Products of Poincaré domains

Abstract

Abstract.A domain O. C RN of finite /V-dimensional Lebesgue measure is a p-Poincare domain (1 < p < oo) if there exists a positive constant K such that the p-Poincare inequality ||u||/>>(£2)

A remark on optimal weighted Poincaré inequalities for convex domains

“…Explicit expression for π p (as (1.6)) can be found for instance in [3,20,24,28,29]. The fact that π 2 = π is consistent with the classical Wirtinger inequality (see [16]) and obviously also with (1.2).…”

Best constants in Poincaré inequalities for convex domains

“…The formula (1.5) for π p can be found for instance in [BB,L,RW,S1,S2]. The fact that π 2 = π is consistent with the classical Wirtinger inequality (see [HL]) and obviously also with (1.4).…”

V.A. Steklov and the Problem of Sharp (Exact) Constants in Inequalities of Mathematical Physics

“…There are also inequalities between the Dirichlet and Neumann eigenvalues (see the recent paper [17] for a brief historical survey). Furthermore, it is shown in [55] that if p = q is arbitrary and ( 5) holds for Ω 1 ⊂ IR m and Ω 2 ⊂ IR n with the sharp constants C 1 and C 2 , respectively, then the sharp constant in the same inequality for Ω 1 × Ω 2 is less than or equal to…”