A critical Moser type inequality with loss of compactness due to infinitesimal shocks

We consider a one-dimensional integral inequality of Moser type: setJ(c)(v) = integral(1)(0) e(c(s)v2(s)) ds and consider sup({integral 01 vertical bar v'vertical bar 2=1,v(0)=0}) J(c)(v)We show that the supremum remains finite up to the optimal coefficient c(1)(s) = 1/s (log e/s + log log e/s). Indeed, for c(gamma) = 1/s (log e/s + gamma log log e/s), with gamma > 1, the supremum is infinite. For c(1) the inequality is critical with loss of compactness: the functional J(c1) fails to be weakly continuous along the infinitesimal Moser sequence w(n)(t) := t root n (0 <= t <= 1/n) w(n)(t) = 1 root n (1/n <= t <= 1). Since w'(t) = root n (0 <= t <= 1/n), one may say that w(n) develops an infinitesimal shock at the origin.