Anisotropic overdetermined problem with capillary boundary condition in a half-space

In this paper, we study a Serrin-type anisotropic overdetermined problem with anisotropic capillary boundary in a half-space R + n \mathbb {R}^{n}_{+} . We prove that a domain on which this partially anisotropic overdetermined problem admits a solution if and only if the domain is a truncated Wulff ball intersecting ∂ R + n \partial \mathbb {R}^{n}_{+} with constant anisotropic capillary angle, which generalizes the isotropic results of Jia-Lu-Xia-Zhang [Calc. Var. Partial Differential Equations 63 (2024), 23 pp.].