Nebenhülle and the Gleason problem

This article concerns the Gleason property as a local phenomenon. We prove that there always exists an open set where the domain $D\Subset \mathbb{C}^{2}$ has the Gleason $\mathcal{B}$ property whenever the boundary of the Nebenhülle of $D$ coincides with a $C^{2}$ smooth part of the boundary $bD$; here $\mathcal{B}$ is either one of the Banach algebras, $H^{\infty}$ or $A$. As an easy consequence of this, we see that if the extremal boundary points are $C^{2}$-smooth, then $D$ has the Gleason $\mathcal{B}$ property close to those points. Also a $\overline{\partial}$-problem for locally supported forms is solved.