# 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 26 to Jan 15 (except vacations and one absence for a conference, see below).
Where: in the campus of Orsay, building 425. Room 225/227 on Thursday; room 121/123 on Friday.
Examination: written exam on Jan 28, 1.30pm-4.30pm.
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).

• 1) (26/11) 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) (27/11) Convexity and weak semi-continuity.
Convexiy 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) (3/12) 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) (4/12) 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.
• 5) (10/12) Harmonic functions, quasi-minima and regularity via comparison
Short memo abour 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: M. Giaquinta L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graph, chapter 5. For quasi-minima, some notions are in the book by Giusti. You can also have a look at the first pages of this (difficult) paper
• 6) (11/12) 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) (7/1) 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. Minimization fo the Raleygh quotient in a fixed Ω and of λ1 under volume constraints on Ω. Symmetrization of a function and Polya-Szego inequality.
-- Unfortunately the proof of the Polya-Szego inequality has been postponed to the next lecture. --
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) (8/1) General Γ-convergence theory
-- Indeed, the first hour of this lecture has been devoted to the Polya-Szego inequality. --
Definitions and properties of Γ-convergence in metric spaces. The example of the location problem (asymptotic density of the optimal N points in facility locations problems as N tends to ∞).
-- Unfortunately, due to RER disruptions and to the Polya-Szego proof, the end of the proof of the Γ-convergence for the location problem is postponed to the next lecture. --
• 9) (14/1) Modica-Mortola and other Γ-convergence problems
-- First, we finished the proof of the Γ-convergence for the location problem --
Γ-convergence of quadratic functional of the form ∫ an│u'│2. The Modica-Mortola approximation of the perimeter functional.
• 10) (15/1) The Mumford-Shah functional and its approximation, examples and exercises
-- End of the Γ-limsup estimate for Modica-Mortola --
An informal introduction to the Mumford-Shah functional in image processing and to its approximation (Ambrosio-Tortorelli).
Exercises.
• References for lessons 8, 9 and 10: 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, the short paper by Bouchitte-Jimenez-Rajesh Asymptotic of an optimal location problem Comptes Rendus Mathematique (I'm looking for the file to put it online). For Modica-Mortola and Ambrosio-Tortorelli, the book by A. Braides Approximation of Free-Discontinuity Problems. Also look at these (incomplete) short notes by G. Leoni.

### Exercise and tutoring

Tutoring: a graduate student, Paul Pegon, will offer some tutoring and exercise classes.
Tutoring meetings: Friday Dec 11, 2pm-4pm, room 113-115; Friday Jan 8, 15 and 22, 2pm-4pm, room 225-227.
Both Paul and me are available for questions about the course or the exercises.

Here is a list of exercises, with exercises related to the whole course.
Here you will find some solutions (currently 13, other solutions will arrive soon).

Here is a mock exam to train for the examination.