Optimal constants in functional inequalities for vector fields with several types of constraints
In this talk we study functional inequalities for vector-valued test functions with several types of constraints (including solenoidal condition) in the N-dimensional Euclidean space. The question we address is what is the optimal value of the constant in each of the inequalities. In Poincaré inequality for tangential vector fields on the sphere, for example, the best constant has physical meaning as the exchange energy in micromagnetics dealing with the prediction of magnetic behaviors at a spherical shell with thicknesses on the sub-micrometer length scale. On the other hand, this value is of pure mathematical interest since such a value depends on the specific form of constraint imposed which could not be found for scalar-valued test functions. As time permits, I would like to introduce some past and recent research findings on this topic.
16:15-17:15
Vitaly Moroz 氏 (Swansea University, UK)
Normalized solutions and limit profiles of the Gross-Pitaevskii-Poisson equation
Gross-Pitaevskii-Poisson (GPP) equation is a nonlocal modification of the Gross-Pitaevskii equation with an attractive Coulomb-like term. It appears in the models of self-gravitating Bose-Einstein condensates proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We investigate the existence of prescribed mass (normalised) solutions to the GPP equation, paying special attention to the shape and asymptotic behaviour of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. In particular, we show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation, or to a compactly supported radial ground state of the integral Thomas-Fermi equation. In different regimes the constructed solutions include global minima, local but not global minima and unstable mountain-pass type solutions. This is a joint work with Riccardo Molle (Rome Tor Vergata) and Giuseppe Riey (Calabria).