Résumé (EN)

The talk given on the occasion of the ISIAM 2016 conference was mainly about rigidity results for nonnegative solutions of semilinear elliptic equation on infinite cylinder-like domains or in the Euclidean espace and as a consequence, about optimal symmetry properties for the optimizers of the Caffarelli–Kohn–Nirenberg inequalities. This text contains the main results presented in that conference. All the results will be stated in the simple case of spherical cylinders, but similar, even if less precise, results can also be stated and proved for general cylinders generated by any compact smooth Riemannian manifold without a boundary. Other consequences from the results below are optimal estimates for the principal eigenvalue of Schrödinger operators on infinite cylinders. The text below is an extended abstract of that talk.