The best constants problem in Sobolev inequalities
Let \((M,g)\) be a smooth compact Riemannian manifold of dimension \(n\). Let \(1< p<\infty\) and let \(p^*= np/(n- p)\). The Sobolev embedding theorem grants the existence of constants \(A,B> 0\) such that for any \(u\) in the \(p\)-th Sobolev space the inequality \(\|u\|_{p^*}\leq A\|\nabla u\|_p+ B\|u\|_p\) holds, where \(\|\cdot\|_p\) is the \(L^p\)-norm. Let \(K(n,p)\) be the best Sobolev constant for \(\mathbb{R}^n\), i.e.: \(K(n,p)^{-1}= \inf_{u\in C^\infty_0(\mathbb{R}^n)\setminus \{0\}}\|\nabla u\|_p/\|u\|_{p^*}\). In the present paper, the author proves that there is \(B>0\) such that \(\|u\|^\theta_{p^*}\leq K(n,p)^\theta\|\nabla u\|^\theta_p+ B\|u\|^\theta_p\) holds for \(\theta= p\) if \(p\leq 2\) and \(\theta= 2\) if \(p\geq 2\). This proves a conjecture of \textit{T. Aubin} [J. Differ. Geom. 11, 573-598 (1976; Zbl 0371.46011)]. The proof proceeds by first reducing the problem to the study of a special type PDE followed by a delicate study of concentration phenomena.
scientific article; zbMATH DE number 1337018