpha }$-potential over any finite ball i.e for any ball $B\in \mathbb R^{n}$, there exists a convex function $u_{B}\in C^{2,\alpha }(B)$ such that

$$\begin{aligned} g=\nabla ^{2}u_{B}\ \text {over}\ B. \end{aligned}$$

Suppose there is a constant $K>0$ such that

$$\begin{aligned} \mathrm{det} g_{kl}=1,\ \frac{1}{K}I\le g_{kl}\le KI\ \text {over}\ \mathbb R^{n}. \end{aligned}$$

(24)

Then, for any $1\le k,l\le n$, $g_{kl}$ is a constant.

Remark 2.12

Actually Calabi and Pogorelov’s original theorems in [5] and [23] are much stronger than Theorem 2.11, but all we need here is Theorem 2.11. In [5], g is assumed to admit a global potential. Though in our new proof of Theorem 1.1 of Caffarelli, we have a global potential, we still want to state the Liouville theorem as Theorem 2.3 to emphasize that it does not need a global potential.

Proposition 2.13

For any constant coefficient Riemannian metric $g_{c}$, there exist a small enough positive number $\delta$ and a big enough constant $C_{g_{c}}$, both depending on the positive lower and upper bounds on the eigenvalues of $g_{c}$, the dimension n, and $\alpha$, with the following properties. Suppose u is a $C^{2,\alpha }$ convex function defined over a unit ball B(1) such that

$$\begin{aligned} \mathrm{det} u_{ij}=e^{f},\ \frac{g_{c}}{1+\delta }\le \nabla ^{2}u\le (1+\delta )g_{c}\ \text {over}\ B(1), \end{aligned}$$

(25)

then the following estimate holds in $B(\frac{1}{4})$.

$$\begin{aligned}{}[\nabla ^{2}u]_{\alpha , B(\frac{1}{4})}\le C_{g_c}(|e^f|_{\alpha ,B(1)}+|\nabla ^{2} u|_{0,B(1)}). \end{aligned}$$

With the above discussion in Appendix, the proof of Theorem 1.1 is complete.