For the planar translating mean curvature equation with zero boundary data, convexity of the domain does not force convexity of sublevel sets, even in dimension two.
Kawohl,Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol
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Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation
For the planar translating mean curvature equation with zero boundary data, convexity of the domain does not force convexity of sublevel sets, even in dimension two.