1. Monge-Ampère equations
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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)= cfor 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. -
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}) = 0on T^n\times[0,1] with convex boundary data. Prove a C^2 estimate for u(\cdot,t) for t fixed. -
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.-
Remark. The Problem solved recently by AN-MIN LI and LI SHENG the paper in arxiv.org/pdf/1809.00824.pdf
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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.