Lisbon Webinar in Analysis and Differential Equations

New Gagliardo-Nirenberg inequalities and applications to biharmonic NLS

Sala P3.10, Instituto Superior Técnico (com transmissão via Zoom)

Por Rainer Mandel (Karlsruher Institut für Technologie).

We present new existence results for nontrivial solutions of some biharmonic Nonlinear Schrödinger equation in $\mathbb{R}^N$ that are based on a constrained minimization approach. Here the main difficulty comes from the fact that spherical rearrangements need not decrease the energy so that more sophisticated arguments are needed to overcome the lack of compactness. A new and intrinsically motivated tool is given by a new class of Gagliardo-Nirenberg inequalities where, essentially, the Laplacian in the classical Gagliardo-Nirenberg inequality is replaced by the Helmholtz operator. Having explained the relevance of such inequalities for our analysis, we comment on their proofs and related questions from Harmonic Analysis. Finally, we shall mention a symmetry-breaking phenomenon related to our results that was recently observed by Lenzmann and Weth. Accordingly, the talk covers topics from the Calculus of Variations as well as Harmonic Analysis or, more specifically, Fourier Restriction Theory.

Transmissão via Zoom.

The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.