Overdetermined problem for optimal transportation

Abstract

In this paper, we establish symmetry results for solutions of overdetermined problems arising in optimal transportation. These problems involve the Monge-Ampère equation with a Dirichlet boundary condition $u=0$ on $\partial \Omega$ and the natural boundary condition $Du(\Omega )={\Omega }^{\ast }$. We show that symmetry holds when the target is the unit ball $B$. For more general target domains, including the cases ${\Omega }^{\ast }=\Omega$ and arbitrary ${\Omega }^{\ast }$, symmetry is retained under an additional volume constraint on $\Omega$ and a boundary condition on $|Du|$. Finally, we introduce a curvature-type overdetermined problem for the Monge-Ampère equation and obtain ellipsoidal or spherical symmetry under an additional integral normalization condition. Our proofs rely on distinct techniques in different contexts: optimal transport and convex analysis for some cases, integral identities and isoperimetric inequalities for others, and P-function methods for the remaining cases. As a byproduct, in the $\tau =2$ case, a new maximum principle for the $P$-function $\varphi (x)={\sum }_{k,l=1}^n\frac{\partial S_{\tau }(D^{2}u)}{\partial u_{kl}}{u}_k{u}_l-2\left(\genfrac{}{}{0ex}{}{n-1}{\tau -1}\right){\int }_0^u{f}^{\frac{\tau }{n}}(t)dt$ is established for arbitrary positive and nondecreasing $f$, which is of independent interest.