# Regularity of free boundaries in optimal transportation

Journal Article

### Abstract

• In this paper, we obtain some regularities of the free boundary in optimal

$C^{1,\alpha}$ regularity of the free boundary for optimal partial transport

between convex domains for densities $f, g$ bounded from below and above. When

$f, g \in C^\alpha$, and $\partial\Omega, \partial\Omega^*\in C^{1,1}$ are far

apart, by adopting our recent results on boundary regularity of Monge-Amp\ere

equations \cite{CLW1}, our second result shows that the free boundaries are

$C^{2,\alpha}$. As an application, in the last we also obtain these

regularities of the free boundary in an optimal transport problem with two

separate targets.

• 2019

### Citation

• Chen, S., & Liu, J. (2019). Regularity of free boundaries in optimal transportation. Retrieved from http://arxiv.org/abs/1911.10790v2

### Abstract

• In this paper, we obtain some regularities of the free boundary in optimal

$C^{1,\alpha}$ regularity of the free boundary for optimal partial transport

between convex domains for densities $f, g$ bounded from below and above. When

$f, g \in C^\alpha$, and $\partial\Omega, \partial\Omega^*\in C^{1,1}$ are far

apart, by adopting our recent results on boundary regularity of Monge-Amp\ere

equations \cite{CLW1}, our second result shows that the free boundaries are

$C^{2,\alpha}$. As an application, in the last we also obtain these

regularities of the free boundary in an optimal transport problem with two

separate targets.

• 2019

### Citation

• Chen, S., & Liu, J. (2019). Regularity of free boundaries in optimal transportation. Retrieved from http://arxiv.org/abs/1911.10790v2