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Moser-Trudinger inequality on compact Riemannian manifolds of dimension two. (English) Zbl 0995.58021
The author studies the Moser-Trudinger type inequality concerning the Orlicz space embeddings of the Sobolev space \(H^{1,2} (M)\) where \(M\) is a two-dimensional Riemannian manifold without boundary. He determines that the best possible constant \(\alpha\) in the inequality \[ \sup \Bigl\{\int_M \exp\bigl(\alpha|u|^2\bigr) :\|u\|_{H^{1,2} (M)} \leq 1\Bigr\} <+\infty \] is equal to \(4\pi\) and shows the existence of an extremal function for which the supremum is attained. The similar results concerning the above inequality with the unit ball in \(H^{1,2}(M)\) replaced by \(\{u\in H^{1,2}(M): \int_Mu=0\), \(\int_M|\nabla u|^2\leq 1\}\) or by \(\{u \in H_0^{1,2}(N): \int_N|\nabla u|^2\leq 1\}\) in case \(N\) is a two-dimensional Riemannian manifold with boundary. The blow-up analysis technique is applied.

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
53C20 Global Riemannian geometry, including pinching