Free energy minimizers with radial densities: classification and quantitative stability

We study the isoperimetric problem with a potential energy $g$ in ${\mathbb{R}}^n$ weighted by a radial density $f$ and analyze the geometric properties of minimizers. Notably, we construct two counterexamples demonstrating that, in contrast to the classical isoperimetric case $g=0$, the condition $\ln (f{)}^{\mathrm{\prime }\mathrm{\prime }}+g^{\mathrm{\prime }}\ge 0$ does not generally guarantee the global optimality of centered spheres. However, we demonstrate that centered spheres are globally optimal when both $f$ and $g$ are monotone. Additionally, we strengthen this result by deriving a sharp quantitative stability inequality.