Calculus of Variations and Elliptic Equations

Master course - Master en Mathématiques Avancées, UCBL and ENS Lyon

Practical Information

Duration: 24h
Schedule: 9 classes of 2h40' each (approximately :-) on Friday morning, from Sep 19 to December (not all Fridays).
Where: La Doua, in the building Braconnier, the main building of the Math Dept. Room 125
Examination: A mid-term exam is scheduled on Nov 7 at 9.30am (30% of the mark) and a final exam (50%) on Dec 19 at 9am; the remaining 20% of the mark will be based on homeworks and participation in class.
Language: the classes are currently in French.
Prerequisites: some functional analysis.

Program

There will be 9 classes of 2h40' each.

  • 1) (19/9) Calculus of Variations in 1D.
    The examples of Geodesics, brachistochrone, economic growth. Techniques for existence and non-existence. Euler-Lagrange equation and boundary conditions. Sufficient conditions in case of convexity.
    References: Sections 1.1, 1.2, 1.3 and the beginning of 4.1 of CCV .
  • 2) (26/9) Multi-dimensional Calculus of Variations; Harmonic functions and distributions
    Techniques for existence in higher dimension. The case of the Dirchlet energy. Mean property and derivatives of harmonic functions. Fundamental solutions and Δu=f. Any distribution u which solves Δu=0 is indeed an analytic function.
    References: Sections 2.1, 2.2 and 2.3 of CCV.
  • 3) (3/10) Convexity and weak semi-continuity; Convex duality.
  • 4) (10/10) Minimal-flow problems; Regularity via duality
    • Warning: no class on October 17.
  • 5) (24/10) Lp theory for Δu=f
  • 6) (14/11) Campanato spaces and Holder regularity for elliptic PDEs with regular coefficients
  • 7) (21/11) De Giorgi's Holder regularity result without regularity of the coefficients
    • Warning: no class on November 28.
  • 8) (5/12) General Γ-convergence theory and examples.
  • 9) (12/12)The perimeter functional and the Modica-Mortola Γ-convergence result.

    • References: CCV means the book A course in the Calculus of Variations that I wrote in 2023 based on my previous courses. The pdf file of the book has been sent to all registered students.

    Exercises and previous examinations

    A list of exercises from CCV will be given, each student should hand three solutions as a homework, according to the rules we discussed.

    The mid-term exam of 2020 is here (with solutions).
    The mid-term exam of 2021 is here (with the solutions).
    The mid-term exam of 2024 is here (with the solutions).
    The final exam of 2020 is here (with its solutions).
    The final exam of 2021 is here (with its solutions).
    The final exam of 2024 is here (with the solutions).

    Evaluation

    The mid-term exam will take place on Friday November 7.

    The final exam will take place on Friday December 19.