Introduction
Minimization Problem
Application of Γ-convergence
Γ-convergence: An Application
Maximilian Wank
LMU München
16.04.2014
Maximilian Wank
Γ-convergence: An Application
1/19
Introduction
Minimization Problem
Application of Γ-convergence
Γ-convergence
X topological space satisfying 1st axiom of countability, Fn , F : X → R.
We say F = Γ- lim Fn iff
n→∞
n→∞
1. ∀(xn )n ⊂ X with xn −−−→ x ∈ X it is
F (x) ≤ lim inf Fn (xn ).
n→∞
n→∞
2. ∀x ∈ X there is (xn )n ⊂ X with xn −−−→ x ∈ X s.t.
F (x) = lim Fn (xn ).
n→∞
The sequence in 2. is called recovering sequence.
Remark: There is a more general definition for arbitrary topol. spaces X
coinciding with this definition.
Maximilian Wank
Γ-convergence: An Application
2/19
Introduction
Minimization Problem
Application of Γ-convergence
Γ-limits – an easy example
Set X := R and Fn (x) := sin(nx). Then,
1. −1 ≤ lim inf sin(nxn ) and
n→∞
2. For x ∈ R define xn as the nearest point with sin(nxn ) = −1.
π
sin(n · ) is 2π
n -periodic =⇒ |xn − x| ≤ n =⇒ xn → x.
Furthermore, −1 = lim sin(nxn ).
n→∞
=⇒ Γ- lim sin(n · ) = −1.
n→∞
We see:
1. Need additional assumptions for comparison
with pointwise convergence.
2. This example points towards minimization problems!
Maximilian Wank
Γ-convergence: An Application
3/19
Introduction
Minimization Problem
Application of Γ-convergence
Coercivity, Γ-limits and minimization
X reflexive Banach space:
kxk→∞
F : X → R coercive w.r.t. weak topology ⇐⇒ F (x) −−−−−→ ∞.
X topological space:
(Fn )n equi-coercive ⇐⇒ ∃ l.sc. and coercive Ψ : X → R s.t. Fn ≥ Ψ.
Theorem
(Fn )n equi-coercive, F = Γ- lim Fn , ∃!x0 ∈ X minimizing F . Choose
n→∞
xn ∈ argmin Fn (x).
x∈X
n→∞
n→∞
Then, xn −−−→ x0 and (Fn (xn ))n −−−→ F (x0 ).
Maximilian Wank
Γ-convergence: An Application
4/19
Introduction
Minimization Problem
Application of Γ-convergence
p-Poisson equation
Strong formulation: u satisfies
−∆p u := − div(|∇u|p−2 ∇u) = f on Ω
u|∂Ω = 0
Weak formulation: u ∈ W01,p (Ω) and for all ξ ∈ C0∞ (Ω)
Z
|∇u|p−2 ∇u · ∇ξ dx = f (ξ).
Ω
u strong solution =⇒ u weak solution ⇐⇒ u is the unique minimizer of
Z
1
J (u) := p |∇u|p dx − f (u).
Ω
Maximilian Wank
Γ-convergence: An Application
5/19
Introduction
Minimization Problem
Application of Γ-convergence
Splitted energy
Consider 1 < p < 2 and minimize
Z
p−2
1
J s (u, a) :=
|∇u|2 + ( p1 − 12 )|a|p dx − f (u)
2 |a|
Ω
Lemma
Let (u, a) be a local minimizer of J s .
Then, u is a weak solution of p-Poisson equation and
|a| = |∇u|.
Sketch of proof: Calculate first variation and use “right” test functions.
Maximilian Wank
Γ-convergence: An Application
6/19
Introduction
Minimization Problem
Application of Γ-convergence
Constrained splitted energy
For stability add constraint for a:
s
J(ε)
(u, a) := J s (u, ε ∨ |a| ∧ 1ε )
Lemma
For fixed u ∈ W01,2 (Ω), the function a = ε ∨ |∇u| ∧
s (u, · ) satisfying a = ε ∨ a ∧ 1 .
minimum of J(ε)
ε
1
ε
is the unique
So we want to use
s
uk+1 := argmin J(ε)
( · , ak )
Maximilian Wank
and
ak+1 := ε ∨ |∇uk+1 | ∧ 1ε .
Γ-convergence: An Application
7/19
Introduction
Minimization Problem
Application of Γ-convergence
Constrained energy
How does ε affect the algorithm?
Define
J(ε) (u) :=
min
1
a=ε∨|a|∧ ε
s
J(ε)
(u, a) = J s (u, ε ∨ |∇u| ∧ 1ε )
Z
κ(ε) (|∇u|) dx − f (u)
=
Ω
Theorem
Let u be minimizer of J and uε be minimizer of J(ε) .
Then, uε * u in W01,p (Ω) as ε & 0.
Maximilian Wank
Γ-convergence: An Application
8/19
Introduction
Minimization Problem
Application of Γ-convergence
Outline of the proof
We need to check:
1. 1st axiom of countability,
2. equi-coercivity of J(ε) and
3. J = Γ- lim J(ε) .
ε&0
4. Finally, apply theorem from slide 4
Arising problems:
• no coercivity w.r.t. norm-topology
=⇒ use weak topology
• no 1st axiom of countability on W01,p (Ω) w.r.t. weak topology
Maximilian Wank
Γ-convergence: An Application
9/19
Introduction
Minimization Problem
Application of Γ-convergence
Apropriate space
Choose the topological space
X := {w ∈ W01,p (Ω) : J(0.5) (w ) ≤ 0} ∪ {w ∈ W01,∞ (Ω) : k∇w kLp (Ω) ≤ γ}.
A priori estimates for uε , monotonicity of J(ε) in ε
=⇒ X is norm-bounded:
0 ≥ J(0.5) (v ) ≥ J (v ) ≥ p1 k∇v kpLp (Ω) − ckf kW −1,p (Ω) · k∇v kLp (Ω)
0
∗
=⇒ Tweak is induced by a metric (since W01,p (Ω) is separable)
=⇒ B 1 (x0 ) are countable basis of neighbourhoods of x0
n
i.e. 1st axiom of countability!
Maximilian Wank
Γ-convergence: An Application
10/19
Introduction
Minimization Problem
Application of Γ-convergence
Equi-coercivity
Need to find l.sc. and coercive Ψ : X → R such that J(ε) ≥ Ψ.
J does the job:
1. J is lower semicontinuous
2. Use
J (v ) ≥ p1 k∇v kpLp (Ω) − ckf kW −1,p (Ω) · k∇v kLp (Ω)
0
k∇v kLp (Ω)
to see J (v ) −−−−−−→ +∞
=⇒ J is coercive w.r.t. Tweak
3. J(ε) monotonic increasing in ε and J = J(0)
=⇒ J(ε) ≥ J
Overall: J(ε) is equi-coercive!
Maximilian Wank
Γ-convergence: An Application
11/19
Introduction
Minimization Problem
Application of Γ-convergence
lim inf-condition
Let v ∈ X and vn * v in W01,p (Ω) as n → ∞. Then,
lim inf J(εn ) (vn ) ≥ lim inf J (vn ) ≥ J (v ).
n→∞
n→∞
=⇒ lim inf-condition is satisfied.
It remains to find recovering sequences.
For v ∈ {w ∈ W01,∞ (Ω) : k∇w kLp (Ω) ≤ γ} take vn ≡ v :
Z
lim J(εn ) (vn ) = lim J(εn ) (v ) = lim
κ(εn ) (|∇v |) dx − f (v )
n→∞
n→∞
n→∞
Ω
Z
=
|∇v |p dx − f (v ) = J (v )
Ω
Maximilian Wank
Γ-convergence: An Application
12/19
Introduction
Minimization Problem
Application of Γ-convergence
More definitions
For v ∈ L1 (Rd ) define the Hardy-Littlewood maximal function
Z
M(v )(x) := sup − |v (y )| dy .
r >0
Br (x)
kMv kLp (Rd ) ≤ ckv kLp (Rd )
M continuous for p > 1
and of weak type (1,1).
sup λ|{M(v ) > λ}| ≤ ckv kL1 (Rd )
λ>0
We call
Oλ (v ) := {x ∈ Rd : M(|∇v |)(x) > λ}
the “bad set”. Due to lower semicontinuity of M(|∇v |), it is open.
Maximilian Wank
Γ-convergence: An Application
13/19
Introduction
Minimization Problem
Application of Γ-convergence
Lipschitz truncation
Theorem (Diening, Kreuzer, Süli)
Let λ > 0 and v ∈ W01,p (Ω). Then, there is vλ ∈ W01,∞ (Ω) with the
following properties:
(a) {v 6= vλ } ⊂ Oλ (v ) ∩ Ω,
(b) kvλ kLp (Ω) ≤ c1 kv kLp (Ω) ,
(c) k∇vλ kLp (Ω) ≤ c2 k∇v kLp (Ω) and
(d) |∇vλ | ≤ cλχOλ (v )∩Ω + |∇v |χOλ (v ){ ∩Ω ≤ c3 λ almost everywhere.
n→∞
For εn −−−→ 0 define λn :=
Maximilian Wank
1
c3 ε n
=⇒
|∇vλn | ≤
Γ-convergence: An Application
1
εn
a.e.
14/19
Introduction
Minimization Problem
Application of Γ-convergence
Recovering sequence
For v ∈ {w ∈ W01,p (Ω) : J(0.5) (w ) ≤ 0} the Lipschitz truncations
vn := vλn
give an admissable recovering sequence. First of all,
k∇vn kLp (Ω) ≤ c2 k∇v kLp (Ω) ≤ c kf kW −1,p (Ω)
1
p−1
=: γ
0
and (vn )n ⊂ W01,∞ (Ω). Hence, (vn )n ⊂ X . It remains to show
Z
n→∞
J(εn ) (vn ) − J (v ) =
κ(εn ) (|∇vn |) − |∇v |p dx − hf , vn − v i −−−→ 0.
Ω
Therefore, split Ω into Ω1 := Oλn (v ){ ∩ Ω and Ω2 := Oλn (v ) ∩ Ω.
Maximilian Wank
Γ-convergence: An Application
15/19
Introduction
Minimization Problem
Application of Γ-convergence
Convergence on good set
On Ω1 we have v ≡ vn .
Furthermore |∇vn | ≤ c3 λn = ε1n and
additionaly κ(εn ) (t) = t p for t ∈ [εn , ε1n ].
Z
Z
p
κ
(|∇v
|)
−
ϕ(|∇v
|)
dx
=
κ
(|∇v
|)
−
|∇v
|
dx
n
n
n
(εn )
(εn )
Ω1
Ω1
=
Z
κ(εn ) (|∇vn |) − |∇vn |p dx Ω1 ∩{|∇vn |<εn }
≤ 2|Ω|εpn
n→∞
−−−→ 0.
Maximilian Wank
Γ-convergence: An Application
16/19
Introduction
Minimization Problem
Application of Γ-convergence
Convergence on bad set
n→∞
Weak type (1,1) implies |Ω2 | −−−→ 0, so
Z
Z
κ(εn ) (|∇vn |) − |∇v |p dx ≤ κ(εn ) (|∇vn |) + |∇v |p dx
Ω2
Ω2
Z
≤
ϕ(c3 λ) + |∇v |p dx
Ω2
Z
≤
Ω2
c3p M(|∇v |)p + |∇v |p dx
|
{z
}
∈L1 (Ω)
n→∞
−−−→ 0.
Maximilian Wank
Γ-convergence: An Application
17/19
Introduction
Minimization Problem
Application of Γ-convergence
Alternative proof via relaxation
1. Define
J˜(ε) : W01,p (Ω) → R
(
J(ε) (u)
u 7→
+∞
for u ∈ W01,2 (Ω)
for u ∈ W01,p (Ω) \ W01,2 (Ω).
and J˜ := J˜(0) .
2. J˜(ε) converges pointwise and decreasing to J˜
=⇒ sc− J˜ = Γ- lim J˜(ε)
ε&0
3.
sc− J˜
=J
4. Use theorem from slide 4
Maximilian Wank
Γ-convergence: An Application
18/19
Introduction
Minimization Problem
Application of Γ-convergence
Thank you for your attention.
Literature:
• Dal Maso: “An Introduction to Γ-convergence”
• Diening, Kreuzer, Süli: “Finite element approximation of steady
flows of incompressible fluids with implicit power-law-like rheology”
Maximilian Wank
Γ-convergence: An Application
19/19