On an inequality on Sobolev spaces involving Nečas inequality and other results

In the years 1946-1947, Friedrichs [9, 10] presented an interesting way to obtain
inequalities in the Sobolev spaces. The application of its method is restricted
to domains which have particular but not too restrictive properties as will be
described in what follows. We found interesting to investigate the possibility to
obtain by this way the well known Nečas’ inequality (equivalent to Lions’ lemma
[1, 2]). We were led to another inequality, not so known but containing Nečas’
inequality as a consequence. The present paper presents the establishment of
this inequality and some of its consequences, including Nečas inequality, Korn’s
inequality, and a lot of related results. As exposed in the frame of Hilbert spaces,
our presentation is as intuitive as possible.