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Improved Poincaré and other classicinequalities: a new approach to prove them
and some generalizations
Ricardo G. Durán
Departamento de MatemáticaFacultad de Ciencias Exactas y Naturales
Universidad de Buenos Airesand IMAS, CONICET
Martin Stynes 60th Birthday CelebrationDresden
November 18, 2011
Ricardo G. Durán Poincaré and related inequalities
COWORKERS
G. Acosta
I. Drelichman
A. L. Lombardi
F. López
M. A. Muschietti
M. I. Prieto
E. Russ
P. Tchamitchian
Ricardo G. Durán Poincaré and related inequalities
Ricardo G. Durán Poincaré and related inequalities
GOALS
This talk is about several classic inequalities(Korn, Poincaré, Improved Poincaré, etc.).
Questions we are interested in:
What are the relations between them?
For which domains are these inequalitiesvalid?
When they are not valid: Are there weakerestimates still useful for variational analysis?
Ricardo G. Durán Poincaré and related inequalities
GOALS
This talk is about several classic inequalities(Korn, Poincaré, Improved Poincaré, etc.).
Questions we are interested in:
What are the relations between them?
For which domains are these inequalitiesvalid?
When they are not valid: Are there weakerestimates still useful for variational analysis?
Ricardo G. Durán Poincaré and related inequalities
GOALS
This talk is about several classic inequalities(Korn, Poincaré, Improved Poincaré, etc.).
Questions we are interested in:
What are the relations between them?
For which domains are these inequalitiesvalid?
When they are not valid: Are there weakerestimates still useful for variational analysis?
Ricardo G. Durán Poincaré and related inequalities
GOALS
This talk is about several classic inequalities(Korn, Poincaré, Improved Poincaré, etc.).
Questions we are interested in:
What are the relations between them?
For which domains are these inequalitiesvalid?
When they are not valid: Are there weakerestimates still useful for variational analysis?
Ricardo G. Durán Poincaré and related inequalities
GOALS
This talk is about several classic inequalities(Korn, Poincaré, Improved Poincaré, etc.).
Questions we are interested in:
What are the relations between them?
For which domains are these inequalitiesvalid?
When they are not valid: Are there weakerestimates still useful for variational analysis?
Ricardo G. Durán Poincaré and related inequalities
GOALS
More generally: in some applications it isinteresting to have “uniformly valid inequalities”for a sequence of domains.
For example: Domain decomposition methodsfor finite elements.
Dohrmann, Klawonn, and Widlund, Domaindecomposition for less regular subdomains:overlapping Schwarz in two dimensions. SIAMJ. Numer. Anal. 46 (2008), no. 4, 2153–2168.
Ricardo G. Durán Poincaré and related inequalities
GOALS
More generally: in some applications it isinteresting to have “uniformly valid inequalities”for a sequence of domains.
For example: Domain decomposition methodsfor finite elements.
Dohrmann, Klawonn, and Widlund, Domaindecomposition for less regular subdomains:overlapping Schwarz in two dimensions. SIAMJ. Numer. Anal. 46 (2008), no. 4, 2153–2168.
Ricardo G. Durán Poincaré and related inequalities
MAIN RESULT
A methodology to prove the inequalities ingeneral domains using analogous inequalities incubes.
Main tool:
Whitney decomposition into cubes!
Ricardo G. Durán Poincaré and related inequalities
NOTATIONS
Ω ⊂ Rn bounded domain 1 ≤ p ≤ ∞
W 1,p(Ω) =
v ∈ Lp(Ω) :∂v∂xj∈ Lp(Ω) ,∀j = 1, · · · ,n
W 1,p
0 (Ω) = C∞0 (Ω) ⊂W 1,p(Ω)
H1(Ω) = W 1,2(Ω) , H10 (Ω) = W 1,2
0 (Ω)
Ricardo G. Durán Poincaré and related inequalities
NOTATIONS
Lp0(Ω) =
f ∈ Lp(Ω) :
∫Ω
f = 0
W−1,p′(Ω) = W 1,p0 (Ω)′
where p′ is the dual exponent of p
C = C(· · · ) constant depending on · · ·
Ricardo G. Durán Poincaré and related inequalities
KORN INEQUALITY
v ∈W 1,p(Ω)n , Dv =(∂vi
∂xj
)
ε(v) symmetric part of Dv, i. e.,
εij(v) =12
(∂vi
∂xj+∂vj
∂xi
)
Ricardo G. Durán Poincaré and related inequalities
KORN INEQUALITY
v ∈W 1,p(Ω)n , Dv =(∂vi
∂xj
)ε(v) symmetric part of Dv, i. e.,
εij(v) =12
(∂vi
∂xj+∂vj
∂xi
)
Ricardo G. Durán Poincaré and related inequalities
KORN INEQUALITY
1 < p <∞, there exists C = C(Ω,p) such that
‖v‖W 1,p(Ω)n ≤ C‖v‖Lp(Ω)n + ‖ε(v)‖Lp(Ω)n×n
Ricardo G. Durán Poincaré and related inequalities
RIGHT INVERSES OF THE DIVERGENCE
f ∈ Lp0(Ω),1 < p <∞
∃u ∈W 1,p0 (Ω)n such that
div u = f , ‖u‖W 1,p(Ω)n ≤ C‖f‖Lp(Ω)
C = C(Ω,p)
Ricardo G. Durán Poincaré and related inequalities
RIGHT INVERSES OF THE DIVERGENCE
f ∈ Lp0(Ω),1 < p <∞
∃u ∈W 1,p0 (Ω)n such that
div u = f , ‖u‖W 1,p(Ω)n ≤ C‖f‖Lp(Ω)
C = C(Ω,p)
Ricardo G. Durán Poincaré and related inequalities
DUAL VERSION
Called Lions Lemma (when p = 2)
∃ C = C(Ω,p) such that∫Ω
f = 0 =⇒ ‖f‖Lp′(Ω) ≤ C‖∇f‖W−1,p′(Ω)n
or equivalently (inf-sup condition for Stokes)
inff∈Lp′
0 (Ω)
supv∈W 1,p
0 (Ω)n
∫Ω f div v
‖f‖Lp′‖v‖W 1,p0 (Ω)n
≥ α > 0
with α = C−1.
Ricardo G. Durán Poincaré and related inequalities
DUAL VERSION
Called Lions Lemma (when p = 2)∃ C = C(Ω,p) such that∫
Ω
f = 0 =⇒ ‖f‖Lp′(Ω) ≤ C‖∇f‖W−1,p′(Ω)n
or equivalently (inf-sup condition for Stokes)
inff∈Lp′
0 (Ω)
supv∈W 1,p
0 (Ω)n
∫Ω f div v
‖f‖Lp′‖v‖W 1,p0 (Ω)n
≥ α > 0
with α = C−1.
Ricardo G. Durán Poincaré and related inequalities
DUAL VERSION
Called Lions Lemma (when p = 2)∃ C = C(Ω,p) such that∫
Ω
f = 0 =⇒ ‖f‖Lp′(Ω) ≤ C‖∇f‖W−1,p′(Ω)n
or equivalently (inf-sup condition for Stokes)
inff∈Lp′
0 (Ω)
supv∈W 1,p
0 (Ω)n
∫Ω f div v
‖f‖Lp′‖v‖W 1,p0 (Ω)n
≥ α > 0
with α = C−1.
Ricardo G. Durán Poincaré and related inequalities
IMPROVED POINCARÉ INEQUALITY
d(x) = distance to ∂Ω
∃C = C(Ω,p) such that
∫Ω
f = 0 =⇒ ‖f‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)n
Ricardo G. Durán Poincaré and related inequalities
IMPROVED POINCARÉ INEQUALITY
d(x) = distance to ∂Ω
∃C = C(Ω,p) such that
∫Ω
f = 0 =⇒ ‖f‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)n
Ricardo G. Durán Poincaré and related inequalities
IMPROVED POINCARÉ INEQUALITY
d(x) = distance to ∂Ω
∃C = C(Ω,p) such that
∫Ω
f = 0 =⇒ ‖f‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)n
Ricardo G. Durán Poincaré and related inequalities
DUAL VERSION
f ∈ Lp′0 (Ω) ⇒ ∃u ∈ Lp′(Ω)n such that
div u = f ,∥∥∥u
d
∥∥∥Lp′(Ω)n
≤ C‖f‖Lp′(Ω)
u · n = 0 en ∂Ω
Ricardo G. Durán Poincaré and related inequalities
DUAL VERSION
f ∈ Lp′0 (Ω) ⇒ ∃u ∈ Lp′(Ω)n such that
div u = f ,∥∥∥u
d
∥∥∥Lp′(Ω)n
≤ C‖f‖Lp′(Ω)
u · n = 0 en ∂Ω
Ricardo G. Durán Poincaré and related inequalities
RELATIONS
Several connections between these inequalitiesare known:
Right Inverse of Div ⇒ Korn
Improved Poincaré ⇒ Korn(Kondratiev-Oleinik)
Explicit relation between constants in 2d for C1
domains (Horgan-Payne)
Ricardo G. Durán Poincaré and related inequalities
RELATIONS
Several connections between these inequalitiesare known:
Right Inverse of Div ⇒ Korn
Improved Poincaré ⇒ Korn(Kondratiev-Oleinik)
Explicit relation between constants in 2d for C1
domains (Horgan-Payne)
Ricardo G. Durán Poincaré and related inequalities
RELATIONS
Several connections between these inequalitiesare known:
Right Inverse of Div ⇒ Korn
Improved Poincaré ⇒ Korn(Kondratiev-Oleinik)
Explicit relation between constants in 2d for C1
domains (Horgan-Payne)
Ricardo G. Durán Poincaré and related inequalities
RELATIONS
Several connections between these inequalitiesare known:
Right Inverse of Div ⇒ Korn
Improved Poincaré ⇒ Korn(Kondratiev-Oleinik)
Explicit relation between constants in 2d for C1
domains (Horgan-Payne)
Ricardo G. Durán Poincaré and related inequalities
A SIMPLE EXAMPLE
THE DOMAIN CANNOT BE ARBITRARY
EXAMPLE (G. ACOSTA):
Ω = (x , y) ∈ R2 : 0 < x < 1 , 0 < y < x2
Ricardo G. Durán Poincaré and related inequalities
A SIMPLE EXAMPLE
THE DOMAIN CANNOT BE ARBITRARY
EXAMPLE (G. ACOSTA):
Ω = (x , y) ∈ R2 : 0 < x < 1 , 0 < y < x2
Ricardo G. Durán Poincaré and related inequalities
f (x , y) =1x2 − 3⇒
∫Ω
f = 0
∫Ω
f 2 =
∫ 1
0
∫ x2
0f 2 dydx '
∫ 1
0
1x2 ⇒ f /∈ L2(Ω)
−2yx−3 ∈ L2(Ω)⇒ ∂f∂x
=∂(−2yx−3)
∂y∈ H−1(Ω)
Ricardo G. Durán Poincaré and related inequalities
A SIMPLE EXAMPLE
THEN:
‖f‖L2(Ω) C‖∇f‖H−1(Ω)n
Ricardo G. Durán Poincaré and related inequalities
A SIMPLE EXAMPLE
SAME EXAMPLE ⇒
‖f‖L2(Ω) C‖d∇f‖L2(Ω)n
Ricardo G. Durán Poincaré and related inequalities
OUR MAIN RESULT
“THEOREM”
‖f‖L1(Ω) ≤ C‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)
⇓“All” the inequalities valid in cubes are valid in Ω(∀p) !!
Ricardo G. Durán Poincaré and related inequalities
MOTIVATION
Why are relations between inequalitiesinteresting?
We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domainThen, we can translate information from oneinequality to other
Ricardo G. Durán Poincaré and related inequalities
MOTIVATION
Why are relations between inequalitiesinteresting?We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)
But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domainThen, we can translate information from oneinequality to other
Ricardo G. Durán Poincaré and related inequalities
MOTIVATION
Why are relations between inequalitiesinteresting?We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domain
Then, we can translate information from oneinequality to other
Ricardo G. Durán Poincaré and related inequalities
MOTIVATION
Why are relations between inequalitiesinteresting?We could think that, if we are not interested instrange domains, we already know that all theseinequalities are valid (for example for Ω aLipschitz domain)But, in many situations it is important to haveinformation of the constants in terms of thegeometry of the domainThen, we can translate information from oneinequality to other
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
For example:
Let Ω be a convex domain with diameter D andinner diameter ρ.Then, it was recently proved by M. Barchiesi, F.Cagnetti, and N. Fusco that
‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
For example:
Let Ω be a convex domain with diameter D andinner diameter ρ.
Then, it was recently proved by M. Barchiesi, F.Cagnetti, and N. Fusco that
‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
For example:
Let Ω be a convex domain with diameter D andinner diameter ρ.Then, it was recently proved by M. Barchiesi, F.Cagnetti, and N. Fusco that
‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
For example:
Let Ω be a convex domain with diameter D andinner diameter ρ.Then, it was recently proved by M. Barchiesi, F.Cagnetti, and N. Fusco that
‖f‖L1(Ω) ≤ CDρ‖d∇f‖L1(Ω)n ∀f ∈ L1
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
Consequently, we obtain from our results that,for Ω convex,
‖f‖Lp(Ω) ≤ CDρ‖∇f‖W−1,p(Ω)n ∀f ∈ Lp
0(Ω)
1 < p <∞
C = C(p,n)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
Consequently, we obtain from our results that,for Ω convex,
‖f‖Lp(Ω) ≤ CDρ‖∇f‖W−1,p(Ω)n ∀f ∈ Lp
0(Ω)
1 < p <∞
C = C(p,n)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
Consequently, we obtain from our results that,for Ω convex,
‖f‖Lp(Ω) ≤ CDρ‖∇f‖W−1,p(Ω)n ∀f ∈ Lp
0(Ω)
1 < p <∞
C = C(p,n)
Ricardo G. Durán Poincaré and related inequalities
OPTIMALITY OF THIS ESTIMATE
The case of a rectangular domain shows thatthe dependence of the constant in terms of theeccentricity D/ρ cannot be improved. Indeed(let us consider p = 2 for simplicity)
Ωε = (x , y) ∈ R2 : −1 < x < 1 , −ε < y < ε
‖f‖L2(Ωε) ≤ Cε‖∇f‖H−1(Ωε)n
⇓‖x‖L2 ≤ Cε‖∇x‖H−1 = Cε‖∇y‖H−1 ≤ Cε‖y‖L2
Ricardo G. Durán Poincaré and related inequalities
OPTIMALITY OF THIS ESTIMATE
The case of a rectangular domain shows thatthe dependence of the constant in terms of theeccentricity D/ρ cannot be improved. Indeed(let us consider p = 2 for simplicity)
Ωε = (x , y) ∈ R2 : −1 < x < 1 , −ε < y < ε
‖f‖L2(Ωε) ≤ Cε‖∇f‖H−1(Ωε)n
⇓‖x‖L2 ≤ Cε‖∇x‖H−1 = Cε‖∇y‖H−1 ≤ Cε‖y‖L2
Ricardo G. Durán Poincaré and related inequalities
OPTIMALITY OF THIS ESTIMATE
The case of a rectangular domain shows thatthe dependence of the constant in terms of theeccentricity D/ρ cannot be improved. Indeed(let us consider p = 2 for simplicity)
Ωε = (x , y) ∈ R2 : −1 < x < 1 , −ε < y < ε
‖f‖L2(Ωε) ≤ Cε‖∇f‖H−1(Ωε)n
⇓‖x‖L2 ≤ Cε‖∇x‖H−1 = Cε‖∇y‖H−1 ≤ Cε‖y‖L2
Ricardo G. Durán Poincaré and related inequalities
OPTIMALITY OF THIS ESTIMATE
‖x‖L2(Ωε) ∼ ε12 , ‖y‖L2(Ωε) ∼ ε
32
⇓
Cε ≥Cε∼ Dρ
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
In many cases it is also possible to go in theopposite way:
For example, if Ω ⊂ R2 hasdiameter D and is star-shaped with respect to aball of radius ρ, we could prove that
‖f‖L2(Ω) ≤ CDρ
(log
Dρ
)‖∇f‖H−1(Ω)n ∀f ∈ L2
0(Ω)
And from this estimate it can be deduced that
‖f‖L2(Ω) ≤ CDρ
(log
Dρ
)‖d∇f‖L2(Ω)n ∀f ∈ L2
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
In many cases it is also possible to go in theopposite way: For example, if Ω ⊂ R2 hasdiameter D and is star-shaped with respect to aball of radius ρ, we could prove that
‖f‖L2(Ω) ≤ CDρ
(log
Dρ
)‖∇f‖H−1(Ω)n ∀f ∈ L2
0(Ω)
And from this estimate it can be deduced that
‖f‖L2(Ω) ≤ CDρ
(log
Dρ
)‖d∇f‖L2(Ω)n ∀f ∈ L2
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
ESTIMATES IN TERMS OF GEOMETRICPROPERTIES
In many cases it is also possible to go in theopposite way: For example, if Ω ⊂ R2 hasdiameter D and is star-shaped with respect to aball of radius ρ, we could prove that
‖f‖L2(Ω) ≤ CDρ
(log
Dρ
)‖∇f‖H−1(Ω)n ∀f ∈ L2
0(Ω)
And from this estimate it can be deduced that
‖f‖L2(Ω) ≤ CDρ
(log
Dρ
)‖d∇f‖L2(Ω)n ∀f ∈ L2
0(Ω)
Ricardo G. Durán Poincaré and related inequalities
MAIN RESULT
Proof of
‖f‖L1(Ω) ≤ C1‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)
⇓∀f ∈ Lp
0(Ω),1 < p <∞
∃u ∈W 1,p0 (Ω)n such that
div u = f , ‖u‖W 1,p(Ω)n ≤ C‖f‖Lp(Ω)
C = C(n,p,C1)
Ricardo G. Durán Poincaré and related inequalities
MAIN RESULT
Proof of
‖f‖L1(Ω) ≤ C1‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)
⇓∀f ∈ Lp
0(Ω),1 < p <∞
∃u ∈W 1,p0 (Ω)n such that
div u = f , ‖u‖W 1,p(Ω)n ≤ C‖f‖Lp(Ω)
C = C(n,p,C1)
Ricardo G. Durán Poincaré and related inequalities
MAIN RESULT
Proof of
‖f‖L1(Ω) ≤ C1‖d∇f‖L1(Ω)n ∀f ∈ L10(Ω)
⇓∀f ∈ Lp
0(Ω),1 < p <∞
∃u ∈W 1,p0 (Ω)n such that
div u = f , ‖u‖W 1,p(Ω)n ≤ C‖f‖Lp(Ω)
C = C(n,p,C1)
Ricardo G. Durán Poincaré and related inequalities
THREE STEPS
Improved Poincaré for p = 1⇒ ImprovedPoincaré for 1 < p <∞
Improved Poincaré for p′ allows us to reducethe problem to “local problems” in cubes
Solve for the divergence in cubes and sum
Ricardo G. Durán Poincaré and related inequalities
THREE STEPS
Improved Poincaré for p = 1⇒ ImprovedPoincaré for 1 < p <∞
Improved Poincaré for p′ allows us to reducethe problem to “local problems” in cubes
Solve for the divergence in cubes and sum
Ricardo G. Durán Poincaré and related inequalities
THREE STEPS
Improved Poincaré for p = 1⇒ ImprovedPoincaré for 1 < p <∞
Improved Poincaré for p′ allows us to reducethe problem to “local problems” in cubes
Solve for the divergence in cubes and sum
Ricardo G. Durán Poincaré and related inequalities
STEP 2
Assume improved Poincaré for p′
Take a Whitney decomposition of Ω, i. e.,
Ω = ∪jQj , Q0j ∩Q0
i = ∅
diam Qj ∼ dist (Qj , ∂Ω) =: dj
For f ∈ Lp0(Ω), there exists a decomposition
f =∑
j
fj
Ricardo G. Durán Poincaré and related inequalities
STEP 2
Assume improved Poincaré for p′
Take a Whitney decomposition of Ω, i. e.,
Ω = ∪jQj , Q0j ∩Q0
i = ∅
diam Qj ∼ dist (Qj , ∂Ω) =: dj
For f ∈ Lp0(Ω), there exists a decomposition
f =∑
j
fj
Ricardo G. Durán Poincaré and related inequalities
STEP 2
Assume improved Poincaré for p′
Take a Whitney decomposition of Ω, i. e.,
Ω = ∪jQj , Q0j ∩Q0
i = ∅
diam Qj ∼ dist (Qj , ∂Ω) =: dj
For f ∈ Lp0(Ω), there exists a decomposition
f =∑
j
fj
Ricardo G. Durán Poincaré and related inequalities
STEP 2
Assume improved Poincaré for p′
Take a Whitney decomposition of Ω, i. e.,
Ω = ∪jQj , Q0j ∩Q0
i = ∅
diam Qj ∼ dist (Qj , ∂Ω) =: dj
For f ∈ Lp0(Ω), there exists a decomposition
f =∑
j
fj
Ricardo G. Durán Poincaré and related inequalities
STEP 2
such that
fj ∈ Lp0(Qj) , Qj :=
98
Qj
and
‖f‖pLp(Ω) ∼
∑j
‖fj‖pLp(Qj )
Ricardo G. Durán Poincaré and related inequalities
STEP 2
such that
fj ∈ Lp0(Qj) , Qj :=
98
Qj
and
‖f‖pLp(Ω) ∼
∑j
‖fj‖pLp(Qj )
Ricardo G. Durán Poincaré and related inequalities
STEP 2
such that
fj ∈ Lp0(Qj) , Qj :=
98
Qj
and
‖f‖pLp(Ω) ∼
∑j
‖fj‖pLp(Qj )
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Proof of the existence of this decomposition:
Take a partition of unity associated with theWhitney decomposition
∑j
φj = 1 , sopφj ⊂ Qj =98
Qj
‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Proof of the existence of this decomposition:
Take a partition of unity associated with theWhitney decomposition
∑j
φj = 1 , sopφj ⊂ Qj =98
Qj
‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Assuming the improved Poincaré for p′, weobtain by duality:
For f ∈ Lp0(Ω) there exists u ∈ Lp(Ω)n such that
div u = f in Ω u · n = 0 on ∂Ω
∥∥∥ud
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Assuming the improved Poincaré for p′, weobtain by duality:
For f ∈ Lp0(Ω) there exists u ∈ Lp(Ω)n such that
div u = f in Ω u · n = 0 on ∂Ω
∥∥∥ud
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Assuming the improved Poincaré for p′, weobtain by duality:
For f ∈ Lp0(Ω) there exists u ∈ Lp(Ω)n such that
div u = f in Ω u · n = 0 on ∂Ω
∥∥∥ud
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Then, we define
fj = div (φju)
and so
f = div u = div u∑
j
φj =∑
j
div (φju) =∑
j
fj
sopφj ⊂ Qj ⇒ sop fj ⊂ Qj ,
∫fj = 0
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Then, we define
fj = div (φju)
and so
f = div u = div u∑
j
φj =∑
j
div (φju) =∑
j
fj
sopφj ⊂ Qj ⇒ sop fj ⊂ Qj ,
∫fj = 0
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Then, we define
fj = div (φju)
and so
f = div u = div u∑
j
φj =∑
j
div (φju) =∑
j
fj
sopφj ⊂ Qj ⇒ sop fj ⊂ Qj ,
∫fj = 0
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
Then, we define
fj = div (φju)
and so
f = div u = div u∑
j
φj =∑
j
div (φju) =∑
j
fj
sopφj ⊂ Qj ⇒ sop fj ⊂ Qj ,
∫fj = 0
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
From finite superposition:
|f (x)|p ≤ C∑
j
|fj(x)|p
and therefore
‖f‖pLp(Ω) ≤ C
∑j
‖fj‖pLp(Qj )
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
From finite superposition:
|f (x)|p ≤ C∑
j
|fj(x)|p
and therefore
‖f‖pLp(Ω) ≤ C
∑j
‖fj‖pLp(Qj )
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
To prove the other estimate we use
‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj
Then,
fj = div (φju) = div uφj +∇φj · u = fφj +∇φj · u
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
To prove the other estimate we use
‖φj‖L∞ ≤ 1 , ‖∇φj‖L∞ ≤ C/dj
Then,
fj = div (φju) = div uφj +∇φj · u = fφj +∇φj · u
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
‖fj‖pLp(Qj )
≤ C‖f‖p
Lp(Qj )+∥∥∥u
d
∥∥∥p
Lp(Qj )
and therefore, using∥∥∥ud
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
we obtain, ∑j
‖fj‖pLp(Qj )
≤ C‖f‖pLp(Ω)
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
‖fj‖pLp(Qj )
≤ C‖f‖p
Lp(Qj )+∥∥∥u
d
∥∥∥p
Lp(Qj )
and therefore, using∥∥∥u
d
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
we obtain, ∑j
‖fj‖pLp(Qj )
≤ C‖f‖pLp(Ω)
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
‖fj‖pLp(Qj )
≤ C‖f‖p
Lp(Qj )+∥∥∥u
d
∥∥∥p
Lp(Qj )
and therefore, using∥∥∥u
d
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
we obtain,
∑j
‖fj‖pLp(Qj )
≤ C‖f‖pLp(Ω)
Ricardo G. Durán Poincaré and related inequalities
DECOMPOSITION OF FUNCTIONS
‖fj‖pLp(Qj )
≤ C‖f‖p
Lp(Qj )+∥∥∥u
d
∥∥∥p
Lp(Qj )
and therefore, using∥∥∥u
d
∥∥∥Lp(Ω)
≤ C‖f‖Lp(Ω)
we obtain, ∑j
‖fj‖pLp(Qj )
≤ C‖f‖pLp(Ω)
Ricardo G. Durán Poincaré and related inequalities
STEP 3
Then, to solve div u = f
solve div uj = fj
uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )
≤ C‖fj‖Lp(Qj )
Observe that C = C(n,p)
andu =
∑j
uj
is the required solution!
Ricardo G. Durán Poincaré and related inequalities
STEP 3
Then, to solve div u = f
solve div uj = fj
uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )
≤ C‖fj‖Lp(Qj )
Observe that C = C(n,p)
andu =
∑j
uj
is the required solution!
Ricardo G. Durán Poincaré and related inequalities
STEP 3
Then, to solve div u = f
solve div uj = fj
uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )
≤ C‖fj‖Lp(Qj )
Observe that C = C(n,p)
andu =
∑j
uj
is the required solution!
Ricardo G. Durán Poincaré and related inequalities
STEP 3
Then, to solve div u = f
solve div uj = fj
uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )
≤ C‖fj‖Lp(Qj )
Observe that C = C(n,p)
andu =
∑j
uj
is the required solution!
Ricardo G. Durán Poincaré and related inequalities
STEP 3
Then, to solve div u = f
solve div uj = fj
uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )
≤ C‖fj‖Lp(Qj )
Observe that C = C(n,p)
and
u =∑
j
uj
is the required solution!
Ricardo G. Durán Poincaré and related inequalities
STEP 3
Then, to solve div u = f
solve div uj = fj
uj ∈W 1,p0 (Qj) , ‖uj‖W 1,p(Qj )
≤ C‖fj‖Lp(Qj )
Observe that C = C(n,p)
andu =
∑j
uj
is the required solution!
Ricardo G. Durán Poincaré and related inequalities
STEP 1: Poincaré in L1 implies Poincaré in Lp,p <∞
To conclude the proof of our theorem we need:
‖f − fΩ‖L1(Ω) ≤ C‖d∇f‖L1(Ω)
⇓
‖f − fΩ‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
STEP 1: Poincaré in L1 implies Poincaré in Lp,p <∞
To conclude the proof of our theorem we need:
‖f − fΩ‖L1(Ω) ≤ C‖d∇f‖L1(Ω)
⇓
‖f − fΩ‖Lp(Ω) ≤ C‖d∇f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
OR MORE GENERALLY
‖f − fΩ‖L1(Ω) ≤ C‖w∇f‖L1(Ω)
⇓
‖f − fΩ‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
∀p <∞
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
1)‖f − fΩ‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
m2)
∀E ⊂ Ω such that |E | ≥ 12|Ω|
f |E = 0⇒ ‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
with C independent of E and f .
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
1)‖f − fΩ‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
m2)
∀E ⊂ Ω such that |E | ≥ 12|Ω|
f |E = 0⇒ ‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
with C independent of E and f .
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
Case p = 1 :
f = f+ − f−
Suppose |f+ = 0| ≥ 12 |Ω|
⇒ ‖f+‖L1(Ω) ≤ C‖w∇f+‖L1(Ω) ≤ C‖w∇f‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
Case p = 1 :
f = f+ − f−
Suppose |f+ = 0| ≥ 12 |Ω|
⇒ ‖f+‖L1(Ω) ≤ C‖w∇f+‖L1(Ω) ≤ C‖w∇f‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
But
∫Ω
f = 0⇒∫
Ω
f+ =
∫Ω
f−
and therefore,
‖f‖L1(Ω) =
∫Ω
f+ + f− = 2∫
Ω
f+ ≤ 2C‖w∇f+‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
But ∫Ω
f = 0⇒∫
Ω
f+ =
∫Ω
f−
and therefore,
‖f‖L1(Ω) =
∫Ω
f+ + f− = 2∫
Ω
f+ ≤ 2C‖w∇f+‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
But ∫Ω
f = 0⇒∫
Ω
f+ =
∫Ω
f−
and therefore,
‖f‖L1(Ω) =
∫Ω
f+ + f− = 2∫
Ω
f+ ≤ 2C‖w∇f+‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
If ∫Ω
f p+ =
∫Ω
f p−
the same argument than for p = 1 applies!
But, from Bolzano’s theorem
∃λ ∈ Rsuch that ∫
Ω
(f − λ)p+ =
∫Ω
(f − λ)p−
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
If ∫Ω
f p+ =
∫Ω
f p−
the same argument than for p = 1 applies!
But, from Bolzano’s theorem
∃λ ∈ Rsuch that ∫
Ω
(f − λ)p+ =
∫Ω
(f − λ)p−
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
If ∫Ω
f p+ =
∫Ω
f p−
the same argument than for p = 1 applies!
But, from Bolzano’s theorem
∃λ ∈ Rsuch that ∫
Ω
(f − λ)p+ =
∫Ω
(f − λ)p−
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
Now,
f |E = 0⇒ |f |p|E = 0
Apply 1−Poincaré to |f |p
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
Now,
f |E = 0⇒ |f |p|E = 0
Apply 1−Poincaré to |f |p
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
∫Ω
|f |p ≤ Cp∫
Ω
|f |p−1|∇f |w
≤(∫
Ω
|f |p)1/p′ (∫
Ω
|∇f |pwp)1/p
therefore,
‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
Poincaré in L1 implies Poincaré in Lp,p <∞
∫Ω
|f |p ≤ Cp∫
Ω
|f |p−1|∇f |w
≤(∫
Ω
|f |p)1/p′ (∫
Ω
|∇f |pwp)1/p
therefore,
‖f‖Lp(Ω) ≤ C‖w∇f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
Proof of the improved Poincaré in L1
Now, to complete our arguments we have toprove the improved Poincaré in L1.
We will show that it can be proved for a verygeneral class of domains by elementarycalculus arguments
Ricardo G. Durán Poincaré and related inequalities
Proof of the improved Poincaré in L1
Now, to complete our arguments we have toprove the improved Poincaré in L1.
We will show that it can be proved for a verygeneral class of domains by elementarycalculus arguments
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
f (y)− f = −∫
Ω
G(x , y) · ∇f (x) dx
Assume Ω star-shaped with respect to a ballB(0, δ)
ω ∈ C∞0 (Bδ/2)
∫Bδ/2
ω = 1 f =
∫Ω
fω
y ∈ Ω , z ∈ Bδ/2
f (y)− f (z) =
∫ 1
0(y − z) · ∇f (y + t(z − y)) dt
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
f (y)− f = −∫
Ω
G(x , y) · ∇f (x) dx
Assume Ω star-shaped with respect to a ballB(0, δ)
ω ∈ C∞0 (Bδ/2)
∫Bδ/2
ω = 1 f =
∫Ω
fω
y ∈ Ω , z ∈ Bδ/2
f (y)− f (z) =
∫ 1
0(y − z) · ∇f (y + t(z − y)) dt
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
f (y)− f = −∫
Ω
G(x , y) · ∇f (x) dx
Assume Ω star-shaped with respect to a ballB(0, δ)
ω ∈ C∞0 (Bδ/2)
∫Bδ/2
ω = 1 f =
∫Ω
fω
y ∈ Ω , z ∈ Bδ/2
f (y)− f (z) =
∫ 1
0(y − z) · ∇f (y + t(z − y)) dt
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
Multiplying by ω(z) and integrating:
f (y)− f =
∫Ω
∫ 1
0(y−z) ·∇f (y + t(z−y))ω(z)dtdz
and changing variables
x = y + t(z − y)
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
Multiplying by ω(z) and integrating:
f (y)− f =
∫Ω
∫ 1
0(y−z) ·∇f (y + t(z−y))ω(z)dtdz
and changing variables
x = y + t(z − y)
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
Multiplying by ω(z) and integrating:
f (y)− f =
∫Ω
∫ 1
0(y−z) ·∇f (y + t(z−y))ω(z)dtdz
and changing variables
x = y + t(z − y)
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
f (y)−f =
∫Ω
∫ 1
0
(y − x)
tn+1 ·∇f (x)ω
(y +
x − yt
)dtdx
that is,
f (y)− f = −∫
Ω
G(x , y) · ∇f (x)dx
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
f (y)−f =
∫Ω
∫ 1
0
(y − x)
tn+1 ·∇f (x)ω
(y +
x − yt
)dtdx
that is,
f (y)− f = −∫
Ω
G(x , y) · ∇f (x)dx
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
G(x , y) =
∫ 1
0
(x − y)
tω
(y +
x − yt
)1tn dt
PROPERTIES OF G(x , y)
|G(x , y)| ≤ C1
|x − y |n−1
|x − y | > C2d(x) ⇒ G(x , y) = 0
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
G(x , y) =
∫ 1
0
(x − y)
tω
(y +
x − yt
)1tn dt
PROPERTIES OF G(x , y)
|G(x , y)| ≤ C1
|x − y |n−1
|x − y | > C2d(x) ⇒ G(x , y) = 0
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
G(x , y) =
∫ 1
0
(x − y)
tω
(y +
x − yt
)1tn dt
PROPERTIES OF G(x , y)
|G(x , y)| ≤ C1
|x − y |n−1
|x − y | > C2d(x) ⇒ G(x , y) = 0
Ricardo G. Durán Poincaré and related inequalities
REPRESENTATION FORMULA
G(x , y) =
∫ 1
0
(x − y)
tω
(y +
x − yt
)1tn dt
PROPERTIES OF G(x , y)
|G(x , y)| ≤ C1
|x − y |n−1
|x − y | > C2d(x) ⇒ G(x , y) = 0
Ricardo G. Durán Poincaré and related inequalities
MORE GENERAL DOMAINS
The representation formula can be generalizedto John domains (Acosta-D.-Muschietti)
G(x , y) =
∫ 1
0
(x − γt
+γ(t , y))ω
(x − γ(t , y)
t
)dttn
where γ(t , y) are “ John curves”
γ(0, y) = y , γ(1, y) = 0
|γ(t , y)| ≤ K , d(γ(t , y)) ≥ δt
Ricardo G. Durán Poincaré and related inequalities
MORE GENERAL DOMAINS
The representation formula can be generalizedto John domains (Acosta-D.-Muschietti)
G(x , y) =
∫ 1
0
(x − γt
+γ(t , y))ω
(x − γ(t , y)
t
)dttn
where γ(t , y) are “ John curves”
γ(0, y) = y , γ(1, y) = 0
|γ(t , y)| ≤ K , d(γ(t , y)) ≥ δt
Ricardo G. Durán Poincaré and related inequalities
MORE GENERAL DOMAINS
The representation formula can be generalizedto John domains (Acosta-D.-Muschietti)
G(x , y) =
∫ 1
0
(x − γt
+γ(t , y))ω
(x − γ(t , y)
t
)dttn
where γ(t , y) are “ John curves”
γ(0, y) = y , γ(1, y) = 0
|γ(t , y)| ≤ K , d(γ(t , y)) ≥ δt
Ricardo G. Durán Poincaré and related inequalities
MORE GENERAL DOMAINS
And the same properties for G(x , y) holds.
|G(x , y)| ≤ C1
|x − y |n−1
|x − y | > C2d(x) ⇒ G(x , y) = 0
with constants depending on δ, K y ω
Ricardo G. Durán Poincaré and related inequalities
PROOF OF IMPROVED POINCARÉ IN L1
(Drelichman-D.)
∫Ω
|f (y)− f |dy ≤∫
Ω
∫Ω
|G(x , y)||∇f (x)|dxdy
=
∫Ω
∫Ω
|G(x , y)| · |∇f (x)|dydx
≤ C∫
Ω
∫|x−y |≤Cd(x)
dy|x − y |n−1 |∇f (x)|dx
≤ C‖d∇f‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
PROOF OF IMPROVED POINCARÉ IN L1
(Drelichman-D.)
∫Ω
|f (y)− f |dy ≤∫
Ω
∫Ω
|G(x , y)||∇f (x)|dxdy
=
∫Ω
∫Ω
|G(x , y)| · |∇f (x)|dydx
≤ C∫
Ω
∫|x−y |≤Cd(x)
dy|x − y |n−1 |∇f (x)|dx
≤ C‖d∇f‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
PROOF OF IMPROVED POINCARÉ IN L1
(Drelichman-D.)
∫Ω
|f (y)− f |dy ≤∫
Ω
∫Ω
|G(x , y)||∇f (x)|dxdy
=
∫Ω
∫Ω
|G(x , y)| · |∇f (x)|dydx
≤ C∫
Ω
∫|x−y |≤Cd(x)
dy|x − y |n−1 |∇f (x)|dx
≤ C‖d∇f‖L1(Ω)
Ricardo G. Durán Poincaré and related inequalities
GENERALIZATION
If Ω satisfies the weaker improved Poincaréinequality
‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)
for some β < 1 then,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−β∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
GENERALIZATION
If Ω satisfies the weaker improved Poincaréinequality
‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)
for some β < 1 then,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−β∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
GENERALIZATION
If Ω satisfies the weaker improved Poincaréinequality
‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)
for some β < 1
then,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−β∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
GENERALIZATION
If Ω satisfies the weaker improved Poincaréinequality
‖f‖L1(Ω) ≤ C1‖dβ∇f‖L1(Ω)n ∀f ∈ L10(Ω)
for some β < 1 then,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−β∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
Ricardo G. Durán Poincaré and related inequalities
APPLICATION TO Hölder α DOMAINS
In this case β = α
that is,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−α∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
The case p = 2 can be used to prove existenceand uniqueness for the Stokes equations.
Ricardo G. Durán Poincaré and related inequalities
APPLICATION TO Hölder α DOMAINS
In this case β = αthat is,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−α∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
The case p = 2 can be used to prove existenceand uniqueness for the Stokes equations.
Ricardo G. Durán Poincaré and related inequalities
APPLICATION TO Hölder α DOMAINS
In this case β = αthat is,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−α∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
The case p = 2 can be used to prove existenceand uniqueness for the Stokes equations.
Ricardo G. Durán Poincaré and related inequalities
APPLICATION TO Hölder α DOMAINS
In this case β = αthat is,
∀f ∈ Lp0(Ω) ∃u ∈W 1,p
0 (Ω)n such that
div u = f , ‖d1−α∇u‖Lp(Ω) ≤ C‖f‖Lp(Ω)
The case p = 2 can be used to prove existenceand uniqueness for the Stokes equations.
Ricardo G. Durán Poincaré and related inequalities
FINAL COMMENTS
We have a general methodology to proveinequalities once they are known in cubes.
The decomposition of functions of zero integralis interesting in itself because it might haveother applications (we are working in theapplication of this decomposition to weighted apriori estimates).
Ricardo G. Durán Poincaré and related inequalities
FINAL COMMENTS
We have a general methodology to proveinequalities once they are known in cubes.
The decomposition of functions of zero integralis interesting in itself because it might haveother applications (we are working in theapplication of this decomposition to weighted apriori estimates).
Ricardo G. Durán Poincaré and related inequalities
THANK YOU FOR YOUR ATTENTION!
HAPPY BIRTHDAY MARTIN!!
Ricardo G. Durán Poincaré and related inequalities