Geometric properties of solutions to elliptic PDE's in Gauss space and related Brunn-Minkowski type inequalities

We prove a Brunn-Minkowski type inequality for the first (nontrivial) Dirichlet eigenvalue of the weighted $p$-operator $$-{\mathrm{\Delta }}_{p,\gamma }u=-\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)+(x,\mathrm{\nabla }u)\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2},$$ where $p>1$, in the class of bounded Lipschitz domains in ${\mathbb{R}}^n$. We also prove that any corresponding positive eigenfunction is log-concave if the domain is convex.