Homogeneous vacuum minimum criterion

For a scalar theory with positive kinetic and gradient energy, a potential attaining a global minimum, and boundary conditions admitting a constant minimizing field, the energy is bounded below by the integral of that potential minimum. A static constant field at the minimum attains the bound. A potential unbounded below, such as $-{\varphi }^4$, invalidates the existence claim; incompatible boundary values can require an inhomogeneous minimizer.