Extremals for Moser inequalities
The authors show that for Moser-Trudinger inequalities \[ \int_Q|\nabla u|^2 \,d\mu\leq 1\quad\text{and}\quad \int_Q u\,d\mu= 0, \] where \(\nabla u\) is the gradient with respect to the usual metric on the sphere and \(d\mu\) the corresponding volume element, there exists an extreme solution involving exponential integrals. The results are obtained for Sobolev spaces when the domain \(Q\) is one of the spheres \(S^2\) and \(S^3\), the unit disk \(Q= D^2\) or the projective space \(\mathbb RP^2\). The method of Carleson and Chang is used for \(n\)-balls, with boundary value zero modified, and applied to functions with mean value zero on these domains. This approach also provides an elementary proof of the Moser inequality for \(S^n\), \(n\geq 2\), which is a special case of the result of Fortuna.
scientific article; zbMATH DE number 2101868