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

Let P be an hyperplane in R , and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an ε > 0, such that if K is a Reifenbergflat set in B(0, 1) ⊂ R that contains the origin, with dH(K,P ) ≤ ε, 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 when K is a ε-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 (ε, ε0)-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 is proved by the same author. Introduction Let B be the unit ball in R and let K be a closed set with locally finite HN−1 measure. We say that u locally minimizes the energy in B\K if for every ball B(x, r) included in B̄ and every function v ∈ W (B\K) such that v = u in B\(B(x, r) ∪K) we have ∫ B(x,r)\K |∇u(x)|dx ≤ ∫ B(x,r)\K |∇v(x)|dx. Then we know that u is harmonic in B\K, and morally has its normal derivative equal to 0 on K, at least if K is smooth enough. 1 ar X iv :0 80 6. 29 87 v1 [ m at h. A P] 1 8 Ju n 20 08 If u is an energy minimizer in B\K, we denote by ω2(x, r) := 1 rN−1 ∫