Boundary density and the Green function (Q805803)

It is shown that for a large class (described below) of domains \(\Omega\) in \({\mathbb{R}}^ m\) (m\(\geq 2)\) the gradient of the Green function has the following integrability property. If G is the Green function of \(\Omega\) with pole Q and \(\Gamma\) is an (m-1)-dimensional hyperplane with \(Q\not\in \Gamma\), then there exists \(p_ 0>1\), depending on \(\Omega\), such that \(\int_{\Gamma \cap \Omega}\| \nabla G\|^ p d\sigma <C\) whenever \(1\leq p\leq p_ 0\); here \(\sigma\) denotes (m-1)-dimensional measure and C depends on p, \(\Omega\), dist(Q,\(\partial \Omega)\) and dist(Q,\(\Gamma\)). The domains \(\Omega\) for which this result is proved are those whose complements satisfy a density condition which we now describe. Let B(x,r) denote the open ball of centre x and radius r, and let \(d(x)=dist(x,\partial \Omega)\). The domain \(\Omega\) is said to have the \(\alpha\)-dimensional density condition (\(\alpha\) DC), where \(0<\alpha \leq m-1\), if there is a fixed \(\zeta >0\) such that \(\Lambda^{\alpha}(\{y/d(x):\) \(y\in B(x,2d(x))\setminus \Omega \})>\zeta\) for all \(x\in \Omega\); here \(\Lambda^{\alpha}(S)\) is the \(\alpha\)-dimensional content of a set S, given by \(\Lambda^{\alpha}(S)=\inf \sum r^{\alpha}_ n\), where the infimum is over all coverings of S by countably many balls of radii \(r_ n\). The integrability result is proved to hold when \(\Omega\) has the (m-1)DC, and an example is given to show that the (m-1)DC cannot be replaced by the \(\alpha\) DC for any \(\alpha\in (0,m-1)\).