Log-lightning computation of capacity and Green's function

See Video Abstract (click the "Video Abstract" button next to the "PDF" button) A basic measure of the size of a set E in the complex plane is the logarithmic capacity cap(E). Capacities are known analytically for a few simple shapes like ellipses, but in most cases they must be computed numerically. We explore their computation by the new "log-lightning'' method based on reciprocal-log approximations in the complex plane. For a sequence of 16 examples involving both connected and disconnected sets E, we compute capacities to 8–15 digits of accuracy at great speed in MATLAB. The convergence is almost-exponential with respect to the number of reciprocal-log poles employed, so it should be possible to compute many more digits if desired in Maple or another extended-precision environment. This is the first systematic exploration of applications of the log-lightning method, which opens up the possibility of solving Laplace problems with an efficiency not achievable by previous methods. The method computes not just the capacity, but also the Green's function and its harmonic conjugate. It also extends to "domains of negative measure" and other Riemann surfaces.

Properties of the ℳ-Harmonic Conjugate Operator

We define the ℳ-harmonic conjugate operator K and prove that it is bounded on the nonisotropic Lipschitz space and on BMO. Then we show K maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of ℳ-harmonic conjugate operator with Lp symbol.