From best constants to critical functions

Abstract

The study of sharp Sobolev inequalities starts with the notion of best constant and leads naturally to the question to know whether or not there exist extremal functions for these inequalities. We restrict ourselves in this paper to the H 2 1 -Sobolev inequality. Then, we extend the notion of best constant to that of critical function, and, with the help of this notion, we answer the question to know whether or not there exist extremal functions for the sharp H 2 1 -Sobolev inequality. Partial answers to the more general question to know whether or not an extremal function always comes with a critical function are also given.Let (M, g) be a smooth compact Riemannian n-manifold with n ≥ 3. By the Sobolev embedding theorem, there exist positive constants A, B such that for any u ∈ H 2 1 (M ),where 2 = 2n/(n − 2) is the critical Sobolev exponent, and H 2 1 (M ) is the usual Sobolev space consisting of functions u ∈ L 2 (M ) such that |∇u| is also in L 2 (M ). The best constant A in this inequality is known to be the best constant K for the Sobolev inequality on the Euclidean spaceIn other words, A opt = K(n, 2) where A opt is the best constant A in (I g ) and K(n, 2), the best constant in the Euclidean Sobolev inequality, is given

Elliptic equations with critical growth and a large set of boundary singularities

“…With (60) and (62), we then get that (58) holds on , , = ( ( (¯)) ∖ ( (¯))) ∩ Ω for large and small. It follows from this last assertion, (19) in Proposition 3.3 and (42) that (58) holds on Ω. □…”

The Problem of Prescribed Critical Functions

“…In addition, the reader may refer to [7] sections 2 and 3 for a sketch of proof of this theorem as stated here.…”

“…Critical functions corresponds to "best functions" in inequality above. More precisely, Definition 1.1 (Hebey, Vaugon [7]) We say that a smooth function f is critical for a metric g if S ′ (K(n, 2) 2 f, g) is true for all u ∈ H 2 1 (M) and if for all smooth function f ′ ≤ f with f ′ = f , inequality S ′ (K(n, 2) 2 f ′ , g) is not true.…”

Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium

“…. , p, where S g is the scalar curvature of g. By combining results in [Brendle 2008a;Brendle and Marques 2009], where noncompactness of the Yamabe equation in the nonconformally flat case is investigated, and those in [Druet and Hebey 2005a;Hebey and Vaugon 2001], where unstability of Yamabe type equations in the conformally flat case is investigated, we obtain the following theorem, in view of the remark above.…”