Extremal functions and the Chang-Marshall inequality

Extremal Functions for Moser’s Inequality

Let Ω be a bounded smooth domain in Rn, and u(x) a C1 function with compact support in Ω. Moser’s inequality states that there is a constant co, depending only on the dimension n, such that 1 |Ω| ∫ Ω e nω 1 n−1 n−1 u n n−1 dx ≤ co, where |Ω| is the Lebesgue measure of Ω, and ωn−1 the surface area of the unit ball in Rn. We prove in this paper that there are extremal functions for this inequalit...

Extremal functions for the sharp L2− Nash inequality

This paper is in the spirit of several works on best constants problems in Sobolev type inequalities. A general reference on this subject is the recent book of Hebey [9]. These questions have many interests. At first, they are at the origin of the resolution of famous geometrical problems as Yamabe problem. More generally, they show how geometry and analysis interact on Riemannian manifolds and...

On the Extremal Functions of Sobolev-poincaré Inequality

where ua = 1 vol(Mn) ∫ Mn u, p∗ = np/(n − p) is the Sobolev conjugate of p. This inequality can be proved by combining Sobolev inequality with Poincaré inequality, see, for example, Hebey’s book [8]. In this paper we are interested in the estimates of the best constant and the existence of extremal functions to the above inequality. Analytically these are naturally motivated questions. On the o...

Extremal functions for the sharp L− Nash inequality

Extremal Cases of the Ahlswede-Cai Inequality

We are concerned with the poset P(n)=P([1, 2, ..., n]). This is the power set of [n]=[1, 2, ..., n], ordered by inclusion. A set system is simply a subset of P(n). A set system is an antichain if no two of its members are comparable. Conversely a chain is a totally ordered set system. We shall often consider maximal chains; those chains which cannot be extended. In particular such chains contai...