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

We consider the Trudinger–Moser type functional

Jλ(v)=12Ωv2λI(logΩeαv)P(dα),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 Ω\Omega is a two-dimensional Riemannian surface without boundary, vH1(Ω)v\in H^1(\Omega), Ωv=0\int_\Omega v=0, I=[1,1]I=[-1,1], P\mathcal P is a Borel probability measure on II and λ>0\lambda>0. The functional JλJ_\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 P\mathcal P. We formulate a Toland non-convex duality principle for JλJ_\lambda and we compute the optimal value of λ\lambda for which JλJ_\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