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 13 to December (not all Fridays).
Where: La Doua, in the building Braconnier, the main building of the Math Dept. Room Seminar 2, in the basement.
Examination: A mid-term exam is scheduled in the week of Nov 4-8 (precise date to be determined, 30% of the mark) and a final exam (50%) end of December or January the remaining 20% of the mark will be based on homeworks and participation in class.
Language: the classes are in English.
Prerequisites: some functional analysis.

Program

There will be 9 classes of 2h40' each.

  • 1) (13/9) 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.1.1
    References: Sections 1.1, 1.2, 1.3 and the beginning of 4.1 of CCV .
  • 2) (27/9)
  • Multi-dimensional Calculus of Variations; Harmonic functions and distributions
    Techniques for existence in higher dimension. Uniqueness of the minimizers. The case of the Dirchlet energy. Mean property and derivatives of harmonic functions. Caccioppoli inequality. Fundamental solutions and Δu=f. Any distribution u which solves Δu=0 is indeed an analytic function.
    References: Again Section 4.1, then 2.1, 2.2 and 2.3 of CCV.
  • 3) (4/10) Convexity and weak semi-continuity; Convex duality.
  • 4) (11/10) Minimal-flow problems; Regularity via duality
  • 5) (18/10) Lp theory for Δu=f
  • 6) (15/11) Campanato spaces and Holder regularity for elliptic PDEs with regular coefficients
  • 7) (22/11) De Giorgi's Holder regularity result without regularity of the coefficients
  • 8) (29/11) General Γ-convergence theory and examples.
  • 10) (13/12)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 final exam of 2020 is here (with its solutions).
    The final exam of 2021 is here (with its solutions).