On a Theorem of Wolff Revisited
Murat Akman, John L. Lewis, Andrew L. Vogel · arXiv (Cornell University) · 2020
We study $p$-harmonic functions, $ 1 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p eq 2 < \infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic function, $ V = V ( r e^{iθ} )$, which is $ 2π/N $ periodic in the $ θ$ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with Lipschitz norm $\leq c N$ on $ \partial B ( 0, 1 )$ satisfying $V(0)=0$ and $ c^{-1} \leq \int_{-π}^π V ( e^{iθ} ) d θ\leq c$. In case $2