On the torsion function with mixed boundary conditions

Let $D$ be a non-empty open subset of ${\mathbb{R}}^m,m\ge 2$, with boundary $\mathrm{\partial }D$, with finite Lebesgue measure $\mathrm{\mid }D\mathrm{\mid }$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading asymptotic behaviour as $\epsilon \downarrow 0$ of the $L^{\mathrm{\infty }}$ norm of the torsion function with a Neumann boundary condition on $\mathrm{\partial }D$, and a Dirichlet boundary condition on $\mathrm{\partial }(\epsilon K)$, in terms of the first eigenvalue of the Laplacian with corresponding boundary conditions. These estimates quantify those of Burdzy, Chen and Marshall who showed that $D\setminus K$ is a non-trap domain.