# 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∈L^{2}⇒ ∇u∈H^{1},

p-Laplacian: Δ_{p}u
=f, f∈W^{1,q}⇒ ∇u^{p/2}∈H^{1} and f∈L^{q}⇒ ∇u^{p/2}∈H^{s},

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
C^{1,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
Δ**_{∞}u=0

Lipschitz extensions. Notions of Absolute
Lipschitz Minimizers and of viscosity solutions. Minimizers of the
L^{p} norm of ∇u tend to AML, solutions of
Δ_{∞}u=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 Δ_{∞}u=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 λ_{1} 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 λ_{1}.
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
∫ a_{n}│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.