# 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

$J_{λ}(v)=21 ∫_{Ω}∣∇v∣_{2}−λ∫_{I}(g∫_{Ω}e_{αv})P(dα),$

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