A sharp weighted Wirtinger inequality Tonia Ricciardi∗ Dipartimento di Matematica e Applicazioni 5 Università di Napoli Federico II 0 Via Cintia, 80126 Naples, Italy 0 2 fax: +39 081 675665 tonia.ricciardi@unina.it n a J 4 Abstract ] P We obtain a sharp estimate for the best constant C > 0 in the Wir- A tinger typeinequality . h 2π 2π at γpw2 ≤C γqw′2 m Z0 Z0 [ where γ is bounded above and below away from zero, w is 2π-periodic and such that 2πγpw = 0, and p+q ≥ 0. Our result generalizes an 1 0 inequality of Piccinini and Spagnolo. v R 4 Let C(a,b) > 0 denote the best constant in the following weighted Wirtinger 4 type inequality: 0 1 0 2π 2π 5 (1) aw2 C(a,b) bw′2, ≤ 0 Z0 Z0 h/ where w H1 (R) is 2π-periodic and satisfies the constraint ∈ loc t a 2π m (2) aw=0, v: Z0 Xi and a,b with ∈B r a = a L∞(R) : a is 2π periodic and infa>0 . B { ∈ − } Hereandinwhatfollows,foreverymeasurablefunctionawedenotebyinfaand supatheessentiallowerboundandtheessentialupperboundofa,respectively. For every L>1, we denote (L)= a L (0,2π) : a is 2π periodic, infa=1 and supa=L . ∞ B { ∈ − } Our aim in this note is to prove: ∗Supported in part by Regione Campania L.R. 5/02 and by the MIUR National Project Variational Methods and Nonlinear DifferentialEquations. 1 Theorem 1. Suppose a=γp and b=γq for some γ (M), M >1, and for some p,q R such that p+q 0. Then ∈B ∈ ≥ 1 2πγ(p q)/2 2 (3) C(γp,γq) 2π 0 − . ≤ π4 arctaRn M−(p+q)/4 ! (cid:0) (cid:1) If p+q>0, then equality holds in (3) if and only if γ(θ)=γ¯ (θ+ϕ) for some p,q ϕ R, where ∈ 1, if 0 θ 1, then ∈B (8) 2 1 C2π(a√,ba)b 1 2 =√a′bs′up(L) 1 C2π(a√′,ab′b) 1 2 =(cid:18)π4 arctanL−1/2(cid:19)− 2π 0 − ∈B 2π 0 ′ ′− (cid:16) (cid:17) (cid:16) (cid:17) R R if and only if the following equation is satisfied: (9) a(θ(τ))b(θ(τ)) =a¯2(τ +ϕ) a.e. τ (0,2π), for some ϕ R, ∈ ∈ where θ(τ) is the homeomorphism of R defined by 1 θ a(θ˜) (10) τ(θ)= dθ˜, c Z0 sb(θ˜) c is defined by 1 2π a(θ˜) (11) c= dθ˜, 2π Z0 sb(θ˜) and a¯ is the function defined in Lemma 1. 2 If b = a 1, then C(a,a 1) = (2π) 1 2πa and equality is attained in − − − 0 (1)–(2) with b=a−1 if and only if w(cid:16)(θ)=CRcos(2(cid:17)π( 02πa)−1 0θa+ϕ) for some C =0 and ϕ R. 6 ∈ R R 3 Proof. Under the change of variables θ = θ(τ) defined by (10)–(11), setting α(τ)=a(θ(τ)), β(τ)=b(θ(τ)), ξ(τ)=w(θ(τ)), we obtain αθ′ =c αβ, βθ′−1 =c−1 αβ, and therefore: p p 2π 2π aw2dθ = αθ ξ2dτ =c αβξ2dτ ′ Z0 Z0 Z 2π 2π p awdθ = αθ ξdτ =c αβξdτ =0 ′ Z0 Z0 Z 2π 2π p bw′2dθ = βθ′−1ξ′2dτ =c−1 αβξ′2dτ. Z0 Z0 Z p Upon substitution, (1)–(2) takes the form: 2π C(a,b) 2π (12) Z0 αβξ2dτ ≤ 1 2π√ab 1 2 Z0 αβξ′2dτ, p 2π 0 − p with constraint (cid:16) R (cid:17) 2π (13) αβξdτ =0. Z0 p If √ab (L), in view of Lemma 1 we obtain ∈B 2 − C(a,b) 4 inf√αβ (14) =C( αβ, αβ) arctan 1 2π√ab 1 2 ≤π ssup√αβ 2π 0 − p p   (cid:16) R (cid:17) 4 infab 1/4 −2 = arctan . π supab (cid:18) (cid:19) ! This yields (7). Moreover, we have C(√αβ,√αβ) = (4/π)arctanL−1/2 −2 if and only if α(τ)β(τ) = a¯(τ +ϕ), for some ϕ R. That is, (8) holds if and ∈ (cid:0) (cid:1) only if (9) holds. p If b=a 1, then (12)–(13) takes the form − 2π C(a,a 1) 2π ξ2dτ − ξ2dτ Z0 ≤ 1 2πa 2 Z0 ′ 2π 0 (cid:16) (cid:17) with constraint R 2π ξdτ =0. Z0 Therefore, by the classical Wirtinger inequality, 1 2π 2 C(a,a−1)= a 2π (cid:18) Z0 (cid:19) andequality holdsin(1)–(2)withb=a 1 ifandonlyif ξ(τ)=Ccos(τ+ϕ)for − someC =0andϕ R,thatis,ifandonlyifw(θ)=Ccos(2π( 2πa) 1 θa+ϕ), 6 ∈ 0 − 0 as asserted. R R 4 Lemma 3. Suppose a,b satisfy √ab (L), L > 1, and (9), where θ(τ) is ∈ B defined in (10) and c is defined by (11). Suppose (15) a=γp, b=γq for some γ (M), with M = L2/(p+q), and for some p,q R such that p+q >0. T∈henBγ(θ)=γ¯ (θ+ϕ) for some ϕ R, where γ¯ ∈is the function p,q p,q ∈ defined in Theorem 1. Proof. When p+q > 0, we have γ(p+q)/2 (L). In view of (9) and (15) we ∈ B have γ(θ(τ))=a¯2/(p+q)(τ +ψ), τ R ∀ ∈ for some ψ R. It follows that ∈ τ b(θ(τ¯)) τ (16) θ(τ)=c dτ¯=c a¯ (p q)/(p+q)(τ¯+ψ)dτ¯ − − Z0 sa(θ(τ¯)) Z0 and, in view of the 2π-periodicity of a and b, 1 1 2π b(θ(τ¯)) − 1 2π −1 c= dτ¯ = a¯ (p q)/(p+q)(τ¯)dτ¯ . − − 2π Z0 sa(θ(τ¯)) ! (cid:18)2π Z0 (cid:19) Setting τ h (τ)=c a¯ (p q)/(p+q)(τ¯)dτ¯, p,q − − Z0 we have θ(τ ψ)=h (τ) h (ψ) for every τ R, and consequently τ(θ)= p,q p,q − − ∈ h 1(θ+h (ψ)) ψ. Inviewofthedefinitionofa¯ withL=M(p+q)/2,wehave: −p,q p,q − τ a¯ (p q)/(p+q)(τ¯)dτ¯= − − Z0 τ, if 0 τ < π ≤ 2 π +M (p q)/2(τ π), if π τ <π =2 − − − 2 2 ≤ . π2(1+M−(p−q)/2)+τ −π, if π ≤τ < 32π π(2+M (p q)/2)+M (p q)/2(τ 3π), if 3π τ <2π 2 − − − − − 2 2 ≤ In particular, we derive 2 c= =c . 1+M (p q)/2 p,q − − It follows that h (τ) is the piecewise linear homeomorphism of R defined in p,q [0,2π) by c τ, if 0 τ < π p,q ≤ 2 c π +M (p q)/2(τ π) , if π τ <π h (τ)= p,q 2 − − − 2 2 ≤ p,q cp,q(cid:2)π2(1+M−(p−q)/2)+τ −(cid:3) π , if π ≤τ < 32π c π(2+M (p q)/2)+M (p q)/2(τ 3π) , if 3π τ <2π p,q(cid:2)2 − − − (cid:3)− − 2 2 ≤  (cid:2) (cid:3) 5 andby h (τ+2πn)=2πn+h (τ), for any τ [0,2π)and for any integern. p,q p,q ∈ Inversion yields c 1θ, if 0 θ 0. In view of Lemma 2 and Lemma 3 we have 1 C2(πγγp(,pγqq))/2 2 =(cid:18)π4 arctanM−(p+q)/4(cid:19)−2 2π 0 − (cid:16) (cid:17) R if and only if γ(θ)=γ¯ (θ+ϕ) for some ϕ R. Equality is attainedin (1)–(2) p,q ∈ witha(θ)=γ¯p (θ+ϕ)andb(θ)=γ¯q (θ+ϕ) ifandonlyif w(θ)=w¯ (θ+ϕ). p,q p,q p,q If p + q = 0, then the conclusion follows by Lemma 2 with a = γp and b=γ p. − Acknowledgements I am grateful to Professor Carlo Sbordone for many useful and stimulating discussions. References [1] P.R.Beesack,IntegralinequalitiesoftheWirtingertype,DukeMath.Jour. 25 (1958), 477–498. [2] G.CroceandB.Dacorogna,OnageneralizedWirtingerinequality,Discrete Cont. Dynam. Systems 9 No. 5 (2003), 1329–1341. [3] B. Dacorogna, W. Gangbo and N. Subía, Sur une généralisation de l’inégalité de Wirtinger, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 29–50. [4] L.C. Piccinini and S. Spagnolo, On the Ho¨lder continuity of solutions of second order elliptic equations in two variables, Ann. Scuola Norm. Sup. Pisa 26 No. 2 (1972), 391–402. 6

A sharp weighted Wirtinger inequality

Similar A sharp weighted Wirtinger inequality