发表机构
Southeast University; Shanghai Institute for Mathematics and Interdisciplinary Sciences（东南大学; 上海数学与交叉学科研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究最优输运中Monge-Ampère方程的超定问题，证明目标为单位球时对称性成立，并在体积约束和边界条件下推广至一般目标域，同时引入曲率型超定问题获得椭球或球对称性，方法涉及最优输运、积分恒等式及P-函数技术。
AI 中文摘要
本文建立了最优输运中出现的超定问题的解的对称性结果。这些问题涉及带有Dirichlet边界条件$u=0$在$\partial \Omega$上以及自然边界条件$Du(\Omega)=\Omega^{*}$的Monge-Ampère方程。我们证明了当目标为单位球$B$时对称性成立。对于更一般的目标域，包括$\Omega^*=\Omega$和任意$\Omega^{*}$的情形，在$\Omega$上附加体积约束以及对$|Du|$的边界条件下，对称性得以保持。最后，我们引入了Monge-Ampère方程的曲率型超定问题，并在附加积分归一化条件下获得椭球或球对称性。我们的证明在不同情形下使用不同技术：某些情形使用最优输运和凸分析，其他情形使用积分恒等式和等周不等式，其余情形使用P-函数方法。作为副产品，在$\tau=2$情形下，对于任意正且非递减的$f$，建立了$P$-函数$\phi(x)=\sum_{k,l=1}^{n}{\frac{\partial{S_{\tau}(D^2{u})}}{\partial{u_{kl}}}u_{k}u_{l}}-2\binom{n-1}{\tau-1}\int_{0}^{u}{f^{\frac{\tau}{n}}(t)\\\\,dt}$的新的最大值原理，这具有独立的意义。
英文摘要
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 $\partial Ω$ and the natural boundary condition $Du(Ω)=Ω^{*}$. We show that symmetry holds when the target is the unit ball $B$. For more general target domains, including the cases $Ω^*=Ω$ and arbitrary $Ω^{*}$, symmetry is retained under an additional volume constraint on $Ω$ 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 $τ=2$ case, a new maximum principle for the $P$-function $ϕ(x)=\sum_{k,l=1}^{n}{\frac{\partial{S_τ(D^2{u})}}{\partial{u_{kl}}}u_{k}u_{l}}-2\binom{n-1}{τ-1}\int_{0}^{u}{f^{\fracτ{n}}(t)\,dt}$ is established for arbitrary positive and nondecreasing $f$, which is of independent interest.
