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 ∂ 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)=∑k,l=1n∂Sτ(D2u)/∂uklukul−2binomn−1τ−1∫0ufτ/n(t) dt is established for arbitrary positive and nondecreasing f, which is of independent interest.
Related
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BibTeX
@misc{zhao2026overdetermined,
title = {{Overdetermined problem for optimal transportation}},
author = {Qing Zhao and Feida Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39782},
eprint = {2609.39782},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39782},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Overdetermined problem for optimal transportation
AU - Qing Zhao
AU - Feida Jiang
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39782
UR - https://arxiv.org/abs/2609.39782
AB - 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 ∂ 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)=∑_k,l=1^n∂S_τ(D^2u)/∂u_klu_ku_l−2binomn−1τ−1∫_0^uf^τ/n(t) dt is established for arbitrary positive and nondecreasing f, which is of independent interest.
ER -