Characterizations and fine properties of functions of bounded variation on metric measure spaces

Panu Lahti

Research output: ThesisDoctoral ThesisCollection of Articles


In this thesis we study functions of bounded variation, abbreviated as BV functions, on metric measure spaces. We always assume the space to be equipped with a doubling measure, and mostly we also assume it to support a Poincaré inequality. A central topic in the thesis are the various characterizations of BV functions. We show that BV functions can be characterized by a pointwise inequality involving the maximal function of a finite measure. Furthermore, we study the Federer-type characterization of sets of finite perimeter, according to which a set is of finite perimeter if and only if the codimension one Hausdorff measure of the set's measure theoretic boundary is finite. Through the study of socalled strong relative isoperimetric inequalities, we establish a slightly weakened version of this characterization. Moreover, we prove the Federer-type characterization on spaces that support a geometric Semmes family of curves. On such spaces, between every pair of points there is a curve family with certain uniformity properties that resemble the behavior of parallel lines on a Euclidean space. Our proof relies on first proving a characterization of BV functions in terms of curves. We also study functionals of linear growth, which give a generalization of BV functions. We consider an integral representation for such functionals by means of the variation measure, but contrary to the Euclidean case, the functional and the integral representation are only comparable instead of being equal. As a by-product of our analysis, we are able to characterize those BV functions that are in fact Newton-Sobolev functions. As an application of the integral representation, we consider a minimization problem for the functionals of linear growth, and show that the boundary values of such a problem can be expressed as a penalty term in which we integrate over the boundary of the domain. For this, we need to study boundary traces and extensions of BV functions. Our analysis of traces also produces novel pointwise results on the behavior of BV functions in their jump sets.
Translated title of the contributionRajoitetusti heilahtelevien funktioiden karakterisaatioita ja ominaisuuksia metrisissä mitta-avaruuksissa
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
  • Kinnunen, Juha, Supervising Professor
  • Kinnunen, Juha, Thesis Advisor
Print ISBNs978-952-60-5633-3
Electronic ISBNs978-952-60-5634-0
Publication statusPublished - 2014
MoE publication typeG5 Doctoral dissertation (article)


  • boundary trace
  • characterization
  • doubling measure
  • extension
  • finite perimeter
  • function of bounded variation
  • functional of linear growth
  • integral representation
  • Poincaré inequality
  • relative isoperimetric inequality
  • Semmes family of curves


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