Right Type Departmental Bulletin Paper

Let us give the $\mathrm{f}\dot{\mathrm{o}}$llowing simple motivation which arises in such a fundamental subject as the Sobolev imbedding theorems. It is well known that for $\Omega\subset \mathrm{R}^{N} $ , a domain, the continuous imbedding (1) $W_{0}^{1,p}(\Omega) $ $\mathrm{c}arrow $ $L^{q}(\Omega)$ holds provided $p\geq 1,q\geq 1 $ and $N\geq 1\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}6^{r} $ certain relations and that under some additional restrictions this imbedding is compact (2) $W_{0}^{1,p}(\Omega)\mapsto\mapsto L^{q}(\Omega)$ (see e.g. Adams [A] or Kuffier, John and Fu\v{c}\’ik [KJF]). Denoting by $||\cdot||_{q} $ and $||\cdot||1,p$ the norm in $L^{q}(\Omega) $ and in $W_{0}^{1_{1}p}(\Omega) $ , respectively, the imbedding (1) expressed in terms of norms reads as follows: there enists $C>0 $ independent of $u\in W_{0}^{1,\mathrm{p}}(\Omega) $ such that (3) $||u||_{q}\leq C||u||_{1,p}$ holds for any $u\in W_{0}^{1,p}(\Omega) $. Due to the Riedrichs inequality (see [A], [KJF]) the last assertion can be restated also as (4) $||u||_{q}\leq C||\nabla u||_{p}$ for any $u\in W_{0}^{1,p}(\Omega) $ , where $C>0 $ does not depend on $u $. To make the notation clear w