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

In this paper, we give a necessary and sufficient condition for a one-dimensional functional I(u)=∫01f(u̇(t))dt to satisfy the so-called (PS)-weak lower semicontinuity property on the space W1,p((0,1);Rm); that is, I(ū)≤lim infk→∞I(uk) for all uk⇀ū in W1,p((0,1);Rm) and I′(uk)→0I′(uk)→0 in W−1,pp−1((0,1);Rm). 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≥2m≥2.