We consider the Trudinger–Moser type functional
where is a two-dimensional Riemannian surface without boundary, , , , is a Borel probability measure on and . The functional arises in the statistical mechanics description of equilibrium turbulence, under the assumption that the intensity and the orientation of the vortices are determined by . We formulate a Toland non-convex duality principle for and we compute the optimal value of for which is bounded from below.
Cite this article
Tonia Ricciardi, Takashi Suzuki, Duality and best constant for a Trudinger–Moser inequality involving probability measures. J. Eur. Math. Soc. 16 (2014), no. 7, pp. 1327–1348DOI 10.4171/JEMS/462