The $L_p$-Brunn-Minkowski inequalities for variational functionals with $0\le p<1$

Abstract

The infinitesimal forms of the $L_p$-Brunn-Minkowski inequalities for variational functionals, such as the $q$-capacity, the torsional rigidity, and the first eigenvalue of the Laplace operator, are investigated for $p\ge 0$. These formulations yield Poincaré-type inequalities related to these functionals. As an application, the $L_p$-Brunn-Minkowski inequalities for torsional rigidity with $0\le p<1$ are confirmed for small $C^2$-perturbations of the unit ball.