Duality and best constant for a Trudinger–Moser inequality involving probability measures

Abstract

We consider the Trudinger–Moser type functional

where $\Omega$ is a two-dimensional Riemannian surface without boundary, $v\in H^1(\Omega )$, ${\int }_{\Omega }v=0$, $I=[-1,1]$, $\mathcal{P}$ is a Borel probability measure on $I$ and $\lambda >0$. The functional $J_{\lambda }$ arises in the statistical mechanics description of equilibrium turbulence, under the assumption that the intensity and the orientation of the vortices are determined by $\mathcal{P}$. We formulate a Toland non-convex duality principle for $J_{\lambda }$ and we compute the optimal value of $\lambda$ for which $J_{\lambda }$ is bounded from below.