We study a variational problem for surfaces with free boundary on given supporting planes in the euclidean three-space. The total energy is the surface area and a “weetting energ”” on the supporting planes , and we impose the volume constraint. The stationary surfaces are surfaces with constant mean curvature which meet each supporting plane with constant contact angle. We determine all minimizers and all local minimizers under a certain assumption for the problem. Our results show that the space of solutions is not continuous with respect to the variation of the boundary condition. We also generalize this subject to higher dimensions and more general energy functional with anisotropy.

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