Introduction
Existence
Characterization
Conclusions
An extremal eigenvalue problem for a
two-phase conductor in ball
Shape derivative for a two-phase eigenvalue problem and
optimal configurations in a ball.
WIPA 2010
Carlos Conca
Rajesh Mahadevan
León Sanz B.
January 4 - 8, Santiago de Chile
1 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
2 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
2 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
2 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
2 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
3 / 28
Introduction
Existence
Characterization
Conclusions
The problem
Introduction
Let Ω be a bounded domain, 0 < α < β.
ω ⊂ Ω region where is placed the material β.
β
α
β
Spectrum problem
If ν := αχΩ\ω + βχω , the spectral conductivity problem
−div (ν∇u) = λu
in Ω
(1)
u = 0
on ∂Ω
has the first eigenvalue
1
λ (ν) =
R
inf
u∈H01 (Ω),u6=0
ν |∇u|2
.
u2
Ω
ΩR
(2)
3 / 28
Introduction
Existence
Characterization
Conclusions
The problem
Optimization Problem
We fix the quantity of material, |ω| = m.
Admissible set:
A = ν | ν = αχΩ\ω + βχω , ω measurable, |ω| = m .
(3)
We are interested in studying
inf λ1 (ν).
ν∈A
(4)
4 / 28
Introduction
Existence
Characterization
Conclusions
The problem
Interesting Questions
Is it possible to find a classical minimizer , ie, which
belongs to A ?
If it is possible, give characterizations.
5 / 28
Introduction
Existence
Characterization
Conclusions
The problem
Interesting Questions
Is it possible to find a classical minimizer , ie, which
belongs to A ?
If it is possible, give characterizations.
5 / 28
Introduction
Existence
Characterization
Conclusions
The problem
Interesting Questions
Is it possible to find a classical minimizer , ie, which
belongs to A ?
If it is possible, give characterizations.
5 / 28
Introduction
Existence
Characterization
Conclusions
Background Research
Background Research
Krein (1955) [8]: Unidimensional case. It exists a
classical solution, completely characterized.
Alvino-Lions-Trombetti (1989) [2]: In the N
dimensional case, they proved the existence of a
classical radially symmetrical solution in a ball.
Murat-Tartar (1970-80) [9]: In general, the
solutions of these kind of problems are coefficients with
micro-structure or homogenized ones: The optimal
materials have to be finely merged.
Cox-Lipton (1996) [5]: They analyzed the relaxed
problem for a optimal solution with micro-structure.
6 / 28
Introduction
Existence
Characterization
Conclusions
Background Research
Background Research
Krein (1955) [8]: Unidimensional case. It exists a
classical solution, completely characterized.
Alvino-Lions-Trombetti (1989) [2]: In the N
dimensional case, they proved the existence of a
classical radially symmetrical solution in a ball.
Murat-Tartar (1970-80) [9]: In general, the
solutions of these kind of problems are coefficients with
micro-structure or homogenized ones: The optimal
materials have to be finely merged.
Cox-Lipton (1996) [5]: They analyzed the relaxed
problem for a optimal solution with micro-structure.
6 / 28
Introduction
Existence
Characterization
Conclusions
Background Research
Background Research
Krein (1955) [8]: Unidimensional case. It exists a
classical solution, completely characterized.
Alvino-Lions-Trombetti (1989) [2]: In the N
dimensional case, they proved the existence of a
classical radially symmetrical solution in a ball.
Murat-Tartar (1970-80) [9]: In general, the
solutions of these kind of problems are coefficients with
micro-structure or homogenized ones: The optimal
materials have to be finely merged.
Cox-Lipton (1996) [5]: They analyzed the relaxed
problem for a optimal solution with micro-structure.
6 / 28
Introduction
Existence
Characterization
Conclusions
Background Research
Background Research
Krein (1955) [8]: Unidimensional case. It exists a
classical solution, completely characterized.
Alvino-Lions-Trombetti (1989) [2]: In the N
dimensional case, they proved the existence of a
classical radially symmetrical solution in a ball.
Murat-Tartar (1970-80) [9]: In general, the
solutions of these kind of problems are coefficients with
micro-structure or homogenized ones: The optimal
materials have to be finely merged.
Cox-Lipton (1996) [5]: They analyzed the relaxed
problem for a optimal solution with micro-structure.
6 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
7 / 28
Introduction
Existence
Characterization
Conclusions
Direct Method
Direct Method of the Calculus of Variations
If we could find a topology in A such that:
{ν | λ1 (ν) ≤ c} were relatively compacts,
and λ1 (ν) were l.s.c
Then λ1 (ν) would reach the minimum in cl(A)
7 / 28
Introduction
Existence
Characterization
Conclusions
Direct Method
Direct Method of the Calculus of Variations
If we could find a topology in A such that:
{ν | λ1 (ν) ≤ c} were relatively compacts,
and λ1 (ν) were l.s.c
Then λ1 (ν) would reach the minimum in cl(A)
7 / 28
Introduction
Existence
Characterization
Conclusions
Direct Method
Weak- ∗ Topology
The feasible set A is relatively compact for the weak-∗
topology in L∞ (Ω).
But λ1 (ν) is not l.s.c for this topology.
If it were, we would have
Z
Z
νn ∇un ∇v −→
ν∇u∇v,
Ω
Ω
∗
For some νn * ν and un * u, where un are the
eigen-functions associated to νn . This statement, a
priori, is not true.
8 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
N -dimensional case −→ We solve it in a ball
It is not easy to find classical solutions in the general
case. (Murat-Tartar, Homogenization theory)
If we had complete symmetry: −→ we could reduce it
to a one-dimensional problem.
Alvino et. al. solved it. We gave a new proof: Could it
work in other kind of domain symmetries.
In the proof we used symmetrization tools: Schwarz
rearrangements.
9 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
Schwarz rearrangement
If D ⊆ Ω, the Schwarz rearrangement D∗ is the ball
centered at the origin such that |D| = |D∗ |.
For f : Ω −→ R, its rearrangements is given by
f ∗ (x) = sup {c | x ∈ {f ≥ c}∗ } .
f ∗ is radially symmetrical.
Equi-measurability:
| {f ≥ c} | = | {f ∗ ≥ c} |.
Z
Z
∗ 2
⇒
(u ) =
u2
Ω∗
Ω
f
r
f∗
10 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
How to diminish the eigenvalue?
R
2
?
ν
|∇u|
ν̃ |∇u∗ |2
ΩR
ΩR
≥
.
u2
(u∗ )2
Ω
Ω
R
(5)
ν̃ some “rearrangement”. Is it feasible?
u∗ ∈ H01 (Ω) ? X
If (5) is true and ν̃ feasible:
−→ We reduce the problem to a one-dimensional.
Besides, we still have the topology problem.
11 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
How to diminish the eigenvalue?
R
2
?
ν
|∇u|
ν̃ |∇u∗ |2
ΩR
ΩR
≥
.
u2
(u∗ )2
Ω
Ω
R
(5)
ν̃ some “rearrangement”. Is it feasible?
u∗ ∈ H01 (Ω) ? X
If (5) is true and ν̃ feasible:
−→ We reduce the problem to a one-dimensional.
Besides, we still have the topology problem.
11 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
Reformulation of A
∗
Feasible set A = ν | (ν −1 )∗ = (θ−1 ) .
A−1 = {ν −1 | ν ∈ A}
cl (A−1 ) is convex and weak-∗ compact.
Extreme points of cl(A− ) = A−1 [2].
12 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
And in fact, we have an inequality like (5):
Theorem (Conca-Mahadevan-Sanz [3], [2])
Let ν ∈ A and u ∈ H01 (Ω). There exists ν̃ −1 ∈ cl (A−1 )
radially symmetrical such that
Z
Z
∗ 2
ν̃ |∇u | ≤
ν |∇u|2 ,
(6)
Ω
Ω
where u∗ is the Schwarz rearrangement of u.
13 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
Topology: weak-∗ convergence of ν −1
λ1 (ν) is continuous for ν radially symmetrical, with
the weak-∗ convergence of the ν −1 .
A is relatively compact for this topology.
The minimizer ν̃ is such that ν̃ −1 ∈ cl(A−1 ).
Classical solution in a Ball
1
= J(ν −1 )
λ(ν)
min
ν −1 ∈cl(A−1 )
λ(ν) ←→
with J(·) convex.
max
ν −1 ∈cl(A−1 )
(7)
J(ν −1 )
The solution ν −1 is an extreme point: It lives in A−1 .
The solution ν of our problem lives in A
14 / 28
Introduction
Existence
Characterization
Conclusions
N dimensions: total symmetry case
Topology: weak-∗ convergence of ν −1
λ1 (ν) is continuous for ν radially symmetrical, with
the weak-∗ convergence of the ν −1 .
A is relatively compact for this topology.
The minimizer ν̃ is such that ν̃ −1 ∈ cl(A−1 ).
Classical solution in a Ball
1
= J(ν −1 )
λ(ν)
min
ν −1 ∈cl(A−1 )
λ(ν) ←→
with J(·) convex.
max
ν −1 ∈cl(A−1 )
(7)
J(ν −1 )
The solution ν −1 is an extreme point: It lives in A−1 .
The solution ν of our problem lives in A
14 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
15 / 28
Introduction
Existence
Characterization
Conclusions
Conjecture
Domain: Ball in RN
λ1 (ν) is minimized in a classical radially symmetric
configuration.
Conjecture
Optimal Solution: Distribute the material β in the center.
α
β
15 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Derivation with respect to the domain
We proceed with the tools of derivation with respect to
the domain: They measure the variations in values
that change when the domain is being perturbed.
The set of admissible domains doesn’t have an
vectorial space structure.
If ω is a region,
θ : RN −→ RN ,
ωt = (Id + tθ) (ω),
If u = u(ω) ⇒ ut := u(ωt )
16 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Formulas
Total Derivative:
ut ◦ (Id + tθ) − u
.
t−→0
t
(8)
ut − u
.
t−→0
t
(9)
u0 = u̇ − θ · ∇u.
(10)
u̇ = lim
Local Derivative:
u0 = lim
Relation:
17 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Formulas
Total Derivative:
ut ◦ (Id + tθ) − u
.
t−→0
t
(8)
ut − u
.
t−→0
t
(9)
u0 = u̇ − θ · ∇u.
(10)
u̇ = lim
Local Derivative:
u0 = lim
Relation:
17 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Formulas
Total Derivative:
ut ◦ (Id + tθ) − u
.
t−→0
t
(8)
ut − u
.
t−→0
t
(9)
u0 = u̇ − θ · ∇u.
(10)
u̇ = lim
Local Derivative:
u0 = lim
Relation:
17 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Formulas
Total Derivative:
ut ◦ (Id + tθ) − u
.
t−→0
t
(8)
ut − u
.
t−→0
t
(9)
u0 = u̇ − θ · ∇u.
(10)
u̇ = lim
Local Derivative:
u0 = lim
Relation:
17 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Perturbed problem
ν0 , ω0 : initial material distribution.
(λ10 , u0 ): eigen-pair of the non perturbed problem.
νt = αχΩ\ωt + βχωt : material distribution.
Spectral perturbed problem:
−div (νt ∇ut ) = λ1t ut
in Ω
(11)
ut = 0
on ∂Ω
(λ1t , ut ): first eigen-pair of the perturbed problem.
18 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Perturbed problem
ν0 , ω0 : initial material distribution.
(λ10 , u0 ): eigen-pair of the non perturbed problem.
νt = αχΩ\ωt + βχωt : material distribution.
Spectral perturbed problem:
−div (νt ∇ut ) = λ1t ut
in Ω
(11)
ut = 0
on ∂Ω
(λ1t , ut ): first eigen-pair of the perturbed problem.
18 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Existence and Formula for λ01 (ω0 ; θ)
Theorem (Conca, Mahadevan, Sanz [4])
λ01 (ω0 ; θ) exists.
λ01 (ω0 ; θ)
Z
=
∂ω0
2
Z
ν0 |∇u0 | θ·ndS+2
[θ · ∇u0 ] ν0 ∇u0 ·ndS,
∂ω0
(12)
where [f ] denotes the jump of f .
In the proof of existence: Implicit Function Theorem.
19 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Existence and Formula for λ01 (ω0 ; θ)
Theorem (Conca, Mahadevan, Sanz [4])
λ01 (ω0 ; θ) exists.
λ01 (ω0 ; θ)
Z
=
∂ω0
2
Z
ν0 |∇u0 | θ·ndS+2
[θ · ∇u0 ] ν0 ∇u0 ·ndS,
∂ω0
(12)
where [f ] denotes the jump of f .
In the proof of existence: Implicit Function Theorem.
19 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Existence and Formula for λ01 (ω0 ; θ)
Theorem (Conca, Mahadevan, Sanz [4])
λ01 (ω0 ; θ) exists.
λ01 (ω0 ; θ)
Z
=
∂ω0
2
Z
ν0 |∇u0 | θ·ndS+2
[θ · ∇u0 ] ν0 ∇u0 ·ndS,
∂ω0
(12)
where [f ] denotes the jump of f .
In the proof of existence: Implicit Function Theorem.
19 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Existence and Formula for λ01 (ω0 ; θ)
Theorem (Conca, Mahadevan, Sanz [4])
λ01 (ω0 ; θ) exists.
λ01 (ω0 ; θ)
Z
=
∂ω0
2
Z
ν0 |∇u0 | θ·ndS+2
[θ · ∇u0 ] ν0 ∇u0 ·ndS,
∂ω0
(12)
where [f ] denotes the jump of f .
In the proof of existence: Implicit Function Theorem.
19 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
Existence Scheme λ01(ω0; θ)
We have seen:
−div(ν(ωt )∇ut ) = λ1t ut
ut = 0
y
in ∂Ω
on ∂∂Ω
ωt = (Id + tθ)(ω) = Φt (ω)
smooth and invertible for t small enough. Change of variables:
−div(ν(ωt ) ◦ Φt )At ∇(ut ◦ Φt ) = λ1t (ut ◦ Φt )J(Φt )
in Ω
ut ◦ Φt = 0
on ∂Ω
−1 T
where At = DΦ−1
t (DΦt ) J(Φt )
We have ν(ωt ) ◦ Φt = ν0 ∀ t. The normalization constraint can
be written in the form
Z
|ut ◦ Φt |2 J(Φt )dx = 1
Ω
20 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
(λ1t , ut ) is a normalized eigen-pair ⇔ (λ1t , ut ◦ Φt ) satisfies
the former equations.
If (λ1t , ut ◦ Φt ) is a smooth zeros curve of the function F
around (0, λ10 , u0 ), where
Z
2
F (t, λ, v) = −div (νAt ∇v) − λv, |v| J(Φt )dx − 1
Ω
⇒
(λ01 (ω; θ), u̇)
exists in R ×
H1 (Ω)
IFT ⇒ The smooth curve exists.
We verify the IFT hypothesis:
F : R × R × H01 (Ω) −→ R is differentiable.
Fλ,v (0, λ1 (ω), u0 ) : R × H01 (Ω) −→ H−1 × R in invertible
with continuous inverse.
21 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
(λ1t , ut ) is a normalized eigen-pair ⇔ (λ1t , ut ◦ Φt ) satisfies
the former equations.
If (λ1t , ut ◦ Φt ) is a smooth zeros curve of the function F
around (0, λ10 , u0 ), where
Z
2
F (t, λ, v) = −div (νAt ∇v) − λv, |v| J(Φt )dx − 1
Ω
⇒
(λ01 (ω; θ), u̇)
exists in R ×
H1 (Ω)
IFT ⇒ The smooth curve exists.
We verify the IFT hypothesis:
F : R × R × H01 (Ω) −→ R is differentiable.
Fλ,v (0, λ1 (ω), u0 ) : R × H01 (Ω) −→ H−1 × R in invertible
with continuous inverse.
21 / 28
Introduction
Existence
Characterization
Conclusions
λ01 :Derivation with respect to the domain
λ01 (ω; θ) Contribution : Its sign
To indicate descent directions of λ1 .
To reinforce the conjecture in a numerical way.
To provide future descent algorithms for other type of
geometries.
22 / 28
Introduction
Existence
Characterization
Conclusions
Numerical Experiments
Experiments in the plane R2
Concentric Rings
Displaced Discs
Concentric Squares
α β
α
β
β
α
23 / 28
Introduction
Existence
Characterization
Conclusions
Numerical Experiments
Experiments in the plane R2
Concentric Rings
Displaced Discs
Concentric Squares
α β
α
β
β
α
23 / 28
Introduction
Existence
Characterization
Conclusions
Numerical Experiments
Experiments in the plane R2
Concentric Rings
Displaced Discs
Concentric Squares
α β
α
β
β
α
23 / 28
Introduction
Existence
Characterization
Conclusions
Numerical Experiments
Experiments in the plane R2
Concentric Rings
Displaced Discs
Concentric Squares
α β
α
β
β
α
23 / 28
Introduction
Existence
Characterization
Conclusions
Numerical Experiments
Matlab and Freefem
24 / 28
Introduction
Existence
Characterization
1
Introduction
The problem
Background Research
2
Existence
Direct Method
N dimensions: total symmetry case
3
Characterization
Conjecture
λ01 :Derivation with respect to the domain
Numerical Experiments
4
Conclusions
Conclusions
25 / 28
Introduction
Existence
Characterization
Conclusions
Conclusions and projections
We gave a new proof for a classical radially
symmetrical solution.
→ We expect to understand the problem in domains
with less symmetries, such as the case of squares and
stars.
The conjecture: It is optimal to distribute β in the
center of the ball.
We gave arguments and evidence:
Derivative with respect to the domain of the first
eigenvalue.
Numerical experiments in the plane.
We expect to provide a deeper view in the analysis of
the λ01 (ω; θ)
25 / 28
Introduction
Existence
Characterization
Conclusions
Conclusions and projections
We gave a new proof for a classical radially
symmetrical solution.
→ We expect to understand the problem in domains
with less symmetries, such as the case of squares and
stars.
The conjecture: It is optimal to distribute β in the
center of the ball.
We gave arguments and evidence:
Derivative with respect to the domain of the first
eigenvalue.
Numerical experiments in the plane.
We expect to provide a deeper view in the analysis of
the λ01 (ω; θ)
25 / 28
Introduction
Existence
Characterization
Conclusions
Conclusions and projections
We gave a new proof for a classical radially
symmetrical solution.
→ We expect to understand the problem in domains
with less symmetries, such as the case of squares and
stars.
The conjecture: It is optimal to distribute β in the
center of the ball.
We gave arguments and evidence:
Derivative with respect to the domain of the first
eigenvalue.
Numerical experiments in the plane.
We expect to provide a deeper view in the analysis of
the λ01 (ω; θ)
25 / 28
Introduction
Existence
Characterization
Conclusions
References
A. Alvino and G. Trombetti, A lower bound for the first eigenvalue of
an elliptic operator, J. Math. Anal. Appl. 94 (1983), no. 2, 328–337.
A. Alvino, P. L. Trombetti, and P. L. Lions, On optimization
problems with prescribed rearrangements, Non-Linear Anal. 13 (1989),
no. 2, 185–220.
C. Conca, R. Mahadevan, and L. Sanz, An extremal eigenvalue
problem for a two-phase conductor in a ball., Appl. Math. Optim.
60(2) (2009), pp. 173–184.
C. Conca, R. Mahadevan, and L. Sanz, Shape derivative for a
two-phase eigenvalue problem and optimal configurations in a ball,
ESAIM: Proceedings 27 (2009), pp. 311–321.
S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase
conductors, Arch. Rational Mech. Anal. 136 (1996), 101–117.
26 / 28
Introduction
Existence
Characterization
Conclusions
A. Henrot and M. Pierre, Variation et optimisation de formes.,
Mathématiques et Applications 48. Springer, 2005.
S. Kesavan, Symmetrization & applications, Series in Analysis, vol. 3,
World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
M. G. Krein, On certain problems on the maximum and minimum of
characteristic values and on the Lyapunov zones of stability, Amer.
Math. Soc. Transl. (2) 1 (1955), 163–187.
F. Murat and L. Tartar, Calculus of variations and homogenization [
MR0844873 (87i:73059)], Topics in the mathematical modelling of
composite materials, Progr. Nonlinear Differential Equations Appl.,
vol. 31, Birkhäuser Boston, Boston, MA, 1997, pp. 139–173.
J. Simon, Differentiation with respect to the domain in boundary value
problems, Numer. Funct. Anal. Optim. 2 (1980), no. 7-8, 649–687
(1981).
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Introduction
Existence
Characterization
Conclusions
Thanks!
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