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 radial function that weights volume and area. We prove compact two-sided hypersurface in B which is stable capillary weighted sense symmetric about some line containing the center of homeomorphic to closed n-dimensional disk. When combined Hsiang symmetrization other stability results this allows deduce interior boundary any isoperimetric region for Gaussian weight n-disk revolution. For n=2 also show surface genus 0 disk