$p$ harmonic measure in simply connected domains revisited

Abstract:

Let $\Omega$ be a bounded simply connected domain in the complex plane, $\mathbb {C}$. Let $N$ be a neighborhood of $\partial \Omega$, let $p$ be fixed, $1 < p < \infty ,$ and let $\hat u$ be a positive weak solution to the $p$ Laplace equation in $\Omega \cap N.$ Assume that $\hat u$ has zero boundary values on $\partial \Omega$ in the Sobolev sense and extend $\hat u$ to $N \setminus \Omega$ by putting $\hat u \equiv 0$ on $N \setminus \Omega .$ Then there exists a positive finite Borel measure $\hat \mu$ on $\mathbb {C}$ with support contained in $\partial \Omega$ and such that \begin{eqnarray*} \int | \nabla \hat u |^{p - 2} \langle \nabla \hat u , \nabla \phi \rangle dA = - \int \phi d \hat \mu \end{eqnarray*} whenever $\phi \in C_0^\infty ( N ).$ In this paper we continue our studies by establishing endpoint type results for the Hausdorff dimension of this measure in simply connected domains. Our results are similar to the well known result of Makarov concerning harmonic measure in simply connected domains.