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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2018: 2.316

CiteScore 2018: 1.77

SCImago Journal Rank (SJR) 2018: 2.350
Source Normalized Impact per Paper (SNIP) 2018: 1.465

Mathematical Citation Quotient (MCQ) 2018: 1.44

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Volume 9, Issue 1


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

Xin Yang Lu
Published Online: 2015-01-10 | DOI: https://doi.org/10.1515/acv-2014-0002


The average-distance problem, in the penalized formulation, involves minimizing (1) Eμλ(Σ) := ∫d d(x,Σ)dμ(x) + λℋ1(Σ), among path-wise connected, closed sets Σ with finite ℋ1-measure, where d ≥ 2, μ is a given measure, λ is a given parameter and d(x,Σ) := infy∈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,λ12)(·):=∫d d(x,·)dμ(x) + λ1Per(·) + λ2Vol(·), where μ is a given measure, λ12 ≥ 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 C1 regular. The aim of this paper is twofold: first, we provide in ℝ2 a second approach in constructing minimizers of (1) which are not C1 regular; then, using the same technique, we provide an example of minimizer of (2) whose border is not C1 regular, under perimeter penalization only.

Keywords: Nonlocal variational problem; average-distance problem; regularity

MSC: 49Q20; 49K10; 49Q10; 35B6

About the article

Received: 2014-02-12

Revised: 2014-09-14

Accepted: 2014-11-27

Published Online: 2015-01-10

Published in Print: 2016-01-01

Funding Source: ICTI and FCT

Award identifier / Grant number: UTA_CMU/MAT/ 0007/2009

Funding Source: NSF

Award identifier / Grant number: DMS-0635983

Citation Information: Advances in Calculus of Variations, Volume 9, Issue 1, Pages 41–63, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2014-0002.

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