(PS)-weak lower semicontinuity in one dimension: A necessary and sufficient condition

Abstract

In this paper, we give a necessary and sufficient condition for a one-dimensional functional I ( u ) = ∫ 0 1 f ( u ̇ ( t ) ) d t to satisfy the so-called (PS)-weak lower semicontinuity property on the space W 1 , p ( ( 0 , 1 ) ; R m ) ; that is, I ( u ̄ ) ≤ lim inf k → ∞ I ( u k ) for all u k ⇀ u ̄ in W 1 , p ( ( 0 , 1 ) ; R m ) and I ′ ( u k ) → 0 in W − 1 , p p − 1 ( ( 0 , 1 ) ; R m ) . The result shows that in this case the property of (PS)-weak lower semicontinuity is in general not equivalent to convexity of the functional if m ≥ 2 .