Calculus of Variations

Lecturer:

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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:

Institut de Mathématique d’Orsay, building 307 :

  • Tuesday from 14h00 to 17h30, room 1A7,
  • Friday from 9h00 ro 12h30, room 2L8.
Day Topic Lecture notes
26/11/2024 Introduction and characterisation of critical points Lecture 1
29/11/2024 Calculus of Variations in 1D: existence, optimality condition and regularity [FS] :sec 1.2, 1.3 look at the boxes “good to know” for some reminders
03/12/2024 Regularity in 1D and Calculus of Variations in high dimension Regularity and reminders [FS]: sec 2.1, 2.2
06/12/2024 Semi-continuity, convexity and general existence thm [FS]: sec 3.1, 3.2, 3.4
10/12/2024 Convexity and duality [FS]: sec 4.3
13/12/2024 Regularity via duality (some degenerate PDEs) and from Optimal Transport to Mean Field Games [FS]: sec 4.4, 4.6 (partially)
17/12/2024 $\Gamma$-convergence I $\Gamma-$convergence I
20/12/2024 $\Gamma$-convergence II $\Gamma-$convergence II
07/01/2024 Surprise: how to disprove a conjecture in mathematical physics! Lecture Ground state
17/01/2024 Exam 9h30 room 0A7 Exam 2022

Ramarks, Typos and beyond!

06/12/2024 :

  • For the existence for the functional $\mathcal{E}(u)\int_\Omega F(x,u)+|\nabla u|^\alpha+\int_{\partial \Omega}\psi(x,Tr[u])d\mathcal H^{d-1}$, the measure of the set ${x\in A;|;g(Tr[u_n])>\ell}$ is at most $C/\ell$. Taking $l=2C/\mathcal{H}^{d-1}(A)$ then we get everything we need to guarantee that there exists an $M$ independent of $n$ such that $|Tr[u_n]|<M$.
  • Lemma For any constant $ M > 0 $, let $ f_M : \mathbb{R}^d \to \mathbb{R}^d $ be defined via $f_M(v) := v 1_{|v|\leq M}(v)$. Given a sequence $ v_n \to v $ in $ L^1(\Omega) $, consider for any $ M > 0 $ the sequence $ v_{n, M} := f_M(v_n) $. Then one can extract a subsequence and to find, for every $ M \in \mathbb{N} $, a function $ v^{(M)} $ such that, for every $ M $, the sequence $ v_{n, M} $ weakly converges in $ L^1 $ to $ v^{(M)} $. Moreover, $ \lim_{M \to \infty} ||v^{(M)} - v||_{L^1(\Omega)} = 0. $

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