  In this manuscript we study the following optimization problem with volume
constraint:
  \[
  \min\left\{\frac{1}{p}\int_{\Omega} |\nabla v|^pdx-
  \int_{\partial \Omega} gv\,dS \colon v \in W^{1, p}
  \left(\Omega\right), \text{ and }
  |\{v>0\}| \leq \alpha\right\}.
  \] Here $g$ is acontinuous function and $\alpha$ is a fixed constant such
that $0< \alpha< |\Omega|$. Under the assumption that $\displaystyle
\int_{\partial \Omega} g(x)dS >0$ we prove that a minimizer exists and
satisfies
  $$
  \left\{
  \begin{array}{ccl}
  -\Delta_p u_p = 0 &\text{in }
  &\{u_p>0\} \cup \{u_p<0\}, \\[5pt]
  |\nabla u_p|^{p-2}\frac{\partial u_p}{\partial \nu} = g
  & \text{on } &\partial \Omega
  \cap\partial(\{u_p>0\} \cup \{u_p<0\} ) ,\\[5pt]
  |\{u_p>0\}| = \alpha. & &
  \end{array}
  \right.
  $$
  Next, we analyze the limit as $p\to \infty$. We obtain that any sequence of
weak solutions converges, up to a subsequence, $\lim\limits_{p_j \to \infty}
u_{p_j}(x)=u_{\infty}(x)$, uniformly in $\overline{\Omega}$, and uniform
limits, $u_\infty$, are solutions to the maximization problem with volume
constraint
  $$
  \max\left\{ \int_{\partial \Omega} gv\,dS \colon v \in W^{1, \infty}
  \left(\Omega\right), \|\nabla v\|_{L^{\infty}(\Omega)}\leq 1
  \text{ and }|\{v>0\}| \leq \alpha\right\}.
  $$
  Furthermore, we obtain the limit equation that is verified by $u_\infty$ in
the viscosity sense. Finally, it turns out that such a limit variational
problem is connected to the Monge-Kantorovich mass transfer problem with the
involved measures are supported on $\partial \Omega$ and along the limiting
free boundary, $\partial \{u_{\infty} \neq 0\}$.
