The best constants of the Sobolev inequalities on the whole Euclidian space are known and even the functions realizing the equality can be computed. Is there any result of this type in bounded domains (e.g. a ball, a cube)? I consider here only functions vanishing on the boundary of the domain.
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4$\begingroup$ Did you google it? a lot seems to come up under "best" or "optimal sobolev inequalities"! $\endgroup$– Matthias LudewigCommented Sep 13, 2013 at 16:00
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By the minimax principle, the optimal constant is just the first eigenvalue of the Laplacian with Dirichlet boundary conditions. And yes, the first eigenvalue of the Laplacian with Dirichlet boundary conditions is known in many different cases, including those of a ball and a cube (or, more generally, a parallelepiped). E.g., the first Dirichlet eigenvalue on a rectangle of sides a,b is $$ \pi^2 (a^{-2}+b^{-2}). $$
