Estimates of $p$-harmonic functions in planar sectors

TL;DR

This paper derives explicit estimates for p-harmonic measure in planar sectors, leading to growth bounds for p-harmonic functions, a sharp Phragmen-Lindel"of theorem, and uniqueness results for positive p-harmonic functions.

Contribution

It provides explicit formulas for p-harmonic measure in sectors and applies these to growth estimates, boundary behavior, and uniqueness of p-harmonic functions.

Findings

Established bounds for p-harmonic measure in sectors.

Derived growth estimates for p-sub- and p-superharmonic functions.

Proved a sharp Phragmen-Lindel"of theorem and uniqueness results.

Abstract

Suppose that $p \in (1,\infty]$, $\nu \in [1/2,\infty)$, $\mathcal{S}_\nu = \left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |\phi| < \frac{\pi}{2\nu}\right\}$, where $\phi$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and $\omega_p(x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_\nu$ at $x$ with respect to $B(0, R)\cap \mathcal{S}_\nu$. We prove that there exists a constant $C$ such that \begin{align*} C^{-1}\left(\frac{|x|}{R}\right)^{k(\nu,p)} \, \leq \omega_p(x) \, \leq C \left(\frac{|x|}{R}\right)^{k(\nu,p)} \end{align*} whenever $x\in B(0,R) \cap \mathcal{S}_{2\nu}$ and where the exponent $k(\nu,p)$ is given explicitly as a function of $\nu$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmen-Lindel\"of theorem for $p$-subharmonic functions in the unbounded sector $\mathcal{S}_\nu$. Moreover, if $p = \infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $\mathbb{R}^n$. Finally, when $\nu \in (1/2, \infty)$ and $p\in (1,\infty)$ we prove uniqueness (modulo normalization) of positive $p$-harmonic functions in $\mathcal{S}_\nu$ vanishing on $\partial\mathcal{S}_\nu$.