On prouve que le problème de Cauchy local pour l’équation d’onde sur-critique dans ${ℝ}^d$, $\square u+u^p=0$, $p$ impair, avec $d\ge 3$ et $p\>(d+2)/(d-2)$, est mal posé dans $H^{\sigma }$ pour tout $\sigma \in ]1,{\sigma }_{\mathrm{crit}}[$, où ${\sigma }_{\mathrm{crit}}=d/2-2/(p-1)$ est l’exposant critique.

In this paper new methods for solving elliptic variational inequalities with weakly coercive operators are considered. The use of the iterative prox-regularization coupled with a successive discretization of the variational inequality by means of a finite element method ensures well-posedness of the auxiliary problems and strong convergence of their approximate solutions to a solution of the original problem. In particular, regularization on the kernel of the differential operator and regularization...