Hardy-Sobolev inequalities in unbounded domains and heat kernel estimates
The author studies Hardy-Sobolev inequalities in unbounded domains. Specifically, new geometric assumptions on the external domains \(\Omega \subset \mathbb{R}^n\), \(n\geq 3\), are introduced and this assumption was used to derive a Hardy inequality with an optimal constant involving the distance on the boundary. Furthermore, it is shown that under some additional assumptions on \(\Omega\), this Hardy inequality can be improved by adding a Sobolev-type term with critical exponent. Several results on Hardy and Hardy-Sobolev-type inequalities in exterior domains are obtained. Finally, the Harnack inequality up to the boundary for the positive solutions of the problem \(u_{t} =\triangle u+\frac1{4}\frac{u}{\operatorname{dist}^{2}(x,\delta,\Omega)}\) and the heat kernel estimates for small times are derived, proved and discussed.
scientific article; zbMATH DE number 6140428