Lisbon Webinar in Analysis in Differential Equations

Action versus energy ground states in nonlinear Schrödinger equations

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Por Simone Dovetta (Università degli Studi di Roma "La Sapienza").

The talk investigates the relations between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold.

First, we show that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre—Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We see that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers, and we exhibit examples of domains where our results apply.

The matter of the talk refers to joint works with Enrico Serra and Paolo Tilli.

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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.