I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, please feel free to post the answer or reference.
By Green-capacity I mean $$Cap_{G}(A)=\inf \{\int\int G(x,y)\mu(dx)\mu(dy):\mu~~\text{ a probability measure on A}\}]^{-1}$$ and $$G(x,y)=\frac{\Gamma(\frac{d}{2}-1)}{2\pi^{\frac{d}{2}}}|x-y|^{d-2}$$ for $A\subset \mathbb{R}^{d}$,with $d\geq 3$.
By isoperimetric I mean, $Cap_{G}(A)\geq Cap_{G}(A^{*})$ where $A^{*}$ is a ball with $vol(A^{*})=vol(A)$.
Thank you
Your terminology is a bit confusing: Green capacity usually refers specifically to using Green's function for a proper subdomain of $\mathbb R^n$.
The case $\alpha=2$ (sometimes called the Newtonian capacity) is classical. What helps in this case is that the potential is a harmonic function on the complement of the support of the measure. This allows one to express capacity in terms of the Dirichlet integral $ \int |\nabla u|^2 $ of the potential over the complement of $A$. The Dirichlet integral is easier to rearrange than equilibrium measures. Basically, one takes the equilibrium potential for $A$, performs symmetric decreasing rearrangement (which does not increase the Dirichlet energy), observes that the rearranged function, though not harmonic, satisfies the appropriate boundary condition for $A^*$, and therefore the harmonic solution of the Dirichlet problem will have even smaller integral.
The above sketch has some holes. One has to cut off the infinity by a large sphere, do the above, and let the radius of the sphere tend to infinity.
For rearrangement of Dirichlet integral, see G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics". For other things mentioned above, see Capacity article of EOM and its references.
To my knowledge, $\alpha>2$ remains open. I don't know of any papers on the subject published since An Isoperimetric Inequality for Riesz Capacities by Méndez-Hernández.