# 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∈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, 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
C^{1,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
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) * (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 λ_{1} 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 λ_{1}.
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 λ_{1} under volume
constraints.

Definitions and properties of Γ-convergence in metric
spaces. Γ-convergence of quadratic functional of the form
∫ a_{n}│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.