Calculus of Variations and Elliptic Equations

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

Practical Information

Duration: 24h
Schedule: 10 classes of 2h24' each (approximately :-) on Monday morning, from Sep 14 to end November. More precisely, we will start at 9.15am and finish at 11.45am, with 6' break somewhere.
Where: La Doua, in the building Braconnier, the main building of the Math Dept. Room 125, 1st floor (with only one exception, on Monday Sept 21st).
Examination: A mid-term exam is scheduled on November 2nd (40% of the mark) and a final exam (60%) in January.
Language: the classes are in English unless everybody speaks fluent French
Prerequisites: some functional analysis.

Program

There will be 10 classes of 2h24' each.

  • 1) (14/9, room 125) Calculus of Variations in 1D.
    The examples of Geodesics, brachistochrone, economic growth. Euler-Lagrange equation and boundary conditions.Techniques for existence.
    References: two easy informal lecture notes on 1D variational problems (originally written for ENSAE engineers): one by Guillaume Carlier on dynamic problems and one about existence; you can also see the book by G. Buttazzo, M. Giaquinta, S. Hildebrandt One-dimensional variational problems (not easy to read)
  • 2) (21/9, room 112) Convexity and weak semi-continuity.
    Convexity and sufficient conditions, strict convexity and uniqueness. Lower-semicontinuous functionals: strong and weak convergence and link with convexity conditions. Integral functionals with L(x,u,Du).
    References: Giusti, Direct Methods in the Calculus of Variations, chapter 4; for Lusin theorem into arbitrary spaces, see these two pages.
  • 3) (28/9, room 125) Convex duality and minimal-flow problems
    Main notions on convex functions, Legendre transform and subdifferentials. Duality between min ∫ H(x,v) : ∇·v = f and min ∫ H*(x,∇u) + fu with proofs.
  • 4) (5/10, room 125) Regularity via duality
    Laplacian: Δu =f, f∈L2⇒ ∇u∈H1, p-Laplacian: Δpu =f, f∈W1,q⇒ ∇up/2∈H1, Very degenerate problems...
    References for lessons 3 and 4: see these short lecture notes, later transformed in a paper (more complete but probably less student-friendly): see here. As a general reference for convex analysis, one can consider the book Convex analysis by R.T. Rockafellar, or Chapter 1 in Functional Analysis by H. Brezis.
  • 5) (12/10, room 125) Harmonic functions and distributions
    Any distribution u which solves Δu=0 is indeed an analytic function.
    References: You can have a look at Chapter 1 in the book Elliptic Partial Differential Equations by Q. Han and F. Lin.
  • 6) (19/10, room 125) Lp theory for Δu=f
    Marcinkiewicz interpolation ; W2,p regularity of Γ*f if f is Lp and Γ is the fundamental solution of the Laplacian.
    References: You can have a look at Sections 9.2, 9.3 and 9.4 in the book Elliptic Partial Differential Equations of Second Order by D. Gilbarg and N. Trudinger.


    • The week of October 26th is an official break of both ENS and UCBL. Classes resume in November, but on November 2 we had a mid-term exam. Because of the new lockdown, classes resumed remotely.

  • 7) (9/11, zoom, from room 125) Campanato spaces and Holder regularity for elliptic PDEs with regular coefficients
    Morrey and Campanato spaces. Solutions of div(aDu)= div(F) are C1,α if a and F are C0,α.
    References: Chapter 5 in the book by M. Giaquinta L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graph.
  • 8) (16/11, zoom) De Giorgi's Holder regularity result without regularity of the coefficients
    Higher-order Holder regularity for equations with variable coefficients. Applications to the 19th Hilbert problem. Moser's proof of the De Giorgi result: div(aDu)=0 with a bounded from below and above implies that u is C0,α. References: the oroginal paper by J. Moser.
  • 9) (23/11, zoom) General Γ-convergence theory and examples.
    Definitions and properties of Γ-convergence in metric spaces. The example of the Γ-convergence of quadratic functional of the form ∫ an│u'│2. The asymptotics of the optimal location problem (without the Γ-limsup part).
  • 10) (30/11, zoom) The perimeter functional and the Modica-Mortola Γ-convergence result.
    After finishing the Γ-limsup proof from the previous class, a short introduction to BV functions and sets of finite perimeter. Then, the Modica-Mortola approximation of the perimeter functional.
    References for lessons 9 and 10: The book by A. Braides Gamma-Convergence for Beginners or the book by G. Dal Maso An Introduction to Γ-Convergence. For the location problem, look at the short paper by Bouchitte-Jimenez-Rajesh Asymptotic of an optimal location problem. For Modica-Mortola, the book by A. Braides Approximation of Free-Discontinuity Problems. Also look at these (incomplete) short notes by G. Leoni. For the BV space, one can look either at the book Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara, or at Measure theory and fine properties of functions by Evans and Gariepy (which also includes the co-area formula we used).

    • Exercises

    A list of exercises is provided along the course. At the end of the course and before the examination, a homework in the form of a mock exam will also be proposed to students around December.

    Here is a list of exercises, with exercises related to the whole course.

    Here you will find some solutions (this file has not been updated after the first four lectures, for discussing the solution of a more recente exercise, just ask me by email and we'll have a zoom meeting).

    Here is the mock exam that you are encouraged to solve in the next weeks. You can find here the solution.

    Evaluation

    The mid-term exam took place (remotely) on Monday November 2nd. Here is the examination text, with the solutions.
    The final exam took place on January 11th, also remotely. Here is the examination text, with the solutions.