Stable capillary hypersurfaces and the partitioning problem in balls with radial weights

In a round ball B ⊂ Rn+1 endowed with an O(n+1)-invariant metric we consider a radial function that weights volume and area. We prove that a compact two-sided hypersurface in B which is stable capillary in weighted sense and symmetric about some line containing the center of B is homeomorphic to a closed n-dimensional disk. When combined with Hsiang symmetrization and other stability results this allows to deduce that the interior boundary of any isoperimetric region in B for the Gaussian weight is a closed n-disk of revolution. For n = 2 we also show that a compact weighted stable capillary surface in B of genus 0 is a closed disk of revolution