散度形式的线性化Monge-Ampère方程的全局Hölder估计及其在双半地转方程和周期均匀化中的应用
Global Hölder estimates for linearized Monge-Ampère equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization
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中文总结 AI 辅助
本文建立了散度形式线性化Monge-Ampère方程的全局Hölder估计，并应用于三维双半地转系统的时间导数正则性及Monge-Ampère方程周期均匀化的线性收敛速率。
中文摘要 AI 辅助
我们在所有维度$n\geq 3$中，当凸Monge-Ampère势的Hessian行列式被正常数上下界所限制且右侧向量场有界时，建立了散度形式的线性化Monge-Ampère方程的内部和全局Hölder估计。这些估计使用任意$p>1$的解的$L^p$范数。一个关键成分是线性化Monge-Ampère算子格林函数梯度在小截面上的$L^1$估计，具有适当的衰减率，且关于极点一致。作为应用，当初始密度远离零和无穷大时，我们获得了三维周期双半地转系统中原始和对偶势时间导数的均匀Hölder估计，并建立了Monge-Ampère方程周期均匀化的线性收敛速率。
英文摘要
We establish interior and global Hölder estimates for linearized Monge-Ampère equations in divergence form in all dimensions $n\geq 3$, when the Hessian determinant of the convex Monge-Ampère potential is bounded above and below by positive constants and the vector field on the right-hand side is bounded. The estimates use the $L^p$ norm of the solution for any $p>1$. A key ingredient is an $L^1$ estimate, with appropriate decay rates, for the gradient of the Green's function of the linearized Monge-Ampère operator in small sections, uniform with respect to the pole. As applications, we obtain uniform Hölder estimates for the time derivatives of the primal and dual potentials in the three-dimensional periodic dual semigeostrophic system when the initial density is bounded away from zero and infinity, and establish a linear convergence rate for periodic homogenization of the Monge-Ampère equation.
发表机构
- Peking University（北京大学）
- Indiana University（印第安纳大学）
- Bocconi University（博科尼大学）
机构由 AI 辅助整理，请以论文原文为准。
