Estimates of $p$-harmonic functions in planar sectors

Suppose that $p \in (1,\infty], \nu \in [1/2,\infty), {S_{\nu }}=\left\{({x_{1}},{x_{2}})\in {\mathbb{R}^{2}}\setminus \{(0,0)\}:|\phi | \lt \frac{\pi }{2\nu }\right\}$, where $\phi$ is the polar angle of $(x_1, x_2)$. Let $R \gt 0$ and $\omega_p (x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_\nu$ at $x$ with respect to $B(0,R) \cap S_\nu$. We prove that there exists a constant $C$ such that\[C^{-1} {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)} \leq \omega_p (x) \leq C {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)}\]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 Phragmén–Lindelöf theorem for $p$-subharmonic functions in the unbounded sector $S_\nu$. 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 $\partial \mathcal{S}_\nu$.

Keywords

Phragmen–Lindelöf principle, growth estimate, Laplace equation, Laplacian, $p$ Laplace equation, infinity Laplace equation, harmonic measure, $p$ harmonic measure, infinity harmonic measure

2010 Mathematics Subject Classification

35B40, 35B50, 35B53, 35J25, 35J60, 35J70

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This work was partially supported by the Swedish research council grant 2018-03743.

Received 2 November 2021

Received revised 1 July 2022

Accepted 1 August 2022

Published 26 April 2023