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## 1. Monge-Ampère equations

1. #### Problem 1.05.

[Xu-Jia Wang] Let $u$ plurisubharmonic satisfy $\mathrm{det}(u_{j \overline k}) = f(u)$ on $B_1(0)$ with Dirichlet boundary conditions, then $u$ is rotationally symmetric
• #### Problem 1.1.

[Xu-Jia Wang] Let $u$ strictly convex with Dirichlet boundary conditions solve $\mathrm{det}(u_{ij}) = 1 + \delta_{x_0},$ then $u=\varphi+w$ for $\varphi$ smooth uniformly convex cone, and $w$ smooth.
• #### Problem 1.15.

[Tristan Collins] For $u$ strictly convex, determine the existence and regularity of solutions to $\mathrm{det}(D^2 u) u^{ij}u_i u_j = 1.$
•     A function $u:\Omega \to \mathbb R$ satisfies the special Lagrangian equation if $\sum_i \mathrm{arctan}(\lambda_i)= c$ for a constant $c$, where $\lambda_i$ are the eigenvalues of $D^2u$.

#### Problem 1.2.

[Tristan Collins] (Conjecture of Nadirashvili-Vlăduţ) Any viscosity solution of the special Lagrangian equation on the unit ball is $C^1(B)$ for sufficiently smooth boundary data.
There exist known examples which are not $C^2$.
• #### Problem 1.25.

[Nam Le] (Conjecture of Nadirashvili-Tkachev-Vlăduţ) A uniformly elliptic equation in divergence form admits a solution in $W^{1,p}$ for some $p>1$.
• #### Problem 1.3.

[Tristan Collins] Let $u$ be a solution to $u_{tt}+\mathrm{det}(u_{ij}) = 0$ on $T^n\times[0,1]$ with convex boundary data. Prove a $C^2$ estimate for $u(\cdot,t)$ for $t$ fixed.
A solution would imply the maximal rank conjecture for complex tori.
• #### Problem 1.35.

[Xu-Jia Wang] Find conditions that allow for $C^2$ regularity for degenerate Monge-Ampère equation (both real and complex).
• #### Problem 1.4.

[Gábor Székelyhidi] Let $u$ be plurisubharmonic solution to $\mathrm{det}(u_{\overline k j}) =1$ on $\mathbb C^n$ satisfying $C^{-1} (|z|^2+1) \leq u \leq C (|z|^2+1)$ then $u$ is quadratic.
(Y. Wang) Answer is yes, if $u=|z|^2+o(1)$.
1. Remark. The Problem solved recently by AN-MIN LI and LI SHENG the paper in arxiv.org/pdf/1809.00824.pdf
• #### Problem 1.45.

[Xiangwen Zhang] Let $u:\mathbb R^2 \to \mathbb R$ be a solution of $u_{11} u_{22} = 1.$ If $u(x) \leq C(1 + |x|^2)$, then $u$ must be quadratic.

Can the growth condition be removed?

If one considers instead solutions $u:\Omega \to \mathbb R$ for $\Omega \subseteq \mathbb R^2$ convex, and bounded, and $u$ with Dirichlet boundary conditions, can one obtain a $C^2$ boundary estimate?
• #### Problem 1.5.

[Bin Zhou] Let $\mathrm{det}(u_{\overline k j}) = \frac {f(z)}{|z|^\alpha}$ for $f(z)$ smooth. What is the optimal regularity of $u$?

Cite this as: AimPL: Nonlinear PDEs in real and complex geometry, available at http://aimpl.org/nonlinpdegeom.