  Nelson Dunford and Billy James Pettis [{\em Trans. Amer. Math. Soc.}, 47
(1940), pp. 323--392] proved that relatively weakly compact subsets of $ L^1 $
coincide with equi-integrable families. We expand it to the case of $ W^{k,1} $
- the non-reflexive Sobolev space - by a tailor-made isometric operator. Herein
we extend an existence theorem of minimizers from reflexive Sobolev spaces to
non-reflexive ones.
