New Poincaré Inequalities from Old

We discuss a geometric method, which we refer to as Coning, for generating new Poincaré type inequalities from old ones. In particular, we derive weighted Poincaré inequalities for star-shaped domains and variant Trudinger inequalities for another class of domains. By a Poincaré type inequality, in the widest sense, we mean a norm inequality in which the variation of a function from its “average” value on a domain is in some way controlled by its gradient (or higher gradients) on that domain. Thus the classical Poincaré and Sobolev–Poincaré inequalities, as well as the Trudinger inequality fall into this category. Such inequalities have been the focus of much study, in particular since the work of Sobolev [So1], [So2]; for accounts of many such results, we refer the reader to the excellent books by Maz’ya [M] and Ziemer [Z]. In this paper, we discuss a geometric method, which we refer to as Coning, for generating new Poincaré type inequalities from old ones. This method takes as input a Poincaré type inequality known to be true for a class of domains with a certain invariance property and generates from it a sequence of related inequalities for the same class of domains. We apply this method to a Poincaré inequality for star-shaped domains and to the Trudinger inequality for the class of QHBC domains (defined later). Rather than being a method for proving all possible Poincaré type inequalities, Coning is a rather unusual and specialized method that produces some new inequalities of Poincaré type that are clearly special cases of more general inequalities which one should try to prove by more conventional methods (in fact, already Buckley and O’Shea have proved new weighted Trudinger inequalities that are motivated by, and generalize, the new Trudinger type inequalities below). Coning can also be used to geometrically link some inequalities already known to be true. To illustrate the method, we begin by stating an unweighted Poincaré inequality for star-shaped domains which was proved by Levi [L] in the planar case, and by 1991 Mathematics Subject Classification: Primary 46E35. The first author was partially supported by Forbairt. The second author was partially supported by NSF Grant DMS-9305742 and the Academy of Finland. 252 Stephen M. Buckley and Pekka Koskela Hurri [H1] and Smith and Stegenga [SS1] in higher dimensions; see also [M, Chapter 2]. We then state a weighted version that follows from it by Coning. Throughout this paper, Ω is a proper subdomain of R and δ(x) = dist(x, ∂Ω). Also, we denote by W the class of monotonic increasing functions w: (0,∞) → (0,∞) which satisfy the weak concavity property w(sr) ≥ sw(r) for all r > 0, 0 < s < 1. We denote by uS,v the mean value of a function u on a set S with respect to the measure v(x) dx ; if v is omitted, it is assumed that v ≡ 1. Theorem A. Suppose that Ω is bounded and star-shaped, and that 1 ≤ p < ∞ . Then there exists a constant C = C(Ω, p) (1) ∫ Ω |u− uΩ| p dx ≤ C ∫ Ω |∇u| dx for all u ∈ C(Ω). Theorem 1. Suppose that Ω is bounded and star-shaped, w ∈ W , 1 ≤ p < ∞ , and k is a positive integer. Then there exists a constant C = C(Ω, p, k) such that (2) ∫