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

What is it about?

We analyze the problem of determining the best constants for the Sobolev inequalities in the limiting case where $p=1$. 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, 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 there are none. The impact of symmetries which appears in the manifold, is also discussed.

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Why is it important?

The case where $p=1$ can be said \emph{limiting} because it can be seen as a limit case as $p$ tends to 1. This case is more complicated due to the lack of compactness of the embedding $H^1_1(M)\hookrightarrow L^1(\partial M)$. However, in the direction of interest in this article, there are some important results and and all cases have been studied.