Calculus of Variations and Elliptic Equations
Master course - Master en Mathématiques Avancées,
UCBL and ENS Lyon
Practical Information
Duration: 24h
Schedule: 9 classes of 2h40' each (approximately :-) on
Friday morning, from Sep 13 to December (not all Fridays).
Where: La Doua, in the building Braconnier, the main building
of the Math Dept. Room Seminar 2, in the basement.
Examination: A mid-term exam is scheduled on Nov 8 at 9.30am (30%
of the mark) and a final exam (50%) end of December or January the
remaining 20% of the mark will be based on homeworks and participation
in class.
Language: the classes are in English.
Prerequisites: some functional analysis.
Program
There will be 9 classes of 2h40' each.
1) (13/9) Calculus of Variations in 1D.
The examples of Geodesics, brachistochrone, economic growth. Techniques for existence and non-existence. Euler-Lagrange equation and boundary conditions. Sufficient conditions in case of convexity.1.1
References: Sections 1.1, 1.2, 1.3 and the beginning of 4.1 of CCV .
2) (27/9) Multi-dimensional Calculus of
Variations; Harmonic functions and
distributions
Techniques for existence in higher dimension. Uniqueness of the minimizers. The case of the Dirchlet energy. 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: Again Section 4.1, then 2.1, 2.2 and 2.3 of CCV.
3) (4/10) Convexity and weak semi-continuity;
Convex duality.
Lower-semicontinuous functionals: strong and weak
convergence and link with convexity conditions. Integral functionals
with L(x,u,Du).
Main notions on convex functions and Legendre transform. Dual problem
and Fenchel-Rockafellar duality.
References: Sections 3.1 and 3.2 of CCV, then Sections 4.2.1,
4.2.4 and 4.5 of CCV.
4) (11/10) Minimal-flow
problems; Regularity via duality
Duality between min ∫ H(x,v) : ∇·v = f and
min ∫ H*(x,∇u) + fu.
Regularity via duality: Laplacian: Δu =f, f∈L2⇒ ∇u∈H1, p-Laplacian: Δpu
=f, f∈W1,q⇒ ∇up/2∈H1, Very degenerate
problems coming from traffic congestion...
References: Sections 4.3 and 4.4 of CCV.
5) (18/10) Lp theory for
Δu=f
Marcinkiewicz interpolation ; W2,p regularity of Γ*f
if f is Lp 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 (pages sent by email). For a short
summary, and for global regularity using
reflections, have a look at Section 5.5 of CCV or at these partial lecture notes.
6) (15/11) Campanato spaces and Holder regularity for
elliptic PDEs with
regular coefficients
Morrey and Campanato spaces. Solutions of div(aDu)= div(F) are
C1,α if a and F are C0,α.
References: Sections 5.1 and 5.2 of CCV.
7) (22/11) 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 C0,α. Same result for div(aDu)=div(F).
References: Sections 5.3 and 5.4 of CCV. Please pay attention
to a mistake in Section 5.4.3, which is corrected here.
8) (29/11) 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
∫ an│u'│2. The asymptotics of
the optimal location problem.
-->
9) (13/12)The perimeter functional and the
Modica-Mortola Γ-convergence result.
After finishing the Γ-convergence proof from the previous class, a
short introduction to vector measures, BV functions and sets of finite perimeter. Then,
the Modica-Mortola
approximation of the perimeter functional.
References for lessons 9 and 10: Sections 7.1, 7.2, 7.3 and
7.4 + Box 6.1 of CCV. For other references on Γ-convergence you
can 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. 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).
References: CCV means the book A course in the Calculus of
Variations that I wrote in 2023 based on my previous courses. The pdf
file of the book has been sent to all registered students.
Exercises and previous examinations
A list of exercises from CCV will be given, each student should hand three
solutions as a homework, according to the rules we discussed.
The mid-term exam of 2020 is here (with solutions).
The mid-term exam of 2021 is here (with the solutions).
The final exam of 2020 is here (with its solutions).
The final exam of 2021 is here (with its solutions).
Evaluation
The mid-term exam took place on Friday November 8.
Here is the examination text, with the solutions.
The final exam took place on Friday December 20.
Here is the examination text, with the solutions.