By G-capacity for capacitable set K I mean:
$Cap(K)=[inf\{\int\int G(x,y)d\mu(y)d\mu(x):\mu$ probability measure on K$\}]^{-1}$.
where G(x,y) is any kernel eg. the Green kernel.
Q1:We've calculated the Capacity for "nice" sets like spheres. But I am wondering what is the current status in more complex sets like compact? Like estimating by open covers? Any conditions, like partial rotational symmetry?Any good books or papers in the subject?
I currently have the "Classical Potential theory" by Gardiner and Armitage, "Foundations of modern potential theory" by Landkoff,"Function Spaces and Potential Theory" by David R. Adams, Lars I. Hedberg and "Brownian motion, Obstacles and Random media" by Sznitman.
I am also looking for RECENT papers or books. Is anyone in that field or at least aware of any progress made in estimating capacities for borel/compact sets?
1)"Comparison Results for Capacity" by Ana Hurtado. Gives bounds for compact subsets of Riemmanian/Hadamard manifolds.
2)
Q2:Can you also give all the alternative formulations of Capacity? It would be nice to have them in one place.
The other one I know is for Newtonian: $Cap(K):=inf\{\int |\bigtriangledown f(x)|^{2}dx:f\in C^{\infty}_{0}\mathbb{R}^{n}$ and $f|_{\partial K}\geq 1\}$
Thank you
Some extra references that might be useful (I have at least found them useful on related topics):
W. Hayman: Subharmonic Functions Volume 2 - in particular, section 7.3, geometric estimates for capacity.
Garnett, Marshall: Harmonic Measure
Doob: Classical Potential Theory and its Probabilistic Counterpart - see Part I, chapter XIII.