Regularity of a free boundary determined by the value of the gradient. Part 1: Third order asymptotics

Abstract

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 $$\Gamma =\partial \{x\in B_1:|\nabla u|>1\}\cap \partial \{x\in B_1:|\nabla u|<1\}.$$ We consider two-phase flat points for which the second derivatives of $F$ jump over $\Gamma$. In this paper, we show that under some regularity and non-degeneracy assumptions, a minimizer $u$ can be expressed as $$u(x)=a+\nu \cdot x+\delta p(x)+\delta \epsilon q(x),$$ where $a\in \mathbb{R}$, $\nu \in {\mathbb{R}}^n$, $0<\delta ,\epsilon <<1$. 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.

81 pages