Energy improvement for energy minimizing functions in the complement of generalized Reifenberg-flat sets

TL;DR

This paper demonstrates energy bounds for functions in the complement of Reifenberg-flat and minimal sets, extending to generalized minimal sets, with implications for regularity in Mumford-Shah functional analysis.

Contribution

It establishes energy improvement results for energy minimizing functions outside Reifenberg-flat and generalized minimal sets, broadening understanding of their regularity properties.

Findings

Normalized energy bounded in smaller balls for Reifenberg-flat sets

Extension of results to epsilon-minimal and (eps; eps_0)-minimal sets

Foundation for regularity analysis of Mumford-Shah functional near minimal cones

Abstract

Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with d_H(K; P) <epsilon, and if u is an energy minimizing function in B(0; 1)\K with restricted values on @B(0; 1)\K, then the normalized energy of u in B(0; r)\K is bounded by the normalized energy of u in B(0; 1)\K. We also prove the same result in R^3 when K is a epsilon-minimal set, that is a generalization of Reifenberg-flat sets with minimal cones of type Y and T. Moreover, the result is still true for a further generalization of sets called (eps; eps_0)-minimal. This article is a preliminary study for a forthcoming paper where a regularity result for the singular set of the Mumford-Shah functional close to minimal cones in R^3 is proved by the same author.