Best constants in the exceptional case of Sobolev inequalities. (English) Zbl 1097.46017
Let \((M,g)\) be a Riemannian \(n\)-dimensional compact manifold, locally conformally flat. The author studies the problem of the best constant in an exponential Sobolev inequality
\[
\int_M e^u dV\, \leq\, C\, \exp{(\mu\| \nabla u\| ^n_n) }
\]
for \(u\in H^n_1(M)\) with \(\int_M u dV = 0 \). It is proved that there exists \(C>0\) such that the above inequality holds for an explicit numerical constant \(\mu=\mu(n, \omega_n)\) where \(\omega_n= \text{ vol}(S_{n-1})\). The constant is the best possible such that the inequality holds for any \(u\in H^n_1(M)\) with \(\int_M u dV = 0 \).
Reviewer: Leszek Skrzypczak (Poznań)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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