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Daily Overview |
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RAR1: Recent advances in regularity theory for PDEs Location: M628 Session Chair: Simon Nowak Session Chair: Marvin Weidner | |
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Nondegeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality Goethe-Universität Frankfurt, Germany The Caffarelli-Kohn-Nirenberg (CKN) inequality is a first-order scale-invariant generalization of the Sobolev inequality. Differently from the latter, as observed by Felli-Schneider (2003), minimizers under a radial symmetry constraint need not be unconstrained local minimizers for the CKN inequality. This induces a remarkable transition from symmetry to symmetry-breaking of optimizers, which has been completely understood by Dolbeault-Esteban-Loss (2016). In this talk, I will present some new results about a fractional-order variant of the CKN inequality, for which to date the above questions remain wide open. More precisely, we show non-degeneracy of minimizers for a certain parameter range and, as a consequence, obtain a quantitative stability inequality. Moreover, we exhibit a new region for which every minimizer must be radially symmetric. Along the way, we rely on careful regularity considerations for distributional solutions to fractional-order equations. This is joint work with Nicola De Nitti (Bari) and Federico Glaudo (Princeton). | |
