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

We consider a one-dimensional integral inequality of Moser type: set J_c(v) = ∫ _0^1 e^c(s) v^2(s) ds and consider sup _{∫ _0^1 |v'|^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(loge/s + logloge/s) . Indeed, for c_γ = 1/s(loge/s + γlogloge/s) , with γ > 1 , the supremum is infinite. For c_1 the inequality is critical with loss of compactness: the functional J_c_1 fails to be weakly continuous along the infinitesimal Moser sequence w_n(t):= t√(n) (0 ≤ t ≤1/n) w_n(t) = 1/√(n) (1/n≤ t ≤ 1) . Since w'(t) = √(n) (0 ≤ t ≤1/n) , one may say that w_n develops an infinitesimal shock at the origin.