On two inequalities of the Serrin type (Q1080546)

Let \(n\geq 3\) and \(B=\{x\in R^ n\); \(0\leq x_ j\leq a_ j\), \(j=1,...,n\}\) with \(a_ j>0\). It is proved that for a function \(u\in C^ 1(B)\) vanishing on \(\partial B\) the integral \(\int_{B}| u(x)|^ pdx\) is estimated by \[ K(\int_{B}| \nabla u(x)|^ Q_ Qdx)^{p/Q}\quad if\quad 1\leq p<Q \] and by \[ L(\int_{B}| u(x)|^{P(p-1)}dx)^{1/P}(\int_{B}| \nabla u(x)|^ Q_ Qdx)^{1/Q}\quad if\quad p\geq 1,\quad Q>1 \] and \(P^{-1}+Q^{-1}=1\). Here \(| \nabla u|^ Q_ Q=\sum^{n}_{i=1}| \partial u/\partial x_ i|^ Q\) and the constants K, L are as follows: \[ K=(\prod^{n}_{i=1}a_ i\sum^{n}_{j=1}a_ j^{Q/(Q-p)})^{(Q-p)/Q}/(2^ pn),\quad L=p(\sum^{n}_{j=1}a^ P_ j)^{1/P}/(2n). \] The proof is very easy and the exponent p general (in contrast to the original one n/(n-1) of J. Serrin), on the other hand the domain B has a very special shape.