An optimization problem with volume constraint with applications to optimal mass transport

Abstract

In this manuscript we study the following optimization problem with volume constraint: \[ \min\left\{\frac{1}{p}\int_Ω |\nabla v|^pdx- \int_{\partial Ω} gv\,dS \colon v \in W^{1, p} \left(Ω\right), \text{ and } |\{v>0\}| \leq α\right\}. \] Here $g$ is acontinuous function and $\alpha$ is a fixed constant such that $0<\alpha <|\Omega |$. Under the assumption that ${\int }_{\partial \Omega }g(x)dS>0$ we prove that a minimizer exists and satisfies $$\{\begin{array}{ccl}-{\Delta }_p{u}_p=0& \text{in }& \{u_p>0\}\cup \{u_p<0\},\\ |\nabla u_p{|}^{p-2}\frac{\partial u_p}{\partial \nu }=g& \text{on }& \partial \Omega \cap \partial (\{u_p>0\}\cup \{u_p<0\}),\\ |\{u_p>0\}|=\alpha .& & \end{array}$$ Next, we analyze the limit as $p\to \infty$. We obtain that any sequence of weak solutions converges, up to a subsequence, $\underset{p_j\to \infty }{\lim }{u}_{p_j}(x)=u_{\infty }(x)$, uniformly in $\overline{\Omega }$, and uniform limits, $u_{\infty }$, are solutions to the maximization problem with volume constraint $$\max \left\{{\int }_{\partial \Omega }gvdS:v\in W^{1,\infty }\left(\Omega \right),\Vert \nabla v{\Vert }_{L^{\infty }(\Omega )}\le 1\text{ and }|\{v>0\}|\le \alpha \right\}.$$ 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, $\partial \{u_{\infty }\ne 0\}$.

23 pages, 2 figures