Calculus of Variations

Program

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

• Characterization of the critical points;
• Existence of minimizers;
• Regularity of minimizers (elliptic case);
• Variational convergence, a.k.a. $\Gamma$-convergence;
• Some links between large deviation principles and $\Gamma$-convergence;
• Applications to problems from Quantum Mechanics (e.g. Density Functional Theory, Coulomb gas) and Machine Learning.

References

• [D] Dacorogna — Direct Methods in the Calculus of Variations.
• [FS] Santambrogio — A Course in the Calculus of Variations.

Schedule & venue

Registration

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