Minimizers of the anisotropic fourth-order Alt-Caffarelli energy in 2D are C^{2,1}, with nonzero gradient on the free boundary.
Ashbaugh and Dorin Bucur, On the isoperimetric inequality for the buckling of a clamped plate , Z
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An anisotropic Alt-Caffarelli problem of higher order
Minimizers of the anisotropic fourth-order Alt-Caffarelli energy in 2D are C^{2,1}, with nonzero gradient on the free boundary.