Uniform rectifiability and singular sets

被引:10
作者
David, G
Semmes, S
机构
[1] INST HAUTES ETUD SCI,F-91440 BURES SUR YVETTE,FRANCE
[2] RICE UNIV,HOUSTON,TX 77251
来源
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE | 1996年 / 13卷 / 04期
关键词
D O I
10.1016/S0294-1449(16)30109-3
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Under what conditions can one say something about the geometric structure of the singular set of a function? A famous result of this type states that the singular set (= set of non-lebesgue points) of a function of bounded variation on R(n) is (countably) rectifiable. In this paper we shall be concerned with quantitative forms of rectifiability, and we give a quantitative version of this theorem. We also give uniform rectifiability results for the singular sets of minimizers of higher-dimensional versions of the Mumford-Shah functional. Along the way we shall encounter some generalizations of the usual topological notion of a set ''separating'' points in the complement, including one which is based on the failure of Poincare inequalities on the complement of the given set.
引用
收藏
页码:383 / 443
页数:61
相关论文
共 29 条
[1]   EXISTENCE THEORY FOR A NEW CLASS OF VARIATIONAL-PROBLEMS [J].
AMBROSIO, L .
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, 1990, 111 (04) :291-322
[2]   EXISTENCE THEOREM FOR A DIRICHLET PROBLEM WITH FREE DISCONTINUITY SET [J].
CARRIERO, M ;
LEACI, A .
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 1990, 15 (07) :661-677
[3]  
Carriero M., 1991, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), V18, P321
[4]   A VARIATIONAL METHOD IN IMAGE SEGMENTATION - EXISTENCE AND APPROXIMATION RESULTS [J].
DALMASO, G ;
MOREL, JM ;
SOLIMINI, S .
ACTA MATHEMATICA, 1992, 168 (1-2) :89-151
[5]   LIPSCHITZ APPROXIMATION TO HYPERSURFACES, HARMONIC MEASURE, AND SINGULAR-INTEGRALS [J].
DAVID, G ;
JERISON, D .
INDIANA UNIVERSITY MATHEMATICS JOURNAL, 1990, 39 (03) :831-845
[6]   QUANTITATIVE RECTIFIABILITY AND LIPSCHITZ MAPPINGS [J].
DAVID, G ;
SEMMES, S .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 1993, 337 (02) :855-889
[7]  
DAVID G, IN PRESS J MATH PURE
[8]  
David G., 1993, MATH SURVEYS MONOGRA, V38
[9]  
DAVID G., 1988, Rev. Mat. Iberoamericana, V4, P73, DOI 10.4171/RMI/64
[10]  
DAVID G, 1995, J FOURIER ANAL APPL, P161