论文标题
严格的凸度和$ c^1 $定期解决方案，以生成雅各布方程。
Strict convexity and $C^1$ regularity of solutions to generated Jacobian equations in dimension two
论文作者
论文摘要
我们提供了2D中严格的$ g $ convexity的证明，用于生成的雅各布方程的解决方案，其$ g $ -monge-ampère措施远离0。随后，这意味着$ c^1 $可怜性在$ g $ - $ mmonge-monge-ampère的措施中。我们的证明是在Monge-Ampère案中由Trudinger和Wang提供的证明。因此，像他们的论点一样，我们的论点是本地的，并且对$ g $ convexity产生了定量估计。结果，即使在最佳运输案例中，我们的不同性结果也是新的：我们削弱了先前所需的域凸状条件。此外，在最佳运输案例和Monge-Ampère情况下，我们的主要假设（即A3W和域凸）是必要的。
We present a proof of strict $g$-convexity in 2D for solutions of generated Jacobian equations with a $g$-Monge-Ampère measure bounded away from 0. Subsequently this implies $C^1$ differentiability in the case of a $g$-Monge-Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge-Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the $g$-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge-Ampère case our key assumptions, namely A3w and domain convexity, are necessary.
