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Tytuł artykułu

Stability and Continuity of Functions of Least Gradient

Treść / Zawartość
Warianty tytułu
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EN
Abstrakty
EN
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.
Twórcy
  • Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland
autor
  • Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland
autor
  • Aalto University, School of Science and Technology, Department of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland
  • Department of Mathematical Sciences, P.O.Box 210025, University of Cincinnati, Cincinnati, OH 452210025, U.S.A.
Bibliografia
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  • [4] L. Ambrosio and S. Di Marino, Equivalent definitions of BV space and of total variation on metric measure spaces, J. Funct. Anal. 266 (2014), no. 7, 4150–4188. [WoS]
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  • [7] L. Ambrosio, A. Pinamonti, and G. Speight, Tensorization of Cheeger energies, the space H1,1 and the area formula for graphs, preprint 2014. [WoS]
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  • [14] P. Hajłasz, Sobolev spaces on metric-measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173–218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.
  • [15] H. Hakkarainen, J. Kinnunen, and P. Lahti, Regularity of minimizers of the area functional in metric spaces, Adv. Calc. Var. 8 (2015), no. 1, 55–68. [WoS]
  • [16] H. Hakkarainen, J. Kinnunen, P. Lahti, and P. Lehtelä, Relaxation and integral representation for functionals of linear growth on metric measure spaces, submitted.
  • [17] J. Heinonen, Lectures on analysis on metric spaces, Universitext. Springer-Verlag, New York, 2001. x+140 pp.
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  • [20] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, The DeGiorgi measure and an obstacle problem related to minimal surfaces in metric spaces, J. Math. Pures Appl. 93 (2010), 599–622. [Crossref][WoS]
  • [21] J. Kinnunen, R. Korte, A. Lorent, and N. Shanmugalingam, Regularity of sets with quasiminimal boundary surfaces in metric spaces, J. Geom. Anal. 23 (2013), 1607–1640. [WoS][Crossref]
  • [22] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Lebesgue points and capacities via the boxing inequality in metric spaces, Indiana Univ. Math. J. 57 (2008), no. 1, 401–430. [WoS]
  • [23] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen, Pointwise properties of functions of bounded variation on metric spaces, Rev. Mat. Complut. 27 (2014), no. 1, 41–67. [Crossref][WoS]
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  • [31] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev.Mat. Iberoamericana 16 (2000), no. 2, 243–279.
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  • [35] P. Sternberg, G. Williams, and W. P. Ziemer, Existence, uniqueness, and regularity for functions of least gradient, J. Reine Angew. Math. 430 (1992), 35–60.
  • [36] W. P. Ziemer, Weakly differentiable functions. Sobolev spaces and functions of bounded variation Graduate Texts in Mathematics, 120. Springer-Verlag, New York, 1989.
  • [37] W. P. Ziemer, Functions of least gradient and BV functions, Nonlinear analysis, function spaces and applications, Vol. 6 (Prague, 1998), 270–312, Acad. Sci. Czech Repub., Prague, 1999.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0009
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