Hardy type inequalities and Gaussian measure
Barbara Brandolini Francesco Chiacchio Cristina Trombetti
Communications on Pure & Applied Analysis 2007, 6(2): 411-428 doi: 10.3934/cpaa.2007.6.411
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
Discrete & Continuous Dynamical Systems - S 2011, 4(4): 825-831 doi: 10.3934/dcdss.2011.4.825
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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