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

### Tonia Ricciardi

Università degli Studi di Napoli Federico II, Italy### Takashi Suzuki

Osaka University, Japan

## 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.

## 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. 1327–1348

DOI 10.4171/JEMS/462