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

In this paper, we give a necessary and sufficient condition for a one-dimensional functional View the MathML sourceI(u)=∫01f(u̇(t))dt to satisfy the so-called (PS)-weak lower semicontinuity property on the space View the MathML sourceW1,p((0,1);Rm); that is, View the MathML sourceI(ū)≤lim infk→∞I(uk) for all View the MathML sourceuk⇀ū in View the MathML sourceW1,p((0,1);Rm) and I′(uk)→0I′(uk)→0 in View the MathML sourceW−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.