论文标题
Monge-Ampere系统：任意维度和编辑中的凸集成
The Monge-Ampere system: convex integration in arbitrary dimension and codimension
论文作者
论文摘要
在本文中，我们通过凸集成对Monge-Ampère系统（MA）的弱解决方案的灵活性。这种新的PDE系统是Monge-ampère方程在$ d = 2 $尺寸中的扩展，自然是由规定的曲率问题引起的，并且与等距沉浸的经典问题密切相关（II）。 我们的主要结果是在hölder$ \ mathcal {c}^{1，α} $ solutions tovonKármán系统（vk）的hölder$ \ mathcal {c}^{1，α} $ solutions中达到密度的密度，这是（ma）的弱公式。规律性指数$α$是满足$α<\ frac {1} {1+的任何指数 d（d+1）/k} $，其中$ d $是任意维度，而$ k $是对问题的任意编成。在$ k = 1 $时，这与（ii）的规则性$ \ natercal {c}^{1，α} $，任何$α<\ frac {1} {1+d（d+1）} $，由conti，delellis，delellis和szekelyhidi证明。在$ d = 2，k = 1 $时，这扩展了作者的初始发现和Pakzad（MA）。 从技术角度来看，对于基于波纹的凸集成方案，我们的结果似乎是最佳的。特别是，它涵盖了\ big in \ big（1，d（d+1）\ big）$到目前为止，即使是系统（ii），因为规律性$ \ nathcal {c}^{c}^{1，α，α} $在\ codiension中都在\ citiSiral中使用的任何$α<1 $ rigities，light catiens^kallen}，我们的第二个主要结果在（ma）的背景下重现了Källen的结果，在$ \ Mathcal {c}^{1，α} $常规解决方案中获得任何$α<1 $时，每当$ k \ geq d（d+1）$时，获得了密度。 作为（VK）的结果的应用，我们得出了一种能量缩放，结合了riemannian指标的定量沉浸性，用于以非线性弹性性薄膜变形的变形能量建模的非线性能量函数。
In this paper, we study flexibility of weak solutions to the Monge-Ampère system (MA) via convex integration. This new system of Pdes is an extension of the Monge-Ampère equation in $d=2$ dimensions, naturally arising from the prescribed curvature problem and closely related to the classical problem of isometric immersions (II). Our main result achieves density in the set of subsolutions, of the Hölder $\mathcal{C}^{1,α}$ solutions to the Von Kármán system (VK) which is the weak formulation of (MA). The regularity exponent $α$ is any exponent satisfying $α<\frac{1}{1+ d(d+1)/k}$ where $d$ is an arbitrary dimension and $k$ an arbitrary codimension of the problem. At $k=1$, this agrees with the regularity $\mathcal{C}^{1,α}$ for (II) with any $α<\frac{1}{1+d(d+1)}$, proved by Conti, Delellis and Szekelyhidi. At $d=2, k=1$, this extends the initial findings by the author and Pakzad for (MA). Our result seems to be optimal, from the technical viewpoint, for the corrugation-based convex integration scheme. In particular, it covers the codimension interval $k\in \big(1, d(d+1)\big)$ so far uncharted even for the system (II), since the regularity $\mathcal{C}^{1,α}$ with any $α<1$ achieved by Källen in \cite{Kallen}, strictly requires a large codimension. Our second main result reproduces Källen's result in the context of (MA), obtaining density in the set of subsolutions, of $\mathcal{C}^{1,α}$ regular solutions for any $α<1$ whenever $k\geq d(d+1)$. As an application of our results for (VK), we derive an energy scaling bound in the quantitative immersability of Riemannian metrics, for nonlinear energy functionals modelled on the energies of deformations of thin prestrained films in the nonlinear elasticity.
