The distance between minimal hypersurfaces is a viscosity supersolution to an elliptic PDE, with extensions to parabolic PDEs when evolving by mean curvature flow.
Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions
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Distance between minimal surfaces and flows
The distance between minimal hypersurfaces is a viscosity supersolution to an elliptic PDE, with extensions to parabolic PDEs when evolving by mean curvature flow.