Some geometric relations for equipotential curves

Abstract

Let $U(\boldsymbol r),\boldsymbol r\inΩ\subset \mathbb R^2$ be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of $U(\boldsymbol r),\boldsymbol r\inΩ$ are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature $κ(\boldsymbol r) $ with the magnitude of gradient $ |\nabla U(\boldsymbol r)|$ on each level set ("equipotential curve"). One of such inequalities is $ \langle [κ(\boldsymbol r)-\langleκ(\boldsymbol r)\rangle][|\nabla U(\boldsymbol r)|-\langle |\nabla U(\boldsymbol r)|\rangle]\rangle\geq0$, where $ \langle \cdot\rangle$ 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 $U(\boldsymbol r)=φ$, and showing that such an entropy is convex in $φ$. The geometric inequality for $κ(\boldsymbol r) $ and $ |\nabla U(\boldsymbol r)|$ then follows from the convexity and monotonicity of our entropy formula. A few other geometric relations for equipotential curves are also built on a convexity argument.