Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions

We prove a compactness result for capillary hypersurfaces with mean curvature prescribed by ambient functions, which generalizes the results of Schätzle and Bellettini to the capillary case. The proof relies on extending the definition of (unoriented) curvature varifolds with capillary boundary introduced by Wang-Zhang to the context of oriented integral varifolds. We also discuss the case when the mean curvature of the boundary is prescribed.