Unified Poincaré and Hardy inequalities with sharp constants for convex domains

Abstract

Abstract Let Ω be an n ‐dimensional convex domain, and let ν ∈ [0, 1/2]. For all f ∈ H 0 1 (Ω) we prove the inequality where δ = dist ( x , ∂ Ω), δ 0 = sup δ. The factor λ ν 2 is sharp for all dimensions, λ ν being the first positive root of the Lamb type equation J ν (λ ν )+ 2λ ν J ν ′ (λ ν ) = 0 for Bessel's functions. In particular, the case ν= 0 with λ 0 = 0,940 ... presents a new sharp form of the Hardy type inequality due to Brezis and Marcus, while in the case ν = 1/2 with λ 1/2 = π/2 we obtain a unified proof of an isoperimetric inequality due to Poincaré for n = 1, Hersch for n = 2 and Payne and Stakgold for n ≥ 3. A generalization, when the latter integral is replaced by the integral ∫ Ω | f | 2 / δ 2‐ m d x , m > 0, is proved, too. As a special case, we obtain the sharp inequality where j ν is the first positive zero of J ν.