Volume mean densities for the heat equation
Let \(D\) be a bounded domain in \(\mathbb R^d\times (-\infty ,0)\), \(d\geq 1\), such that \(z_0=(0,0) \in \overline D\). If \(0<a<1/e\) and \(D=\{ (x,-t)\in \mathbb R^{d+1}; |x|<\sqrt{2dt \log(1/t)}, 0<t<a\} \), then there exists a Borel measurable function \(w\) on \(D\) such that \(\inf w(D)>0\) and \(\int_D w(z)v(z)\;dz\leq v(z_0)\) for every supertemperature \(v\) on a neighbourhood of \(\overline D\). On the other hand, if there exist \(t_0,r\in (0,\infty )\) such that \(\{ (x,t)\in \mathbb R^{d+1}; 0<t<t_0, |x|<(d+1)d^{-1}\sqrt{2dt\log (r/t)}\} \subset D\), then there is no Borel measurable function \(w\) on \(D\) with these properties.
scientific article; zbMATH DE number 6369345
scientific article