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 13 to end of November. More precisely, we will start at 9.30am and finish at 12pm, 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 27st).
Examination: A mid-term exam is scheduled on October 25 (30% of the mark) and a final exam (50%) on December 13; the remaining 20% of the mark wil lbe based on homeworks and participation in class.
Language: the classes are in English.
Prerequisites: some functional analysis.

Program

There will be 10 classes of 2h24' each.

1) (13/9, room 125) Calculus of Variations in 1D.
The examples of Geodesics, brachistochrone, economic growth. Euler-Lagrange equation and boundary conditions.Techniques for existence.
References: I wrote some partial lecture notes here. You can also have a look at these notes in French (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) (20/9, room 125) 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; I also wrote some partial lecture notes here

3) (27/9, room 112) 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) (4/10, room 125) Regularity via duality
Laplacian: Δu =f, f∈L²⇒ ∇u∈H¹, p-Laplacian: Δ_pu =f, f∈W^1,q⇒ ∇u^p/2∈H¹, Very degenerate problems...
References for lessons 3 and 4: some partial lecture notes for the beginning of lesson 3 are here; for the duality result, see the beginning of 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) (11/10, room 125) Harmonic functions and distributions
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: You can have a look at Chapter 1 in the book Elliptic Partial Differential Equations by Q. Han and F. Lin.
but you'll also find some partial lecture notes here.

6) (18/10, room 125) L^p theory for Δu=f
Marcinkiewicz interpolation ; W^2,p regularity of Γ*f if f is L^p 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. In what concerns global regularity using reflections, have a look at these partial lecture notes

7) Campanato spaces and Holder regularity for elliptic PDEs with regular coefficients
Morrey and Campanato spaces. Solutions of div(aDu)= div(F) are C^1,α if a and F are C^0,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) De Giorgi's Holder regularity result without regularity of the coefficients
The 19th Hilbert problem and the path to obtain C^∞ solutions. Moser's proof of the De Giorgi result: div(aDu)=0 with a bounded from below and above implies that u is C^0,α. Local L^∞ estimates for div(aDu)=div(F) for bounded F.
References: the original paper by J. Moser; partial notes here for div(aDu)=div(F).

9) 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 ∫ a_n│u'│². The asymptotics of the optimal location problem (without the Γ-limsup part).

10) 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: Have a look at these three sets of partial lecture notes general theory, optimal location, and perimeter approximation. You can also have a look at the books by A. Braides Gamma-Convergence for Beginners or by G. Dal Maso An Introduction to Γ-Convergence and 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 available. Each student should hand three solutions as a homework, according to the rules we discussed.

Full list of exercises.

Here you will find some solutions (exercises with a written solution cannot be chosen as a homework ; exercise 43 has been done in class and cannot be chosen neither).

The mid-term exam of last year is here, (with solutions).

Evaluation

The mid-term exam took place on Monday October 25th. Here is the examination text, with the solutions.
The final exam took place on December 13th. You can find the examination text, with its solutions.