# Calculus of Variations

## Master course - AMS and Optimization programs, Université Paris-Saclay

### Practical Information

Duration: 30h (5 ECTS)
Schedule: 3h on thursday afternoon (2-5pm) + 3h on Friday morning (9am-12pm), from Nov 24 to Jan 13 (except vacations and one week absence). Calendar: Nov 24 and 25, Dec 1, 2, 15, 16, Jan 5, 6, 12, 13.
Where: in the campus of Orsay, building 450, room 447, on Thursdays and building 440, room 229, on Fridays (I asked to change the rooms because of the blackboard).
Examination: written exam on Jan 27 in room 229, bdg 440.
Language: the classes are in English
Prerequisites: some functional analysis.

### Program

There will be 10 classes of 3h each (with a small break in the middle). Classes are tentatively organized as follows.

• 1) (24/11) Calculus of Variations in 1D.
Geodesics, brachistochrone, economic growth, and examples from mechanics. Techniques for existence and non-existence. Euler-Lagrange equation and boundary conditions.
References: two easy informal lecture notes on 1D variational problems (originally written for ENSAE engineers) : Notes by Guillaume Carlier on dynamic problem; about existence; the book by G. Buttazzo, M. Giaquinta, S. Hildebrandt One-dimensional variational problems (not easy to read)
• 2) (25/11) 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) (1/12) 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) (2/12) Regularity via duality
Laplacian: Δu =f, f∈L2⇒ ∇u∈H1, p-Laplacian: Δpu =f, f∈W1,q⇒ ∇up/2∈H1 and f∈Lq⇒ ∇up/2∈Hs, 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.
• 5) (15/12) Harmonic functions, quasi-minima and regularity via comparison
Main properties of harmonic functions. Different definitions of quasi-minima and examples of functionals where the minimizers are quasi-minima of the Dirichlet energy. Campanato spaces. Proof of Lipschitz and C1,α regularity.
References: You can have a look at these handwritten notes. Computations are essentially taken from the book M. Giaquinta L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graph, chapter 5 and from the first pages of this (difficult) paper
• 6) (16/12) Infinity-harmonic functions Δu=0
Lipschitz extensions. Notions of Absolute Lipschitz Minimizers and of viscosity solutions. Minimizers of the Lp norm of ∇u tend to AML, solutions of Δu=0. Uniqueness of the solution of the PDE with given boundary data.
References: some slides by P. Juutinen (read them all, they are short); the short proof of uniqueness for Δu=0 by S. Armstrong and C. Smart.
• 7) (5/1) The isoperimetric problem, shape optimization issues, and the BV space
Dido's legend about the isoperimetric problem. Proof of the isoperimetric inequality by Fourier series in 2D and statement in higher dimension. Introduction to the BV space and to the space of measures. Existence of a minimizer for the perimeter in a box. Minimization fo the Raleygh quotient in a fixed Ω and of λ1 under volume constraints on Ω.
References: The book by L. C. Evans and F. Gariepy, Measure theory and fine properties of functions, chapter 5; for the Fourier proof of the isoperimetric inequality, look at this paper by B. Fuglede; finally, here are some notes containing (Section 4.2) the rearrangement inequality that we used for λ1.
• 8) (6/1) General Γ-convergence theory
Indeed, the first hour of this lecture has been devoted to the Polya-Szego inequality and the application to the minimization of λ1 under volume constraints.
Definitions and properties of Γ-convergence in metric spaces. Γ-convergence of quadratic functional of the form ∫ an│u'│2.
• 9) (12/1) Two examples Γ-convergence problems: optimal location and Modica-Mortola
The location problem (asymptotic density of the optimal N points in facility locations problems as N tends to ∞).
. The Modica-Mortola approximation of the perimeter functional.
• 10) (13/1) Exercises
Some exercises from the list below.
• References for lessons 8 and 9: The book by A. Braides Gamma-Convergence for Beginners or the book by G. Dal Maso An Introduction to Γ-Convergence (available online). 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.

### Exercises

Here is a list of exercises, with exercises related to the whole course.
You should now be able to consider more or less all the exercises (Exercises 7, 21, 23, 24, 26, 31, 33, 34 have been shortly adressed in class, or solved during the last lecture). Here you will find some solutions.

Here is a mock exam that I gave last year to train for the examination.
On Friday Dec 2 a homework has been handed, graded for those who did during vacations, and on Jan 5 we corrected it after the lecture.
Finally, here are the examination tests of last year: first session and second session.