Radial $Φ^4$ models in supercritical regimes

We consider the $Φ^4_d$ measure under radial symmetry on the unit ball, formally given by $$dρ(u) = \frac 1 Z \exp\Big( - \frac 14 \int_{B_1} |u|^4 dx - \frac 12 \int_{B_1} |\nabla u|^2 dx\Big) d u, $$ where $u$ is restricted to radially symmetric functions with 0 boundary conditions. We prove that