More counterexamples to regularity for minimizers of the average-distance problem

Abstract

Abstract The average-distance problem, in the penalized formulation, involves minimizing (1) E μ λ(Σ) := ∫ℝ d d(x,Σ)dμ(x) + λℋ 1 (Σ), among path-wise connected, closed sets Σ with finite ℋ 1 -measure, where d ≥ 2, μ is a given measure, λ is a given parameter and d(x,Σ) := inf y∈Σ|x - y|. The average-distance problem can be also considered among compact, convex sets with perimeter and/or volume penalization, i.e. minimizing (2) ℰ(μ,λ1,λ2)(·):=∫ℝ d d(x,·)dμ(x) + λ1Per(·) + λ2Vol(·), where μ is a given measure, λ1,λ2 ≥ 0 are given parameters with λ1 + λ2 > 0, and the unknown varies among compact, convex sets. Very little is known about the regularity of minimizers of (2). In particular it is unclear if minimizers of (2) are in general C 1 regular. The aim of this paper is twofold: first, we provide in ℝ2 a second approach in constructing minimizers of (1) which are not C 1 regular; then, using the same technique, we provide an example of minimizer of (2) whose border is not C 1 regular, under perimeter penalization only.