Abstract:The main goal of this paper is to establish a general Lorentzian Lasry-Lions regularization theorem: let $u$ be a function defined on a globally hyperbolic spacetime. Assume that its forward Lax--Oleinik evolution $Tu$ is locally semiconcave in a neighbourhood of $(t_0,y_0)$ and has future-directed timelike superdifferentials there. Then, for $t$ close to $t_0$ and sufficiently small $s>0$, the function $\hat T_s\circ T_tu$ is of class $C_{\mathrm{loc}}^{1,1}$ in a neighbourhood of $y_0$. We give sufficient conditions ensuring the assumptions of the theorem and present an application to optimal transport: under quite general assumptions, for any two intermediate measures along a displacement interpolation, there exists a $C^{1,1}_{loc}$-regular maximizing pair in the dual formulation.
From: Alec Metsch [view email]
[v1]
Mon, 22 Jun 2026 22:07:32 UTC (37 KB)
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