Calculus of Variations and Geometric Measure Theory

L. Ambrosio - E. Paolini

Partial regularity for quasi minimizers of perimeter

created on 12 Oct 1998
modified by paolini on 10 Dec 2013


Published Paper

Inserted: 12 oct 1998
Last Updated: 10 dec 2013

Journal: Ricerche di Matematica
Volume: 48
Number: supplemento
Pages: 167-186
Year: 1999


Let $E\subset R^n$ be a quasi minimizer of perimeter, that is, a set such that $P(E,$ $B_\rho(x))\le (1+\omega(\rho))P(F,B_\rho(x))$ for all variations $F$ with $F\Delta E \subset\subset$ $B_\rho(x)$ and for a given function $\omega$ with $\lim_{\rho\to 0}\omega(\rho)=0$. We prove that, up to a closed set with dimension at most $n-8$, for all $\alpha<1$ the set $\partial E$ is an $(n-1)$-dimensional $C^{0,\alpha}$ manifold. This result is obtained combining the De Giorgi and Reifenberg regularity theories for area minimizers. Moreover we prove that, in the case $n=2$, $\partial E$ is a bi-lipschitz curve.

Keywords: regularity, minimal surfaces, perimeter