For every phase between pi/2 and pi, a uniformly convex domain in R3 exists whose zero-Dirichlet special Lagrangian solution has a nonconvex negative sublevel set.
Gradient estimates for the Lagrangian mean curvature equation with critical and supercritical phase.Amer
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Nonconvex Sublevel Sets for the Three-Dimensional Special Lagrangian Equation
For every phase between pi/2 and pi, a uniformly convex domain in R3 exists whose zero-Dirichlet special Lagrangian solution has a nonconvex negative sublevel set.