Some geometric relations for equipotential curves

Abstract

Let $U\left(r\right),r\in \Omega \subset R^2$ be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of $U\left(r\right),r\in \Omega$ are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature $\kappa \left(r\right)$ with the magnitude of gradient $\left|\nabla U\right(r\left)\right|$ on each level set (“equipotential curve”). One of such inequalities is $〈\left[\kappa \right(r)-〈\kappa (r)〉]\left[\right|\nabla U\left(r\right)|-〈|\nabla U\left(r\right)|〉]〉\ge 0$, where $〈\cdot 〉$ denotes average over a level set, weighted by the arc length of the Jordan curve. We prove such a geometric inequality by constructing an entropy for each level set $U\left(r\right)=\phi$, and showing that such an entropy is convex in φ. The geometric inequality for $\kappa \left(r\right)$ and $\left|\nabla U\right(r\left)\right|$ then follows from convexity and monotonicity of our entropy formula. A few other geometric relations for equipotential curves are also built on a convexity argument.