Optimal shapes for positivity preserving

We are looking for an optimal convex domain on which the boundary value problem $$\{\begin{array}{cc}(-\mathrm{\Delta }{)}^2{u}_{\gamma }-\gamma \mathrm{\Delta }{u}_{\gamma }=f,& in\mathrm{\Omega }\\ u_{\gamma }={\mathrm{\partial }}_{\nu }{u}_{\gamma }=0,& on\mathrm{\partial }\mathrm{\Omega }\end{array}$$ admits a nonnegative solution for the most $\gamma$, if $f$ is a given nonnegative function.