BIRS Workshop Lecture Videos
Symmetry in interpolation inequalities Dolbeault, Jean
In presence of radially symmetric weights in an Euclidean space, it is well known that symmetry breaking may occur: the minimizing functions of functionals which are invariant under rotation are, in some cases, not radially symmetric. This usually follows from a linear stability analysis of the minimizers. The goal of this lecture is to investigate the reverse property and establish, in some interpolation inequalities, when the local linear stability of radial optimal functions means that the global optimal functions are in fact radially symmetric. The main tool is a flow: spectral properties of the linearized operator around radial optimizers can be interpreted in terms of large time asymptotics of the solution to the evolution equation and related with the optimal constant in the inequality. The symmetry range depends on a parameter, which can be used to classify the solutions of the Euler-Lagrange equations. A singular limit can be identified when the parameter takes large values. The interpolation inequality has a spectral counterpart for Schrödinger operators, which allows to quantify the distance to a semi-classical regime. Various equivalent problems on spheres and cylinders can also be considered. This is joint work with various collaborators, among which Maria J. Esteban, Michal Kowalczyk, Michael Loss, and Matteo Muratori.
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