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For compact subsets $E$ of the unit disk $ mathbb{D}$ we study the capacity of the condenser ${rm cap}( mathbb{D},E)$ by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the equilateral triangle has the least capacity.
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Given a compact connected set $E$ in the unit disk $mathbb{B}^2$, we give a new upper bound for the conformal capacity of the condenser $(mathbb{B}^2, E),$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$ we construct a set of diameter $t$ and show by numerical computation that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is of constant hyperbolic width equal to $t$, the so called hyperbolic Reuleaux triangle.
We apply domain functionals to study the conformal capacity of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.
In light of the Suita conjecture, we explore various rigidity phenomena concerning the Bergman kernel, logarithmic capacity, Greens function, and Euclidean distance and volume.
For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capacity associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.
We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic properties of the condenser quasicentral modulus and compute a simple example. Some of the results are in the more general setting of a semifinite von Neumann algebra.
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