Duality and best constant for a Trudinger-Moser inequality involving probability measures
logarithmic Hardy-Littlewood-Sobolev inequality
Let \(\Omega\) be a two-dimensional Riemannian surface without boundary and assume that \(\lambda\) is a positive parameter. Denote \(I=[-1,1]\) and suppose that \(\mathcal P\) is a Borel probability measure on \(I\). For all \(v\in H^1(\Omega)\) satisfying \(\int_\Omega v=0\), consider the Trudinger-Moser energy functional NEWLINE\[NEWLINE J_\lambda(v)=\frac{1}{2}\int_\Omega|\nabla v|^2-\lambda\int_I\left(\log\int_\Omega e^{\alpha v}\right)\,\mathcal P(d\alpha).NEWLINE\]NEWLINE The main result establishes a Toland non-convex duality principle for \(J_\lambda\) and a computation of the optimal value of \(\lambda\) for which \(J_\lambda\) is bounded from below. The study of such problems is motivated by models arising in the statistical mechanics description of equilibrium turbulence, under the assumption that the intensity and the orientation of the vortices are determined by \(\mathcal P\).
scientific article; zbMATH DE number 6353603