improved poincaré and other classic inequalities: a new
TRANSCRIPT
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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
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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
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Ricardo G. Durán Poincaré and related inequalities
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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
![Page 5: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/5.jpg)
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
![Page 6: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/6.jpg)
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
![Page 7: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/7.jpg)
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
![Page 8: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/8.jpg)
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
![Page 9: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/9.jpg)
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
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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
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MAIN RESULT
A methodology to prove the inequalities ingeneral domains using analogous inequalities incubes.
Main tool:
Whitney decomposition into cubes!
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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)
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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 · · ·
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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
)
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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
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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
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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)
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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)
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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.
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
![Page 29: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/29.jpg)
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
![Page 30: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/30.jpg)
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
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A SIMPLE EXAMPLE
THE DOMAIN CANNOT BE ARBITRARY
EXAMPLE (G. ACOSTA):
Ω = (x , y) ∈ R2 : 0 < x < 1 , 0 < y < x2
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A SIMPLE EXAMPLE
THE DOMAIN CANNOT BE ARBITRARY
EXAMPLE (G. ACOSTA):
Ω = (x , y) ∈ R2 : 0 < x < 1 , 0 < y < x2
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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)
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A SIMPLE EXAMPLE
THEN:
‖f‖L2(Ω) C‖∇f‖H−1(Ω)n
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A SIMPLE EXAMPLE
SAME EXAMPLE ⇒
‖f‖L2(Ω) C‖d∇f‖L2(Ω)n
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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) !!
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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
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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
![Page 39: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/39.jpg)
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
![Page 40: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/40.jpg)
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
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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)
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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
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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
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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
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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)
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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)
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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)
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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
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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
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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
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OPTIMALITY OF THIS ESTIMATE
‖x‖L2(Ωε) ∼ ε12 , ‖y‖L2(Ωε) ∼ ε
32
⇓
Cε ≥Cε∼ Dρ
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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STEP 2
such that
fj ∈ Lp0(Qj) , Qj :=
98
Qj
and
‖f‖pLp(Ω) ∼
∑j
‖fj‖pLp(Qj )
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STEP 2
such that
fj ∈ Lp0(Qj) , Qj :=
98
Qj
and
‖f‖pLp(Ω) ∼
∑j
‖fj‖pLp(Qj )
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STEP 2
such that
fj ∈ Lp0(Qj) , Qj :=
98
Qj
and
‖f‖pLp(Ω) ∼
∑j
‖fj‖pLp(Qj )
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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
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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
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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)
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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)
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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)
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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
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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
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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
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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
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DECOMPOSITION OF FUNCTIONS
From finite superposition:
|f (x)|p ≤ C∑
j
|fj(x)|p
and therefore
‖f‖pLp(Ω) ≤ C
∑j
‖fj‖pLp(Qj )
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DECOMPOSITION OF FUNCTIONS
From finite superposition:
|f (x)|p ≤ C∑
j
|fj(x)|p
and therefore
‖f‖pLp(Ω) ≤ C
∑j
‖fj‖pLp(Qj )
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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
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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
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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)
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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)
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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)
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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)
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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
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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
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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)
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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)
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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
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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
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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
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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−
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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)
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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)
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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)
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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)
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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)
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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)
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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
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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
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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
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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
![Page 137: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/137.jpg)
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
![Page 138: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/138.jpg)
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
![Page 139: Improved Poincaré and other classic inequalities: a new](https://reader031.vdocuments.site/reader031/viewer/2022012513/618cd4ed49c5663600290e27/html5/thumbnails/139.jpg)
THANK YOU FOR YOUR ATTENTION!
HAPPY BIRTHDAY MARTIN!!
Ricardo G. Durán Poincaré and related inequalities