Some ancient and modern shape optimization problems
Ilaria Fragalà
AIM meeting
January 19, 2017
PoliMi
Ilaria Fragalà
Some ancient and modern shape optimization problems
A naı̈ve presentation of:
Calculus of Variations
n
o
min F (u) : u ∈ H
Shape Optimization
n
o
min J(Ω) : Ω ∈ A
Strictly related fields: functional inequalities and geometrical inequalities
F (u) ≥ F (u)
Ilaria Fragalà
J(Ω) ≥ J(Ω)
Some ancient and modern shape optimization problems
Souvent, il n’existe pas de solution !
Mathematical questions
Contre-exemple 1 : En elasticité :
trouver une structure de poids donné la
. Existence of a solution
plus résistante possible.
Not always guaranteed!
Mathématiquement,
les suites
minimisantes convergent vers une
Minimizing sequences may converge to homogenized configurations
structure homogénéisée.
Contre-exemple
thermique :
Counterexample2in: En
thermics:
Mettre
de
la
glace
dans
un récipient
find the optimal
configuration
Existence
d’une
solution ? proche
pour
aboutir
à
une
température
of a given quantity of ice into a glass
d’une
température
Ou trouver
to reach
a desired donnée.
temperature.
la forme
optimale
d’un pas
radiateur
!
Souvent,
il n’existe
de solution
!
Contre-exemple 1 : En elasticité :
Counterexample
in elasticity:
trouver une structure
de poids donné la
find
the
optimal
configuration
plus résistante possible.
ofMathématiquement,
a given amount of les
material
suites
tominimisantes
resist to a given
load. vers une
convergent
structure homogénéisée.
Contre-exemple 2 : En thermique :
Mettre de la glace dans un
récipient
Ilaria Fragalà
Some ancient and modern shape optimization problems
. Regularity and geometrical properties of solutions
Related to free boundary problems, symmetrization methods.
. Optimality conditions
Need to make shape derivatives, with respect to domain variations.
. Calculus of the solution
Not always it is possible to determine explicit solutions,
often the target is to obtain numerical approximations.
Ilaria Fragalà
Some ancient and modern shape optimization problems
A sample of problems with a long history
I. From Queen Dido to Geometric Measure Theory
II. From Lord Rayleigh to Spectral Optimization
III. From Varro to Optimal Partition Problems
Ilaria Fragalà
Some ancient and modern shape optimization problems
I. From Queen Dido to Geometric Measure Theory
Inégalité isopérimétrique (1)
Dido problem is probably the oldest problem in the Calculus of Variations.
D
Ω?
sea
Figure:
Problème
de la Reine
Didon
It is an example of what
is called
an “isoperimetric
problem”.
Ilaria Fragalà
Some ancient and modern shape optimization problems
The isoperimetric problem
n
o
max |Ω| : Ω ⊂ Rn , |∂ Ω| = c
or equivalently
n
o
min |∂ Ω| : Ω ⊂ Rn , |Ω| = c
Theorem. Solutions are balls:
|∂ Ω|
|Ω|
n−1
n
≥
|∂ B|
|B|
n−1
n
Mathematicians are well acquainted with this property of balls since more that
2500 years, but the first attempts to give a rigorous proof are relatively recent.
Ilaria Fragalà
Some ancient and modern shape optimization problems
. XIX th century: Steiner, Edler
- Wikipedia
https://en.wikipedia.org/wiki/Ennio_de_Giorgi
. XX th century: Hurwitz, Minkowski, Lebesgue, Caratheodory, Blaschke,
Bonnesen.
nnio de Giorgi
Wikipedia, the free encyclopedia
. De Giorgi (Lecce 1928 - Pisa 1996)
Geometric Measure Theory [http://cvgmt.sns.it/]
o De Giorgi (8 February
1928finite
– 25 perimeter,
Sets with
ber 1996) was an Italian
ematician, member of the House of
gi, who worked on partial differential
ions and the foundations of
ematics.
Ennio De Giorgi
ontents
1
2
3
4
5
6
Mathematical work
Quotes
Selected publications
See also
Notes
Born
8 February 1928
References
you can’tandprove
shifting parts of the conclusion to the
Lecce, Italy
6.1“If
Biographical
generalyour theorem, keep
assumptions,
until you can”
references
Died
25 October 1996 (aged 68)
6.2 Scientific references
Pisa, Italy Some ancient and modern shape optimization problems
Ilaria Fragalà
7 External links
Reverse isoperimetric inequality
Is it possible to reverse the isoperimetric inequality, namely to maximize the
isoperimetric quotient?
The supremum of the isoperimetric quotient is +∞ ... BUT:
sup
inf
K ∈K n T ∈An
|∂ T (K )|
|T (K )|
n−1
n
sup
inf
K ∈K∗n T ∈GLn
|∂ T (K )|
|T (K )|
n−1
n
.
are well-posed. Solutions are n-dimensional simplexes and cubes.
[Ball, 1991]
Ilaria Fragalà
Some ancient and modern shape optimization problems
II. From Lord Rayleigh to Spectral Optimization
Question [Lord Rayleigh, 1894]
Which is the shape of a drum which gives the lowest note?
The vibration frequencies of a drum whose membrane’s shape is Ω are the
eigenvalues of the Laplacian on Ω with Dirichlet boundary condition:

−∆uk = λk uk inΩ

uk = 0
on∂ Ω
The problem is then:
n
o
min λ1 (Ω) : |Ω| = c
Ilaria Fragalà
Some ancient and modern shape optimization problems
Results in spectral optimization
413r7qi5c2l-_sl500_aa300_.jpg (JPEG Image, 300 × 300 pixels)
. Theorem [Faber-Krahn, 1923]
Balls minimize λ1 among domains with given volume.
λ1 (Ω)|Ω| ≥ λ1 (B)|B|
. Theorem [Krahn-Szegö, 1926]
The union of two equal balls minimize λ2 among sets with given volume.
. Theorem [Bucur, Mazzoleni-Pratelli, 2012-2013]
There exists a domain which minimizes λk among sets with given volume
(in the class of quasi-open domains).
Ilaria Fragalà
Some ancient and modern shape optimization problems
https://momentspass
Candidate minimizers for λk [Oudet, 2004]
No
Optimal union of discs
Computed shapes
3
46.125
46.125
4
64.293
64.293
5
82.462
78.47
6
92.250
88.96
7
110.42
107.47
8
127.88
119.9
9
138.37
133.52
10
154.62
143.45
Figure 5.1: A table showing, for each eigenvalue λk , 3 ≤ k ≤ 10, the optimal union
of disks (left) and the optimal shape Ilaria
(right)
this one
is ancient
obtained
numerically.
Fragalà
Some
and modern
shape optimization problems
III. From Varro to Optimal partition problems
Question [Varro 36 b.C., Pappus of Alexandria, Fejes Tóth, Morgan]
Why the bees’ honecomb has hexagonal form?
Theorem [Hales, 2001]
Any partition of the plane into regions of equal area has perimeter at least that
of the regular hexagonal honeycomb tiling.
.MG/9906042 v2 20 May 2002
THE HONEYCOMB CONJECTURE
mk (Ω) = inf
n
∑
i=1,...,k
o
Per(Ei ) : Ei ⊆ Ω , |Ei | ∈ (0, +∞) , |Ei ∩ Ej | = 0 .
Thomas C. Hales
Abstract. This article gives a proof of the classical honeycomb conjecture: any
partition of the plane into regions of equal area has perimeter at least that of the
regular hexagonal honeycomb tiling.
lim
k→+∞
|Ω|
m (Ω) = Per(H) ,
kγ k
H = regular hexagon
1. Introduction
Fragalà
ancient
and modern wrote
shape optimization
problems
Around 36 B.C., Marcus Ilaria
Terentius
Varro, in Some
his book
on agriculture,
about
Conjecture [Caffarelli-Lin, 2007]
The hexagonal tiling is optimal also for the principal frequency
mk (Ω) = inf
n
∑
i=1,...,k
lim
k→+∞
o
λ1 (Ei ) : Ei ⊆ Ω , |Ei | ∈ (0, +∞) , |Ei ∩ Ej | = 0 .
|Ω|
m (Ω) = λ1 (H) ,
k2 k
H = regular hexagon
STILL OPEN!
Ilaria Fragalà
Some ancient and modern shape optimization problems
A couple of recent results and related open problems
Problem 1: In the same spirit of the reverse isoperimetric inequality
inf
K
|∂ K |
|K |
n−1
n
→ ball ,
|∂ T (K )|
sup inf
K
|T (K )|
T
n−1
n
→ simplex or cube
is it possible to reverse the Faber-Krahn inequality ?
2
inf λ1 (K )|K | n → ball ,
K
Ilaria Fragalà
2
sup inf λ1 (T (K ))|T (K )| n → ???
K
T
Some ancient and modern shape optimization problems
Simplification: work within the family of convex axisymmetric octagons.
We have a two parameter problem:
Numerical evidence:
Plot of the map (x1 , x2 ) 7→ λ1 (Ω(x1 ,x2 ) )|Ω(x1 ,x2 ) | and its level sets
Ilaria Fragalà
Some ancient and modern shape optimization problems
Theorem [Bucur-F., 2015] For every convex axisymmetric convex octagon Ω
with two vertices at (0, 1) and (1, 0), it holds
λ1 (Ω)|Ω| ≤ λ1 (Q)|Q| = 2π 2 .
Open problem: extend the result to more general settings.
Ilaria Fragalà
Some ancient and modern shape optimization problems
Problem 2: In the same spirit of Caffarelli-Lin conjecture,
given an optimal partition problem of the form
mk (Ω) = inf
n
o
max F (Ei ) : {Ei } ∈ Pk (Ω) .
i=1,...,k
which kind of shape functionals F satisfy the honeycomb conjecture ?
lim
k→+∞
|Ω| γ
k
mk (Ω) = F (H) ???
Ilaria Fragalà
Some ancient and modern shape optimization problems
Simplification: assume that the cells of the partitions are convex.
Theorem [Bucur-F.-Velichkov-Verzini, 2016]
If the cells are assumed to be convex, the honeycomb conjecture holds true
for a large class of functionals of the Calculus of Variations,
including the Cheeger constant and the logarithmic capacity.
Open problem: extend the result to more general settings.
Ilaria Fragalà
Some ancient and modern shape optimization problems
Many thanks for your attention
Ilaria Fragalà
Some ancient and modern shape optimization problems