On a Restricted Weak Lower Semicontinuity for Smooth Functionals on Sobolev Spaces

This paper is motivated by a problem suggested in Müller [11] that concerns the weak lower semicontinuity of a smooth integral functional I(u) on a Sobolev space along all its weakly convergent minimizing sequences. Here we study a restricted weak lower semicontinuity of I(u) along all weakly convergent Palais-Smale sequences (that is, sequences {uk} satisfying I′(uk)→ 0). In view of Ekeland’s variational principle, this restricted weak lower semicontinuity, replacing the usual (unrestricted) weak lower semicontinuity in the direct method of calculus of variations, is sufficient for the existence of minimizers under the standard coercivity assumption. The main purpose of the paper is to study the relationships of this restricted weak lower semicontinuity condition with the usual weak lower semicontinuity condition that is known to be equivalent to the Morrey quasiconvexity in the calculus of variations. We show that the two conditions are not equivalent in general, but are equivalent in certain interesting cases.