Some geometric relations for equipotential curves
Yajun Zhou
TL;DR
This work analyzes how the curvature of equipotential curves and the gradient magnitude of a harmonic function are statistically correlated in a 2D exterior Dirichlet problem. It introduces four level-set functionals $\mathscr H(\varphi)$, $\mathscr E(\varphi)$, $\mathscr F(\varphi)$, and $\mathscr L(\varphi)$ and proves their convexity in the level parameter $\varphi$, from which sharp covariance-type inequalities between $\kappa$ and $E$ on each level set follow; a conservation law $\mathscr E(\varphi)\equiv \frac{4\pi^2}{\Phi}$ further ties curvature and gradient statistics. The approach is entropy-monotonicity driven, aligning with ideas of Colding and Colding–Minicozzi to provide monotone isoperimetric-type trends and supporting heuristics for dendritic patterns in Hele-Shaw flow and diffusion-limited aggregation; a complementary 2-inD analysis using Green's functions yields reversed-sign inequalities. The paper also discusses extensions to higher dimensions, highlighting challenges like potential critical points and the absence of a global mapping tool, and outlines avenues for conditional generalizations and future work. Overall, it provides a rigorous geometric-entropic framework to quantify how level-set geometry constrains or correlates with field intensities in harmonic exterior problems, with potential implications for electrostatics, fluid dynamics, and pattern formation.
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)=\varphi $, and showing that such an entropy is convex in $\varphi$. 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.