Calculus of Variations

Lecturer:

• Luca Nenna (luca.nenna@universite-paris-saclay.fr)

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Program:

The calculus of variations is the study of the minimizers or critical points of “functionals”, which are functions defined in spaces of infinite dimensions, typically functional spaces. Why is it interesting? (1) it provides sometimes a very simple tool for showing existence of (weak) solutions to a problem; (2) many PDEs come from problems in physics, mechanics, etc, and precisely from “variational” principles and are therefore (often minimizing) critical points of some physical energy. (3) many problems in the industry (or finance, etc) are designed as finding the “best” state according to some criterion, and their solution is precisely a minimizer, or maximizer, of this criterion (“optimization”). In particular we will focus on:

• Characterization of the critical points;
• Existence of minimizers;
• Regularity for minimizers (elliptic case);
• The variational convergence aka the $\Gamma-$convergence;
• Some links between large deviation principles and $\Gamma-$convergence;
• Applications to some problems arising in Quantum Mechanics (e.g. Density Functional Theory, Coulomb gas etc) and Machine learning.

References:

• [D] Dacorogna: Direct methods in the calculus of variations.
• [FS]Santambrogio: A Course in the Calculus of Variations.

Schedule and venue:

TBD

Registration: