Narrow Results

The paper reviews some parts of classical potential theory with applications to two-dimensional fluid dynamics, in particular vortex motion. Energy and forces within a system of point vortices are similar to those for point charges when the vortices are kept fixed, but the dynamics is different in the case of free vortices. Starting from Euler’s and Bernoulli’s equations we derive these laws. Letting the number of vortices tend to infinity leads in the limit to considerations of equilibrium distributions, capacity, harmonic measure and many other notions in potential theory. In particular various kinds of Green functions have a central role in the paper, and we make a distinction between electrostatic and hydrodynamic Green functions. We also consider the corresponding concepts in the case of closed Riemann surfaces provided with a metric. From a canonically defined monopole Green function we rederive much of the classical theory of harmonic and analytic forms on Riemann surfaces. In the final section of the paper we return to the planar case, which then reappears in the form of a symmetric Riemann surface, the Schottky double. Bergman kernels, electrostatic and hydrodynamic, come up naturally as well as other kernels, and associated to the Green function there is a certain Robin function which is important for vortex motion, and which also relates to capacity functions in classical potential theory.

In several phenomena of practical interest, such as catalyst deactivation, fouling in heat transfer and other systems of technological and scientific relevance, an irregular surface accessed by diffusion can be progressively passivated. In a diffusion limited situation, an interface that works unevenly due to Laplacian screening is simultaneously and unevenly passivated. To study this phenomenon, we describe a process in which the regions of the surface that are initially working, are transformed into passive, reflecting zones. As a consequence, at each step, a new part of the interface becomes active. In turn, this new active zone is passivated, and so on. It is found that the length of the successive active zones remains approximately constant for a prefractal interface. The concept of active zone in Laplacian transport can then be successfully extended to elucidate this self-limiting behavior of the passivation process. A conjecture is then proposed which states that, in D=2, the information dimension of the harmonic measure on a fractal supporting a "passivated or reflecting subfractal" (of smaller dimension) is equal to 1. This constitutes an extension of Makarov theorem. From our results, fractal geometry is revealed as a potential candidate to engineer substrate morphologies that are robust to Laplacian passivation.

We study two actions of a stochastic flow ${\phi }_t$ on the space of $0$-currents $T$ of a differentiable manifold $M$. In particular, we give conditions on a current $T$ to be invariant under these actions. Also, we apply our results to the case of $0$-currents that come from harmonic measures associated to certain type of foliations on homogeneous spaces.

This is a brief survey of results related to planar harmonic measure, roughly from Makarov’s results of the 1980’s to recent applications involving 4-manifolds, dessins d’enfants and transcendental dynamics. It is non-chronological and rather selective, but I hope that it still illustrates various areas in analysis, topology and algebra that are influenced by harmonic measure, the computational questions that arise, the many open problems that remain, and how these questions bridge the gaps between pure/applied and discrete/continuous mathematics.

We survey some results on non-uniform hyperbolicity, geometric pressure and equilibrium states in one-dimensional real and complex dynamics. We present some relations with Hausdorff dimension and measures with refined gauge functions of limit sets for geometric coding trees for rational functions on the Riemann sphere. We discuss fluctuations of iterated sums of the potential −t log |f′| and of radial growth of derivative of univalent functions on the unit disc and the bound-aries of range domains preserved by a holomorphic map f repelling towards the domains.

A central question in Potential Theory is the extent to which the geometry of a domain influences the boundary regularity of solutions to divergence form elliptic operators. To answer this question one studies the properties of the corresponding elliptic measure. On the other hand one of the central questions in Geometric Measure Theory (GMT) is the extent to which the regularity of a measure determines the geometry of its support. The goal of this paper is to present a few instances in which techniques from GMT and Harmonic Analysis come together to produce new results in both of these areas. In particular, the work described in section 3 makes it clear that for this type of problems in higher dimensions, GMT is the right alternative to complex analysis in dimension 2.