Consider a field $\phi(x,y)$ in the 2D plane. The minimum of the following Functional leads to formation of a 2D disk in which $\phi(x,y)=1$ deep inside the disk and $\phi(x,y)=0$ outside the disk.The boundary of the disk has a thickness so that $\phi$ continuously gors from 1 to 0. $$\DeclareMathOperator{\Dm}{\operatorname{d\!}} F=\int \Dm x \Dm y \big(\phi^2(1-\phi)^2 + K (\nabla \phi)^2 \big) $$ with $K>0$ and a constraint on $\phi$ as following: $\int \phi dx dy =c$ where $c>>1$.
Is it possible to add a term to the above functional so that the minimum of $F$ becomes a disk with a hollow circle at the centre? I thought about considering a term which favours negative curvature like $-\nabla^2 \phi$, but since this term can be written as a divergence, its integral is zero and would not change the $F$.
I would appreciate it if some one could clarify if this question has a solution or not.