Some geometric relations for equipotential curves

Abstract

Let $U(\mathbf{r}),\mathbf{r}\in \Omega \subset {\mathbb{R}}^2$ be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of $U(\mathbf{r}),\mathbf{r}\in \Omega$ are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature $\kappa (\mathbf{r})$ with the magnitude of gradient $|\nabla U(\mathbf{r})|$ on each level set ("equipotential curve"). One of such inequalities is $⟨[\kappa (\mathbf{r})-⟨\kappa (\mathbf{r})⟩][|\nabla U(\mathbf{r})|-⟨|\nabla U(\mathbf{r})|⟩]⟩\ge 0$, 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 $U(\mathbf{r})=\phi$, and showing that such an entropy is convex in $\phi$. The geometric inequality for $\kappa (\mathbf{r})$ and $|\nabla U(\mathbf{r})|$ 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.

(v1) 12 pages, 1 TikZ figure; (v2) 14 pages, 1 TikZ figure. New results added. Typos corrected; (v3) 14 pages, 1 table, 1 TikZ figure. Revised according to reviewer's comments

References in corpus (2)

• Scaling Limits for Internal Aggregation Models with Multiple Sources
• Two inequalities for convex equipotential surfaces

Cited by in corpus (1)

• Spectral structure of electromagnetic scattering on arbitrarily shaped dielectrics