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Register a new user$C^{2,alpha}$ regularity of free boundaries in optimal transportation
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In this paper we establish the $C^{2,alpha}$ regularity for free boundary in the optimal transport problem in all dimensions.
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In this paper, we obtain some regularities of the free boundary in optimal transportation with the quadratic cost. Our first result is about the $C^{1,alpha}$ 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 $partialOmega, partialOmega^*in C^{1,1}$ 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.
The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^infty$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $nleq4$, the free boundary is generically a $C^infty$ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions $nleq4$.
This work builds the connection between the regularity theory of optimal transportation map, Monge-Amp`{e}re equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Amp`{e}re equation, if the support of the target measure is disconnected or just non-convex, the optimal transportation mapping is discontinuous. General DNNs can only approximate continuous mappings. This intrinsic conflict leads to the convergence difficulty and mode collapse in GANs. We test our hypothesis that the supports of real data distribution are in general non-convex, therefore the discontinuity is unavoidable using an Autoencoder combined with discrete optimal transportation map (AE-OT framework) on the CelebA data set. The testing result is positive. Furthermore, we propose to approximate the continuous Brenier potential directly based on discrete Brenier theory to tackle mode collapse. Comparing with existing method, this method is more accurate and effective.
We give a simple proof of a recent result in [1] by Caffarelli, Soria-Carro, and Stinga about the $C^{1,alpha}$ regularity of weak solutions to transmission problems with $C^{1,alpha}$ interfaces. Our proof does not use the mean value property or the maximum principle, and also works for more general elliptic systems. Some extensions to $C^{1,text{Dini}}$ interfaces and to domains with multiple sub-domains are also discussed.
This paper intents to present the state of art and recent developments of the optimal transportation theory with many marginals for a class of repulsive cost functions. We introduce some aspects of the Density Functional Theory (DFT) from a mathematical point of view, and revisit the theory of optimal transport from its perspective. Moreover, in the last three sections, we describe some recent and new theoretical and numerical results obtained for the Coulomb cost, the repulsive harmonic cost and the determinant cost.
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