Rigidity of critical points of hydrophobic capillary functionals

Abstract

We prove the rigidity, among sets of finite perimeter, of volume-preserving critical points of the capillary energy in the half space, in the case where the prescribed interior contact angle is between $90^{\circ }$ and $120^{\circ }$. No structural or regularity assumption is required on the finite perimeter sets. Assuming that the ``tangential'' part of the capillary boundary is ${\mathcal{H}}^n$-null, this rigidity theorem extends to the full hydrophobic regime of interior contact angles between $90^{\circ }$ and $180^{\circ }$. Furthermore, we establish the anisotropic counterpart of this theorem under the assumption of lower density bounds.