Asymptotic Analysis

We consider resonances in the semi‐classical limit (h→0), generated by a single closed hyperbolic orbit, for an operator on R². We determine all such resonances in an h‐independent domain. As an application we determine all resonances generated by a saddle point in a fixed disc around the critical energy.

Résumé. Nous considérons les résonances engendrées par une trajectoire fermée hyperbolique pour un opérateur sur R² dans la limite semi‐classique (h→0). Nous déterminons toutes les résonances dans un domaine indépendant de h. Comme une application nous obtenons toutes les résonances engendrées par un point selle, dans un disque fixé autour de l'énérgie critique.

By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi‐spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which a Pucci–Serrin type identity plays an important role.

The stable Dirichlet observability and exact controllability from boundary for evolution, variable coefficients Lamé system is stated. The observability is proved first for a special sequence of solutions by the use of semiclassical measure analysis. Then one concept coming from G. Lebeau is applied. The controllability is achieved by the use of H.U.M.

We consider the system of equations of Koiter thin shell theory in a slightly simplified form, in the case when the limit (for small thickness) problem is elliptic, i.e., the principal curvatures of the middle surface are everywhere of the same sign. Under singular loadings, which are not in the dual of the energy space of the limit problem, the energy of the solutions grows without limit as the thickness tends to zero. Moreover, it concentrates on boundary layers along the singularities of the loading. We define and prove the convergence to the leading order term in that layers.