Hardy type inequalities and Gaussian measure
Barbara Brandolini Francesco Chiacchio Cristina Trombetti
In this paper we prove some improved Hardy type inequalities with respect to the Gaussian measure. We show that they are strictly related to the well-known Gross Logarithmic Sobolev inequality. Some applications to elliptic P.D.E.'s are also given.
keywords: degenerate elliptic P.D.E.'s. Gaussian symmetrization Hardy type inequalities
Shape optimization for Monge-Ampère equations via domain derivative
Barbara Brandolini Carlo Nitsch Cristina Trombetti
In this note we prove that, if $\Omega$ is a smooth, strictly convex, open set in $R^n$ $(n \ge 2)$ with given measure, the $L^1$ norm of the convex solution to the Dirichlet problem $\det D^2 u=1$ in $\Omega$, $u=0$ on $\partial\Omega$, is minimum whenever $\Omega$ is an ellipsoid.
keywords: Monge-Ampère equation domain derivative affine isoperimetric inequalities

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