Uniform partial regularity of quasi minimizers for the perimeter

Abstract.

Quasi minimizers for the perimeter are measurable subsets G of $\mathbb{R}^n$ such that

\[ \int_{B(x,r)}|D \varphi_G|\leq (1+\omega(r))\int_{B(x,r)}|D \varphi_{G'}|, \]

for all variations $G'$ of G with $G'\triangle G\Subset B(x,r)$ and for a given increasing function $\omega$ such that $\lim_{r\rightarrow 0}\omega(r)=0$. We prove here that, given $\alpha<1$, G a reduced quasi minimizer, $x\in \partial G$ and $r\leq 1$, there are $y\in \partial G$, $t\geq C^{-1}r$ with $B(y,t)\subset B(x,r)$, and $\tilde S\subset \mathbb{R}^n$, $C^{0,\alpha}-$homeomorphic to a closed ball with radius t in $\mathbb{R}^{n-1}$, such that $\partial G\cap \overline B(y,\eta t)\subset \tilde S\subset \partial G\cap \overline B(y,t)$ for some absolute constant $\eta<1$. The constant $C>1$ above depends only on n, $\omega$ and $\alpha$. If moreover $\omega(r)=C'r^{2\alpha}$ for some $\alpha\in(0,\frac{1}{2}]$, we prove that we can find such a ball $B(y,t)$ such that $\partial G\cap B(y,t)$ is a $(n-1)-$dimensional graph of class $C^{1,\alpha}$. This will be obtained proving that a quasi minimizer is equivalent to some set which satisfies the condition B. This condition gives some kind of uniform control on the flatness of the boundary and then criterions proven by Ambrosio-Paolini and Tamanini can be applied to get the required regularity properties.