由梯度值确定的自由边界的正则性。第一部分：三阶渐近展开
Regularity of a free boundary determined by the value of the gradient. Part 1: Third order asymptotics
AI总结：
本文研究由梯度值决定的自由边界正则性，通过建立极小化子的三阶渐近展开，证明余项弱极限满足特定偏微分方程且具有C^3正则性，为自由边界正则性提供基础。
AI中文摘要：
这是一个关于函数自由边界正则性研究项目的第一部分，其中自由边界依赖于函数的梯度。我们研究表达式\begin{equation*} J(u): = \int_{B_1} F(\nabla u) \\ dx, \end{equation*}的极小化问题，其中$F(x)$是一个一致凸函数，其二阶导数可能在$|x|=1$处跳跃。这导致欧拉-拉格朗日方程在自由边界上变化，我们将自由边界定义为$$ \Gamma = \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. $$我们考虑$F$的二阶导数在$\Gamma$上跳跃的两相平坦点。在本文中，我们证明在一些正则性和非退化假设下，极小化子$u$可以表示为$$ u(x)= a + \nu \cdot x+ \delta p(x) + \delta \epsilon q(x), $$其中$a \in \mathbb{R}$，$\nu \in \mathbb{R}^n$，$0 < \delta, \epsilon << 1$。这里$p$是一个$C^1$函数，由上半个球中的一个多项式和下半个球中的另一个多项式组成。函数$q$是一个具有有界$L^2$范数的余项。此外，假设我们有一列极小化子$u^j$，我们证明$q^j$弱收敛到一个满足某个偏微分方程的函数$q^0$。另外，我们证明$q^0$在$B_r^+$和$B_r^-$中分别属于$C^3$，其中$0<r<1$。这是本文的主要结果，旨在用于证明自由边界的正则性。
英文摘要：
This is the first part of a project concerning the regularity of the free boundary of a function, where the free boundary depends on the gradient of the function. We study the minimizer of the expression \begin{equation*} J(u) : = \int_{B_1} F(\nabla u) \ dx, \end{equation*} where $F(x)$ is a uniformly convex function whose second derivatives might jump at $|x|=1$. This results in an Euler-Lagrange equation that varies over the free boundary, which we define as $$ Γ= \partial \{x \in B_1: |\nabla u| >1 \} \cap \partial \{x \in B_1: |\nabla u| <1 \}. $$ We consider two-phase flat points for which the second derivatives of $F$ jump over $Γ$. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$ u(x)= a + ν\cdot x+ δp(x) + δεq(x), $$ where $a \in \mathbb{R}$, $ν\in \mathbb{R}^n$, $0 < δ, ε<< 1$. Here $p$ is a $C^1$ function, which consists of one polynomial in the upper half ball and another polynomial in the lower half ball. The function $q$ is a rest term with bounded $L^2$ norm. Furthermore, assuming that we have a sequence, $u^j$, of minimizers, we show that $q^j$ converges weakly to a function $q^0$ that satisfies a certain PDE. In addition, we show that $q^0$ is $C^3$ in $B_r^+$ and $B_r^-$, respectively, for $0<r<1$. This is the main result of this paper which is intended to be used to show regularity of the free boundary.
