Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

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.