Asymptotically sharp multiplicative inequalities. (English) Zbl 0840.46018

Let \(\Omega\) be a domain in \(\mathbb{R}^n\). The limiting case of the Sobolev embedding theorem states that the Sobolev space \(W^{1, n}_0(\Omega)\) is continuously embedded in the Lebesgue space \(L_q(\Omega)\) for any \(q\geq n\) and the inequality \(|u|_q\leq c(n, q, \Omega)|u|_{1, n}\) holds for functions \(u\in W^{1, n}_0(\Omega)\). If \(\Omega\) is bounded, then the best constant satisfies \(c(n, q, \Omega)\leq c(n)|\Omega|^{1/q} q^{1-1/n}\) and it follows from the theory of Orlicz spaces that the exponent \(1- 1/n\) is the best possible. However, the ratio \(|u|_q/|\nabla u|_n\) is not scale-invariant. The scale-invariant form of the embedding is the Gagliardo-Nirenberg inequality \(|u|_q\leq c(n,q, r)|\nabla u|^{1-r/q}_n|u|^{r/q}_r\), where \(r\in [1,\infty)\), \(q\in [r, \infty)\) and \(u\in W^{1,n}_0 \cap L_r (\Omega)\).
The authors proved that the best possible constant in the above Gagliardo-Nirenberg inequality satisfies the estimate \(c(n, q, r)\leq c'(n, r)q^{1- 1/n}\) and that the exponent \(1-1/n\) here is the best possible. This result improves the estimate \(c(n, q, r)\leq c(n, r)'q\) given by O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva [“Linear and quasilinear equations of parabolic type”, Trans. Math. Monographs 23 (1968; Zbl 0174.15403); Russian original (1967; Zbl 0164.12302)].


46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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