Sharp constants in the Sobolev embedding theorem and a derivation of the Brezis-Gallouet interpolation inequality

Sharp estimates are obtained for the constants appearing in the Sobolev embedding theorem on the two-dimensional torus. Furthermore, a version of the Brezis-Gallouet interpolation inequality is obtained with an explicit but not necessarily optimal constant in the leading term, namely, the logarithmic term. The constants are expressed in terms of the Riemann zeta-function and the Dirichlet beta-series. (C) 2011 American Institute of Physics. [doi:10.1063/1.3638056]