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Improved Poincar and other classic inequalities: a new approach to prove them and some generalizations Ricardo G. Durn Departamento de Matemtica Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires and IMAS, CONICET

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Improved Poincaré and other classic inequalities: a new approach to prove them and some generalizations

Ricardo G. Durán

Departamento de Matemática Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires and IMAS, CONICET

Martin Stynes 60th Birthday Celebration Dresden 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.).

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:

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?

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 inequalities valid?

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 inequalities valid? When they are not valid: Are there weaker estimates still useful for variational analysis?

Ricardo G. Durán Poincaré and related inequalities

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GOALS More generally: in some applications it is interesting to have “uniformly valid inequalities” for a sequence of domains. For example: Domain decomposition methods for finite elements.

Ricardo G. Durán Poincaré and related inequalities

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GOALS More generally: in some applications it is interesting to have “uniformly valid inequalities” for a sequence of domains. For example: Domain decomposition methods for finite elements. Dohrmann, Klawonn, and Widlund, Domain decomposition for less regular subdomains:

  • verlapping Schwarz in two dimensions. SIAM
  • J. 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 in general domains using analogous inequalities in cubes. Main tool: Whitney decomposition into cubes!

Ricardo G. Durán Poincaré and related inequalities

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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) , H1

0(Ω) = W 1,2 0 (Ω)

Ricardo G. Durán Poincaré and related inequalities

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NOTATIONS Lp

0(Ω) =

  • f ∈ Lp(Ω) :
  • Ω

f = 0

  • W −1,p′(Ω) = W 1,p

0 (Ω)′

where p′ is the dual exponent of p C = C(· · · ) constant depending on · · ·

Ricardo G. Durán Poincaré and related inequalities

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KORN INEQUALITY v ∈ W 1,p(Ω)n , Dv = ∂vi ∂xj

  • Ricardo G. Durán

Poincaré and related inequalities

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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) = 1 2 ∂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 vW 1,p(Ω)n ≤ C

  • vLp(Ω)n + ε(v)Lp(Ω)n×n
  • Ricardo G. Durán

Poincaré and related inequalities

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RIGHT INVERSES OF THE DIVERGENCE f ∈ Lp

0(Ω), 1 < p < ∞

Ricardo G. Durán Poincaré and related inequalities

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RIGHT INVERSES OF THE DIVERGENCE f ∈ Lp

0(Ω), 1 < p < ∞

∃u ∈ W 1,p

0 (Ω)n

such that div u = f , uW 1,p(Ω)n ≤ CfLp(Ω) C = C(Ω, p)

Ricardo G. Durán Poincaré and related inequalities

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DUAL VERSION Called Lions Lemma (when p = 2)

Ricardo G. Durán Poincaré and related inequalities

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DUAL VERSION Called Lions Lemma (when p = 2) ∃ C = C(Ω, p) such that

  • Ω

f = 0 = ⇒ fLp′(Ω) ≤ C∇fW −1,p′(Ω)n

Ricardo G. Durán Poincaré and related inequalities

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DUAL VERSION Called Lions Lemma (when p = 2) ∃ C = C(Ω, p) such that

  • Ω

f = 0 = ⇒ fLp′(Ω) ≤ C∇fW −1,p′(Ω)n

  • r equivalently (inf-sup condition for Stokes)

inf

f∈Lp′

0 (Ω)

sup

v∈W 1,p (Ω)n

  • Ω f div v

fLp′vW 1,p

(Ω)n

≥ α > 0 with α = C−1.

Ricardo G. Durán Poincaré and related inequalities

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IMPROVED POINCARÉ INEQUALITY d(x) = distance to ∂Ω

Ricardo G. Durán Poincaré and related inequalities

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IMPROVED POINCARÉ INEQUALITY d(x) = distance to ∂Ω ∃C = C(Ω, p) such that

Ricardo G. Durán Poincaré and related inequalities

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IMPROVED POINCARÉ INEQUALITY d(x) = distance to ∂Ω ∃C = C(Ω, p) such that

  • Ω

f = 0 = ⇒ fLp(Ω) ≤ Cd∇fLp(Ω)n

Ricardo G. Durán Poincaré and related inequalities

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DUAL VERSION f ∈ Lp′

0 (Ω)

⇒ ∃u ∈ Lp′(Ω)n such that

Ricardo G. Durán Poincaré and related inequalities

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DUAL VERSION f ∈ Lp′

0 (Ω)

⇒ ∃u ∈ Lp′(Ω)n such that div u = f ,

  • u

d

  • Lp′(Ω)n ≤ CfLp′(Ω)

u · n = 0 en ∂Ω

Ricardo G. Durán Poincaré and related inequalities

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RELATIONS Several connections between these inequalities are known:

Ricardo G. Durán Poincaré and related inequalities

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RELATIONS Several connections between these inequalities are known: Right Inverse of Div ⇒ Korn

Ricardo G. Durán Poincaré and related inequalities

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RELATIONS Several connections between these inequalities are known: Right Inverse of Div ⇒ Korn Improved Poincaré ⇒ Korn (Kondratiev-Oleinik)

Ricardo G. Durán Poincaré and related inequalities

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RELATIONS Several connections between these inequalities are known: Right Inverse of Div ⇒ Korn Improved Poincaré ⇒ Korn (Kondratiev-Oleinik) Explicit relation between constants in 2d for C1 domains (Horgan-Payne)

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A SIMPLE EXAMPLE THE DOMAIN CANNOT BE ARBITRARY

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}

Ricardo G. Durán Poincaré and related inequalities

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f(x, y) = 1 x2 − 3 ⇒

  • Ω

f = 0

  • Ω

f 2 = 1 x2 f 2 dydx ≃ 1 1 x2 ⇒ f / ∈ L2(Ω) −2yx−3 ∈ L2(Ω) ⇒ ∂f ∂x = ∂(−2yx−3) ∂y ∈ H−1(Ω)

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A SIMPLE EXAMPLE THEN:

fL2(Ω) C∇fH−1(Ω)n

Ricardo G. Durán Poincaré and related inequalities

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A SIMPLE EXAMPLE SAME EXAMPLE ⇒

fL2(Ω) Cd∇fL2(Ω)n

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OUR MAIN RESULT “THEOREM” fL1(Ω) ≤ Cd∇fL1(Ω)n ∀f ∈ L1

0(Ω)

⇓ “All” the inequalities valid in cubes are valid in Ω (∀p) !!

Ricardo G. Durán Poincaré and related inequalities

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MOTIVATION Why are relations between inequalities interesting?

Ricardo G. Durán Poincaré and related inequalities

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MOTIVATION Why are relations between inequalities interesting? We could think that, if we are not interested in strange domains, we already know that all these inequalities are valid (for example for Ω a Lipschitz domain)

Ricardo G. Durán Poincaré and related inequalities

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MOTIVATION Why are relations between inequalities interesting? We could think that, if we are not interested in strange domains, we already know that all these inequalities are valid (for example for Ω a Lipschitz domain) But, in many situations it is important to have information of the constants in terms of the geometry of the domain

Ricardo G. Durán Poincaré and related inequalities

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MOTIVATION Why are relations between inequalities interesting? We could think that, if we are not interested in strange domains, we already know that all these inequalities are valid (for example for Ω a Lipschitz domain) But, in many situations it is important to have information of the constants in terms of the geometry of the domain Then, we can translate information from one inequality to other

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES For example:

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES For example: Let Ω be a convex domain with diameter D and inner diameter ρ.

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES For example: Let Ω be a convex domain with diameter D and inner diameter ρ. Then, it was recently proved by M. Barchiesi, F. Cagnetti, and N. Fusco that

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES For example: Let Ω be a convex domain with diameter D and inner diameter ρ. Then, it was recently proved by M. Barchiesi, F. Cagnetti, and N. Fusco that fL1(Ω) ≤ CD ρ d∇fL1(Ω)n ∀f ∈ L1

0(Ω)

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES Consequently, we obtain from our results that, for Ω convex,

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES Consequently, we obtain from our results that, for Ω convex, fLp(Ω) ≤ CD ρ ∇fW −1,p(Ω)n ∀f ∈ Lp

0(Ω)

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES Consequently, we obtain from our results that, for Ω convex, fLp(Ω) ≤ CD ρ ∇fW −1,p(Ω)n ∀f ∈ Lp

0(Ω)

1 < p < ∞ C = C(p, n)

Ricardo G. Durán Poincaré and related inequalities

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OPTIMALITY OF THIS ESTIMATE The case of a rectangular domain shows that the dependence of the constant in terms of the eccentricity D/ρ cannot be improved. Indeed (let us consider p = 2 for simplicity)

Ricardo G. Durán Poincaré and related inequalities

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OPTIMALITY OF THIS ESTIMATE The case of a rectangular domain shows that the dependence of the constant in terms of the eccentricity D/ρ cannot be improved. Indeed (let us consider p = 2 for simplicity) Ωε = {(x, y) ∈ R2 : −1 < x < 1 , −ε < y < ε}

Ricardo G. Durán Poincaré and related inequalities

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OPTIMALITY OF THIS ESTIMATE The case of a rectangular domain shows that the dependence of the constant in terms of the eccentricity D/ρ cannot be improved. Indeed (let us consider p = 2 for simplicity) Ωε = {(x, y) ∈ R2 : −1 < x < 1 , −ε < y < ε} fL2(Ωε) ≤ Cε∇fH−1(Ωε)n ⇓ xL2 ≤ Cε∇xH−1 = Cε∇yH−1 ≤ CεyL2

Ricardo G. Durán Poincaré and related inequalities

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OPTIMALITY OF THIS ESTIMATE xL2(Ωε) ∼ ε

1 2

, yL2(Ωε) ∼ ε

3 2

⇓

Cε ≥ C ε ∼ D ρ

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES In many cases it is also possible to go in the

  • pposite way:

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES In many cases it is also possible to go in the

  • pposite way: For example, if Ω ⊂ R2 has

diameter D and is star-shaped with respect to a ball of radius ρ, we could prove that fL2(Ω) ≤ CD ρ

  • log D

ρ

  • ∇fH−1(Ω)n

∀f ∈ L2

0(Ω)

Ricardo G. Durán Poincaré and related inequalities

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ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES In many cases it is also possible to go in the

  • pposite way: For example, if Ω ⊂ R2 has

diameter D and is star-shaped with respect to a ball of radius ρ, we could prove that fL2(Ω) ≤ CD ρ

  • log D

ρ

  • ∇fH−1(Ω)n

∀f ∈ L2

0(Ω)

And from this estimate it can be deduced that fL2(Ω) ≤ CD ρ

  • log D

ρ

  • d∇fL2(Ω)n

∀f ∈ L2

0(Ω)

Ricardo G. Durán Poincaré and related inequalities

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MAIN RESULT Proof of fL1(Ω) ≤ C1d∇fL1(Ω)n ∀f ∈ L1

0(Ω)

Ricardo G. Durán Poincaré and related inequalities

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MAIN RESULT Proof of fL1(Ω) ≤ C1d∇fL1(Ω)n ∀f ∈ L1

0(Ω)

⇓ ∀f ∈ Lp

0(Ω), 1 < p < ∞

∃u ∈ W 1,p

0 (Ω)n

such that div u = f , uW 1,p(Ω)n ≤ CfLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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MAIN RESULT Proof of fL1(Ω) ≤ C1d∇fL1(Ω)n ∀f ∈ L1

0(Ω)

⇓ ∀f ∈ Lp

0(Ω), 1 < p < ∞

∃u ∈ W 1,p

0 (Ω)n

such that div u = f , uW 1,p(Ω)n ≤ CfLp(Ω) C = C(n, p, C1)

Ricardo G. Durán Poincaré and related inequalities

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THREE STEPS Improved Poincaré for p = 1 ⇒ Improved Poincaré for 1 < p < ∞

Ricardo G. Durán Poincaré and related inequalities

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THREE STEPS Improved Poincaré for p = 1 ⇒ Improved Poincaré for 1 < p < ∞ Improved Poincaré for p′ allows us to reduce the problem to “local problems” in cubes

Ricardo G. Durán Poincaré and related inequalities

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THREE STEPS Improved Poincaré for p = 1 ⇒ Improved Poincaré for 1 < p < ∞ Improved Poincaré for p′ allows us to reduce the 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′

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.,

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 , Q0

j ∩ Q0 i = ∅

diam Qj ∼ dist (Qj, ∂Ω) =: dj

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 , Q0

j ∩ Q0 i = ∅

diam Qj ∼ dist (Qj, ∂Ω) =: dj For f ∈ Lp

0(Ω), there exists a decomposition

f =

  • j

fj

Ricardo G. Durán Poincaré and related inequalities

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STEP 2 such that

Ricardo G. Durán Poincaré and related inequalities

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STEP 2 such that fj ∈ Lp

0(

Qj) ,

  • Qj := 9

8Qj and

Ricardo G. Durán Poincaré and related inequalities

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STEP 2 such that fj ∈ Lp

0(

Qj) ,

  • Qj := 9

8Qj and fp

Lp(Ω) ∼

  • j

fjp

Lp( Qj)

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Proof of the existence of this decomposition:

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Proof of the existence of this decomposition: Take a partition of unity associated with the Whitney decomposition

  • j

φj = 1 , sop φj ⊂ Qj = 9 8Qj φjL∞ ≤ 1 , ∇φjL∞ ≤ C/dj

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Assuming the improved Poincaré for p′, we

  • btain by duality:

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Assuming the improved Poincaré for p′, we

  • btain by duality:

For f ∈ Lp

0(Ω) there exists u ∈ Lp(Ω)n such that

div u = f in Ω u · n = 0 on ∂Ω

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Assuming the improved Poincaré for p′, we

  • btain by duality:

For f ∈ Lp

0(Ω) there exists u ∈ Lp(Ω)n such that

div u = f in Ω u · n = 0 on ∂Ω

  • u

d

  • Lp(Ω) ≤ CfLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Then, we define

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Then, we define fj = div (φju)

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS Then, we define fj = div (φju) and so

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 sop φj ⊂ Qj ⇒ sop fj ⊂ Qj ,

  • fj = 0

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS From finite superposition: |f(x)|p ≤ C

  • j

|fj(x)|p

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS From finite superposition: |f(x)|p ≤ C

  • j

|fj(x)|p and therefore fp

Lp(Ω) ≤ C

  • j

fjp

Lp( Qj)

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DECOMPOSITION OF FUNCTIONS To prove the other estimate we use φjL∞ ≤ 1 , ∇φjL∞ ≤ C/dj

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS To prove the other estimate we use φjL∞ ≤ 1 , ∇φjL∞ ≤ C/dj Then, fj = div (φju) = div uφj + ∇φj · u = fφj + ∇φj · u

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS fjp

Lp( Qj) ≤ C

  • fp

Lp( Qj) +

  • u

d

  • p

Lp( Qj)

  • Ricardo G. Durán

Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS fjp

Lp( Qj) ≤ C

  • fp

Lp( Qj) +

  • u

d

  • p

Lp( Qj)

  • and therefore, using
  • u

d

  • Lp(Ω) ≤ CfLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS fjp

Lp( Qj) ≤ C

  • fp

Lp( Qj) +

  • u

d

  • p

Lp( Qj)

  • and therefore, using
  • u

d

  • Lp(Ω) ≤ CfLp(Ω)

we obtain,

Ricardo G. Durán Poincaré and related inequalities

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DECOMPOSITION OF FUNCTIONS fjp

Lp( Qj) ≤ C

  • fp

Lp( Qj) +

  • u

d

  • p

Lp( Qj)

  • and therefore, using
  • u

d

  • Lp(Ω) ≤ CfLp(Ω)

we obtain,

  • j

fjp

Lp( Qj) ≤ Cfp Lp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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STEP 3 Then, to solve div u = f

Ricardo G. Durán Poincaré and related inequalities

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STEP 3 Then, to solve div u = f solve div uj = fj

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,p

0 (

Qj) , ujW 1,p(

Qj) ≤ CfjLp( Qj)

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STEP 3 Then, to solve div u = f solve div uj = fj uj ∈ W 1,p

0 (

Qj) , ujW 1,p(

Qj) ≤ CfjLp( Qj)

Observe that C = C(n, p)

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,p

0 (

Qj) , ujW 1,p(

Qj) ≤ CfjLp( Qj)

Observe that C = C(n, p) and

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,p

0 (

Qj) , ujW 1,p(

Qj) ≤ CfjLp( Qj)

Observe that C = C(n, p) and u =

  • j

uj is the required solution!

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STEP 1: Poincaré in L1 implies Poincaré in Lp, p < ∞ To conclude the proof of our theorem we need:

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 − fΩL1(Ω) ≤ Cd∇fL1(Ω) ⇓ f − fΩLp(Ω) ≤ Cd∇fLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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Poincaré in L1 implies Poincaré in Lp, p < ∞ OR MORE GENERALLY f − fΩL1(Ω) ≤ Cw∇fL1(Ω) ⇓ f − fΩLp(Ω) ≤ Cw∇fLp(Ω) ∀p < ∞

Ricardo G. Durán Poincaré and related inequalities

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Poincaré in L1 implies Poincaré in Lp, p < ∞ 1) f − fΩLp(Ω) ≤ Cw∇fLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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Poincaré in L1 implies Poincaré in Lp, p < ∞ 1) f − fΩLp(Ω) ≤ Cw∇fLp(Ω)

  • 2)

∀E ⊂ Ω such that |E| ≥ 1 2|Ω| f|E = 0 ⇒ fLp(Ω) ≤ Cw∇fLp(Ω) with C independent of E and f.

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Poincaré in L1 implies Poincaré in Lp, p < ∞ Case p = 1 :

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Poincaré in L1 implies Poincaré in Lp, p < ∞ Case p = 1 : f = f+ − f− Suppose |{f+ = 0}| ≥ 1

2|Ω|

⇒ f+L1(Ω) ≤ Cw∇f+L1(Ω) ≤ Cw∇fL1(Ω)

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Poincaré in L1 implies Poincaré in Lp, p < ∞ But

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 99

Poincaré in L1 implies Poincaré in Lp, p < ∞ But

  • Ω

f = 0 ⇒

  • Ω

f+ =

  • Ω

f−

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SLIDE 100

Poincaré in L1 implies Poincaré in Lp, p < ∞ But

  • Ω

f = 0 ⇒

  • Ω

f+ =

  • Ω

f− and therefore, fL1(Ω) =

  • Ω

f+ + f− = 2

  • Ω

f+ ≤ 2Cw∇f+L1(Ω)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 101

Poincaré in L1 implies Poincaré in Lp, p < ∞ If

  • Ω

f p

+ =

  • Ω

f p

−

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 102

Poincaré in L1 implies Poincaré in Lp, p < ∞ If

  • Ω

f p

+ =

  • Ω

f p

−

the same argument than for p = 1 applies!

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 103

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 ∈ R such that

  • Ω

(f − λ)p

+ =

  • Ω

(f − λ)p

−

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 104

Poincaré in L1 implies Poincaré in Lp, p < ∞ Now, f|E = 0 ⇒ |f|p|E = 0

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 105

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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SLIDE 106

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

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 107

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, fLp(Ω) ≤ Cw∇fLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 108

Proof of the improved Poincaré in L1 Now, to complete our arguments we have to prove the improved Poincaré in L1.

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 109

Proof of the improved Poincaré in L1 Now, to complete our arguments we have to prove the improved Poincaré in L1. We will show that it can be proved for a very general class of domains by elementary calculus arguments

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 110

REPRESENTATION FORMULA f(y) − f = −

  • Ω

G(x, y) · ∇f(x) dx

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 111

REPRESENTATION FORMULA f(y) − f = −

  • Ω

G(x, y) · ∇f(x) dx Assume Ω star-shaped with respect to a ball B(0, δ)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 112

REPRESENTATION FORMULA f(y) − f = −

  • Ω

G(x, y) · ∇f(x) dx Assume Ω star-shaped with respect to a ball B(0, δ) ω ∈ C∞

0 (Bδ/2)

  • Bδ/2

ω = 1 f =

  • Ω

fω y ∈ Ω , z ∈ Bδ/2 f(y) − f(z) = 1 (y − z) · ∇f(y + t(z − y)) dt

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 113

REPRESENTATION FORMULA Multiplying by ω(z) and integrating:

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 114

REPRESENTATION FORMULA Multiplying by ω(z) and integrating: f(y)−f =

  • Ω

1 (y −z)·∇f(y +t(z −y))ω(z)dtdz

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 115

REPRESENTATION FORMULA Multiplying by ω(z) and integrating: f(y)−f =

  • Ω

1 (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

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SLIDE 116

REPRESENTATION FORMULA f(y)−f =

  • Ω

1 (y − x) tn+1 ·∇f(x) ω

  • y + x − y

t

  • dtdx

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 117

REPRESENTATION FORMULA f(y)−f =

  • Ω

1 (y − x) tn+1 ·∇f(x) ω

  • y + x − y

t

  • dtdx

that is, f(y) − f = −

  • Ω

G(x, y) · ∇f(x)dx

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 118

REPRESENTATION FORMULA G(x, y) = 1 (x − y) t ω

  • y + x − y

t 1 tndt

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 119

REPRESENTATION FORMULA G(x, y) = 1 (x − y) t ω

  • y + x − y

t 1 tndt PROPERTIES OF G(x, y)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 120

REPRESENTATION FORMULA G(x, y) = 1 (x − y) t ω

  • y + x − y

t 1 tndt PROPERTIES OF G(x, y) |G(x, y)| ≤ C1 |x − y|n−1

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 121

REPRESENTATION FORMULA G(x, y) = 1 (x − y) t ω

  • y + x − y

t 1 tndt 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

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SLIDE 122

MORE GENERAL DOMAINS The representation formula can be generalized to John domains (Acosta-D.-Muschietti)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 123

MORE GENERAL DOMAINS The representation formula can be generalized to John domains (Acosta-D.-Muschietti) G(x, y) = 1 x − γ t +˙ γ(t, y)

  • ω

x − γ(t, y) t dt tn

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 124

MORE GENERAL DOMAINS The representation formula can be generalized to John domains (Acosta-D.-Muschietti) G(x, y) = 1 x − γ t +˙ γ(t, y)

  • ω

x − γ(t, y) t dt tn 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

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SLIDE 125

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

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SLIDE 126

PROOF OF IMPROVED POINCARÉ IN L1 (Drelichman-D.)

  • Ω

|f(y) − f|dy ≤

  • Ω
  • Ω

|G(x, y)||∇f(x)|dxdy

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 127

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

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 128

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 ≤ Cd∇fL1(Ω)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 129

GENERALIZATION If Ω satisfies the weaker improved Poincaré inequality

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 130

GENERALIZATION If Ω satisfies the weaker improved Poincaré inequality fL1(Ω) ≤ C1dβ∇fL1(Ω)n ∀f ∈ L1

0(Ω)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 131

GENERALIZATION If Ω satisfies the weaker improved Poincaré inequality fL1(Ω) ≤ C1dβ∇fL1(Ω)n ∀f ∈ L1

0(Ω)

for some β < 1

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 132

GENERALIZATION If Ω satisfies the weaker improved Poincaré inequality fL1(Ω) ≤ C1dβ∇fL1(Ω)n ∀f ∈ L1

0(Ω)

for some β < 1 then, ∀f ∈ Lp

0(Ω)

∃u ∈ W 1,p

0 (Ω)n

such that div u = f , d1−β∇uLp(Ω) ≤ CfLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 133

APPLICATION TO Hölder α DOMAINS In this case β = α

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 134

APPLICATION TO Hölder α DOMAINS In this case β = α that is, ∀f ∈ Lp

0(Ω)

∃u ∈ W 1,p

0 (Ω)n

such that

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 135

APPLICATION TO Hölder α DOMAINS In this case β = α that is, ∀f ∈ Lp

0(Ω)

∃u ∈ W 1,p

0 (Ω)n

such that div u = f , d1−α∇uLp(Ω) ≤ CfLp(Ω)

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 136

APPLICATION TO Hölder α DOMAINS In this case β = α that is, ∀f ∈ Lp

0(Ω)

∃u ∈ W 1,p

0 (Ω)n

such that div u = f , d1−α∇uLp(Ω) ≤ CfLp(Ω) The case p = 2 can be used to prove existence and uniqueness for the Stokes equations.

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 137

FINAL COMMENTS We have a general methodology to prove inequalities once they are known in cubes.

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 138

FINAL COMMENTS We have a general methodology to prove inequalities once they are known in cubes. The decomposition of functions of zero integral is interesting in itself because it might have

  • ther applications (we are working in the

application of this decomposition to weighted a priori estimates).

Ricardo G. Durán Poincaré and related inequalities

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SLIDE 139

THANK YOU FOR YOUR ATTENTION!

HAPPY BIRTHDAY MARTIN!!

Ricardo G. Durán Poincaré and related inequalities