# Oberseminar: Calculus of Variations and Applications

## Winter Semester 2021

The online seminar takes place on Wednesday, starting from 4:15 pm, unless indicated otherwise.Zoom link (Meeting ID: 952 0698 8894, Passcode: 800511).

## Speakers

- 10.11.2021:
**David Gontier**(Paris Dauphine University).

Title: Density Functional Theory for two-dimensional homogeneous materials.

Abstract: In this talk, we consider DFT models, when applied to semi-infinite systems, that is systems which are homogeneous in s dimensions, and localized in d other dimensions. We prove that one can reduce the DFT equations to the remaining d variables, and show how the different terms in the energy are modified. We prove in particular that the Pauli principle constraint is replaced by a penalization term in the energy. This is joint work with Salma Lahbabi and Abdallah Maichine.

- 02.12.2021:
**Hoai-Minh Nguyen**(Sorbonne University).

Title: Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenbergâ€™s inequalities associated with the Coulomb terms.

Abstract: I will discuss the full range of Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenbergâ€™s inequalities associated with the Coulomb terms when the derivative s is between 0 and 1. Their applications in deriving new one body and many body Hardy-Lieb-Thirringâ€™s inequalities are also mentioned. This is joint work with Arka Mallick.

- 08.12.2021:
**Andreas Deuchert**(University of Zurich).

Title: Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons.

Abstract: We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional - a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose-Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions. The above is joint work with Robert Seiringer.

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