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

Tonia Ricciardi; Takashi Suzuki

doi:10.4171/jems/462

JournalsjemsVol. 16, No. 7pp. 1327–1348

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

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Abstract

We consider the Trudinger–Moser type functional

J_\lambda(v)=\frac{1}{2}\int_\Omega|\nabla v|^2-\lambda\int_I\left(\log\int_\Omega e^{\alpha v}\right)\,\mathcal P(d\alpha),

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