Improved Poincar and other classic inequalities: a new approach to - PowerPoint PPT Presentation

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

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 � f � L 1 (Ω) ≤ CD ∀ f ∈ L 1 ρ � d ∇ f � L 1 (Ω) n 0 (Ω) Ricardo G. Durán Poincaré and related inequalities

ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES Consequently, we obtain from our results that, for Ω convex, Ricardo G. Durán Poincaré and related inequalities

ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES Consequently, we obtain from our results that, for Ω convex, � f � L p (Ω) ≤ CD ∀ f ∈ L p ρ �∇ f � W − 1 , p (Ω) n 0 (Ω) Ricardo G. Durán Poincaré and related inequalities

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

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) Ω ε = { ( x , y ) ∈ R 2 : − 1 < x < 1 , − ε < y < ε } Ricardo G. Durán Poincaré and related inequalities

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) Ω ε = { ( x , y ) ∈ R 2 : − 1 < x < 1 , − ε < y < ε } � f � L 2 (Ω ε ) ≤ C ε �∇ f � H − 1 (Ω ε ) n ⇓ � x � L 2 ≤ C ε �∇ x � H − 1 = C ε �∇ y � H − 1 ≤ C ε � y � L 2 Ricardo G. Durán Poincaré and related inequalities

OPTIMALITY OF THIS ESTIMATE 1 3 � x � L 2 (Ω ε ) ∼ ε , � y � L 2 (Ω ε ) ∼ ε 2 2 ⇓ C ε ≥ C ε ∼ D ρ Ricardo G. Durán Poincaré and related inequalities

ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES In many cases it is also possible to go in the opposite way: Ricardo G. Durán Poincaré and related inequalities

ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES In many cases it is also possible to go in the opposite way: For example, if Ω ⊂ R 2 has diameter D and is star-shaped with respect to a ball of radius ρ , we could prove that � � � f � L 2 (Ω) ≤ CD log D ∀ f ∈ L 2 �∇ f � H − 1 (Ω) n 0 (Ω) ρ ρ Ricardo G. Durán Poincaré and related inequalities

ESTIMATES IN TERMS OF GEOMETRIC PROPERTIES In many cases it is also possible to go in the opposite way: For example, if Ω ⊂ R 2 has diameter D and is star-shaped with respect to a ball of radius ρ , we could prove that � � � f � L 2 (Ω) ≤ CD log D ∀ f ∈ L 2 �∇ f � H − 1 (Ω) n 0 (Ω) ρ ρ And from this estimate it can be deduced that � � � f � L 2 (Ω) ≤ CD log D ∀ f ∈ L 2 � d ∇ f � L 2 (Ω) n 0 (Ω) ρ ρ Ricardo G. Durán Poincaré and related inequalities

MAIN RESULT Proof of ∀ f ∈ L 1 � f � L 1 (Ω) ≤ C 1 � d ∇ f � L 1 (Ω) n 0 (Ω) Ricardo G. Durán Poincaré and related inequalities

MAIN RESULT Proof of ∀ f ∈ L 1 � f � L 1 (Ω) ≤ C 1 � d ∇ f � L 1 (Ω) n 0 (Ω) ⇓ ∀ f ∈ L p 0 (Ω) , 1 < p < ∞ ∃ u ∈ W 1 , p 0 (Ω) n such that div u = f � u � W 1 , p (Ω) n ≤ C � f � L p (Ω) , Ricardo G. Durán Poincaré and related inequalities

MAIN RESULT Proof of ∀ f ∈ L 1 � f � L 1 (Ω) ≤ C 1 � d ∇ f � L 1 (Ω) n 0 (Ω) ⇓ ∀ f ∈ L p 0 (Ω) , 1 < p < ∞ ∃ u ∈ W 1 , p 0 (Ω) n such that div u = f � u � W 1 , p (Ω) n ≤ C � f � L p (Ω) , C = C ( n , p , C 1 ) Ricardo G. Durán Poincaré and related inequalities

THREE STEPS Improved Poincaré for p = 1 ⇒ Improved Poincaré for 1 < p < ∞ Ricardo G. Durán Poincaré and related inequalities

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

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

STEP 2 Assume improved Poincaré for p ′ Ricardo G. Durán Poincaré and related inequalities

STEP 2 Assume improved Poincaré for p ′ Take a Whitney decomposition of Ω , i. e., Ricardo G. Durán Poincaré and related inequalities

STEP 2 Assume improved Poincaré for p ′ Take a Whitney decomposition of Ω , i. e., Q 0 j ∩ Q 0 Ω = ∪ j Q j , i = ∅ diam Q j ∼ dist ( Q j , ∂ Ω) =: d j Ricardo G. Durán Poincaré and related inequalities

STEP 2 Assume improved Poincaré for p ′ Take a Whitney decomposition of Ω , i. e., Q 0 j ∩ Q 0 Ω = ∪ j Q j , i = ∅ diam Q j ∼ dist ( Q j , ∂ Ω) =: d j For f ∈ L p 0 (Ω) , there exists a decomposition � f = f j j Ricardo G. Durán Poincaré and related inequalities

STEP 2 such that Ricardo G. Durán Poincaré and related inequalities

STEP 2 such that Q j := 9 f j ∈ L p 0 ( � � Q j ) , 8 Q j and Ricardo G. Durán Poincaré and related inequalities

STEP 2 such that Q j := 9 f j ∈ L p 0 ( � � Q j ) , 8 Q j and � � f � p � f j � p L p (Ω) ∼ L p ( � Q j ) j Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Proof of the existence of this decomposition: Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Proof of the existence of this decomposition: Take a partition of unity associated with the Whitney decomposition � Q j = 9 sop φ j ⊂ � φ j = 1 , 8 Q j j � φ j � L ∞ ≤ 1 �∇ φ j � L ∞ ≤ C / d j , Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Assuming the improved Poincaré for p ′ , we obtain by duality: Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Assuming the improved Poincaré for p ′ , we obtain by duality: 0 (Ω) there exists u ∈ L p (Ω) n such that For f ∈ L p div u = f in Ω u · n = 0 on ∂ Ω Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Assuming the improved Poincaré for p ′ , we obtain by duality: 0 (Ω) there exists u ∈ L p (Ω) n such that For f ∈ L p div u = f in Ω u · n = 0 on ∂ Ω � � � � u � L p (Ω) ≤ C � f � L p (Ω) � d Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Then, we define Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Then, we define f j = div ( φ j u ) Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Then, we define f j = div ( φ j u ) and so Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS Then, we define f j = div ( φ j u ) and so � � � f = div u = div u div ( φ j u ) = φ j = f j j j j � sop φ j ⊂ � Q j ⇒ sop f j ⊂ � f j = 0 , Q j Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS From finite superposition: � | f ( x ) | p ≤ C | f j ( x ) | p j Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS From finite superposition: � | f ( x ) | p ≤ C | f j ( x ) | p j and therefore � � f � p � f j � p L p (Ω) ≤ C L p ( � Q j ) j Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS To prove the other estimate we use � φ j � L ∞ ≤ 1 �∇ φ j � L ∞ ≤ C / d j , Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS To prove the other estimate we use � φ j � L ∞ ≤ 1 �∇ φ j � L ∞ ≤ C / d j , Then, f j = div ( φ j u ) = div u φ j + ∇ φ j · u = f φ j + ∇ φ j · u Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS � � � � p � � u � � f j � p � f � p Q j ) ≤ C Q j ) + � L p ( � L p ( � d L p ( � Q j ) Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS � � � � p � � u � � f j � p � f � p Q j ) ≤ C Q j ) + � L p ( � L p ( � d L p ( � Q j ) and therefore, using � � � � � u L p (Ω) ≤ C � f � L p (Ω) � d Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS � � � � p � � u � � f j � p � f � p Q j ) ≤ C Q j ) + � L p ( � L p ( � d L p ( � Q j ) and therefore, using � � � � � u L p (Ω) ≤ C � f � L p (Ω) � d we obtain, Ricardo G. Durán Poincaré and related inequalities

DECOMPOSITION OF FUNCTIONS � � � � p � � u � � f j � p � f � p Q j ) ≤ C Q j ) + � L p ( � L p ( � d L p ( � Q j ) and therefore, using � � � � � u L p (Ω) ≤ C � f � L p (Ω) � d we obtain, � � f j � p Q j ) ≤ C � f � p L p ( � L p (Ω) j Ricardo G. Durán Poincaré and related inequalities

STEP 3 Then, to solve div u = f Ricardo G. Durán Poincaré and related inequalities

STEP 3 Then, to solve div u = f solve div u j = f j Ricardo G. Durán Poincaré and related inequalities

STEP 3 Then, to solve div u = f solve div u j = f j u j ∈ W 1 , p 0 ( � Q j ) , � u j � W 1 , p ( � Q j ) ≤ C � f j � L p ( � Q j ) Ricardo G. Durán Poincaré and related inequalities

STEP 3 Then, to solve div u = f solve div u j = f j u j ∈ W 1 , p 0 ( � Q j ) , � u j � W 1 , p ( � Q j ) ≤ C � f j � L p ( � Q j ) Observe that C = C ( n , p ) Ricardo G. Durán Poincaré and related inequalities

STEP 3 Then, to solve div u = f solve div u j = f j u j ∈ W 1 , p 0 ( � Q j ) , � u j � W 1 , p ( � Q j ) ≤ C � f j � L p ( � Q j ) Observe that C = C ( n , p ) and Ricardo G. Durán Poincaré and related inequalities

STEP 3 Then, to solve div u = f solve div u j = f j u j ∈ W 1 , p 0 ( � Q j ) , � u j � W 1 , p ( � Q j ) ≤ C � f j � L p ( � Q j ) Observe that C = C ( n , p ) and � u = u j j is the required solution! Ricardo G. Durán Poincaré and related inequalities

STEP 1: Poincaré in L 1 implies Poincaré in L p , p < ∞ To conclude the proof of our theorem we need: Ricardo G. Durán Poincaré and related inequalities

STEP 1: Poincaré in L 1 implies Poincaré in L p , p < ∞ To conclude the proof of our theorem we need: � f − f Ω � L 1 (Ω) ≤ C � d ∇ f � L 1 (Ω) ⇓ � f − f Ω � L p (Ω) ≤ C � d ∇ f � L p (Ω) Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ OR MORE GENERALLY � f − f Ω � L 1 (Ω) ≤ C � w ∇ f � L 1 (Ω) ⇓ � f − f Ω � L p (Ω) ≤ C � w ∇ f � L p (Ω) ∀ p < ∞ Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ 1) � f − f Ω � L p (Ω) ≤ C � w ∇ f � L p (Ω) Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ 1) � f − f Ω � L p (Ω) ≤ C � w ∇ f � L p (Ω) � 2) | E | ≥ 1 such that ∀ E ⊂ Ω 2 | Ω | f | E = 0 ⇒ � f � L p (Ω) ≤ C � w ∇ f � L p (Ω) with C independent of E and f . Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ Case p = 1 : Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ Case p = 1 : f = f + − f − Suppose |{ f + = 0 }| ≥ 1 2 | Ω | ⇒ � f + � L 1 (Ω) ≤ C � w ∇ f + � L 1 (Ω) ≤ C � w ∇ f � L 1 (Ω) Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ But Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ But � � � f = 0 ⇒ f + = f − Ω Ω Ω Ricardo G. Durán Poincaré and related inequalities

Poincaré in L 1 implies Poincaré in L p , p < ∞ But � � � f = 0 ⇒ f + = f − Ω Ω Ω and therefore, � � f + + f − = 2 f + ≤ 2 C � w ∇ f + � L 1 (Ω) � f � L 1 (Ω) = Ω Ω Ricardo G. Durán Poincaré and related inequalities