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We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions $u\in SB{V}^2\left(\Omega \right)$ for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for $E(·)$ is proved and the asymptotic behaviour of the solutions is investigated.

We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions ∈ ²(Ω) for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for (.) is proved and the asymptotic behaviour of the solutions is investigated.