Rigidity of critical points of hydrophobic capillary functionals

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.