On the best constants in Sobolev inequalities on the solid torus in the limit case p=1

In this paper, we analyze the problem of determining the best constants for the Sobolev inequalities in the limiting case vvherep = 1. Firstly, the special case of the solid torus is studied, whenever it is proved that the solid torus is an extremal domain with respect to the second best constant and totally optimal with respect to the best constants in the trace Sobolev inequality. Secondly, in the spirit of Andreu, Mazon and Rossi [3], a Neumann problem involving the 1-Laplace operator in the solid torus is solved. Finally, the existence of both best constants in the case of a manifold with boundary is studied, when they exist. Further examples are provided where they do not exist. The impact of symmetries which appear in the manifold is also discussed.