Harmonic radius and concentration of energy; Hyperbolic radius and Liouville's equations Delta U=e(U) and Delta U=Un-2/n+2

The conformal radius of a simply connected planar domain Omega is defined as r(x) = 1/\f'(x)\, where f denotes a Riemann map from Omega to the unit disk mapping the given point x to the origin. Beyond geometric function theory the conformal radius has various applications in partial differential equations. Therefore it is desirable to extend it to multiply connected and higher-dimensional domains. We discuss two natural extensions. 1. The harmonic radius is defined in terms of the Green's function. Its maximum points, the harmonic centers, play an important role in variational problems involving concentration of energy. They are of the following form: sup{integral(Omega) f(u) : u is an element of H-0(1)(Omega), integral(Omega) \del u\(2) less than or equal to epsilon(2)}. As epsilon --> 0 the solutions concentrate at a harmonic center. Our examples include Bernoulli's free-boundary problem, the plasma problem, and the elastic membrane with a movable obstacle. An efficient method for the numerical approximation of harmonic centers is developed. Thus we can easily construct approximate low energy solutions of the above problems. 2. The hyperbolic radius is defined in terms of the maximal solution of Liouville's equation. It is used for the construction of metrics of constant negative curvature. The two radii are compared analytically and numerically.