On a generalized Wirtinger inequality

Let α (p, q, r) = inf (ku0kp kukq : u ∈W 1,p per (−1, 1) \ {0} , Z 1 −1 |u|r−2 u = 0 ) . We show that α (p, q, r) = α (p, q, q) if q ≤ rp+ r − 1 α (p, q, r) < α (p, q, q) if q > (2r − 1) p generalizing results of Dacorogna-Gangbo-Subía and others. 1 The main result In the present article we discuss the following minimization problem α (p, q, r) = inf ( kukp kukq : u ∈W 1,p per (−1, 1) \ {0} , Z 1 −1 |u|r−2 u = 0 ) where p > 1, q ≥ r − 1 ≥ 1 and kukq = μZ 1 −1 |u| ¶1/q W 1,p per (−1, 1) = © u : u ∈W 1,p (−1, 1) and u (−1) = u (1)a . We will denote by p0 the conjugate exponent of p (i.e. 1 p + 1 p0 = 1) and the Beta function B (p, q) = Γ (p)Γ (q) Γ (p+ q) = Z 1 0 tp−1 (1− t)q−1 dt . Our main result will be