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A Notion of Fine Continuity for BV Functions on Metric Spaces

Abstract

In the setting of a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that BV functions are, in the sense of multiple limits, continuous with respect to a 1-fine topology, at almost every point with respect to the codimension 1 Hausdorff measure.

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References

  1. 1.

    Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces, Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10(2-3), 111–128 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems., Oxford Mathematical Monographs. The Clarendon Press, p xviii+434. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. 3.

    Ambrosio, L., Miranda Jr., M., Pallara, D.: Special functions of bounded variation in doubling metric measure spaces, Calculus of variations: topics from the mathematical heritage of E. De Giorgi, 1–45, Quad Mat., vol. 14. Dept. Math., Seconda Univ. Napoli, Caserta (2004)

  4. 4.

    Ambrosio, L., Tilli, P.: Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, 25, p viii+133. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  5. 5.

    Björn, A., Björn, J.: Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17, p xii+403. European Mathematical Society (EMS), Zürich (2011)

    Book  MATH  Google Scholar 

  6. 6.

    Björn, A., Björn, J., Shanmugalingam, N.: Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces. Houston J. Math. 34(4), 1197–1211 (2008)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Björn, A., Björn, J., Shanmugalingam, N.: Sobolev extensions of Hölder continuous and characteristic functions on metric spaces. Canad J. Math. 59(6), 1135–1153 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Björn, J.: Fine continuity on metric spaces. Manuscripta Math. 125(3), 369–381 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Björn, J., MacManus, P., Shanmugalingam, N.: Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces. J. Anal. Math. 85, 339–369 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Carriero, M., Dal Maso, G., Leaci, A., Pascali, E.: Relaxation of the nonparametric plateau problem with an obstacle. J. Math. Pures Appl. (9) 67(4), 359–396 (1988)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cartan, H.: Théorie générale du balayage en potentiel newtonien. Ann. Univ. Grenoble. Sect. Sci. Math. Phys. (N.S.) 22, 221–280 (1946)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions, Studies in Advanced Mathematics, p viii+268. CRC Press, Boca Raton, FL (1992)

    MATH  Google Scholar 

  13. 13.

    Franchi, B., Serapioni, R., Serra Cassano, F.: On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13(3), 421–466 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Giusti, E.: Minimal surfaces and functions of bounded variation, Monographs in Mathematics, 80, p xii+240. Birkhäuser Verlag, Basel (1984)

    Book  Google Scholar 

  15. 15.

    Hakkarainen, H., Kinnunen, J.: The BV-capacity in metric spaces. Manuscripta Math. 132(1-2), 51–73 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Hakkarainen, H., Kinnunen, J., Lahti, P.: Regularity of minimizers of the area functional in metric spaces. Adv. Calc. Var. 8(1), 55–68 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Hakkarainen, H., Korte, R., Lahti, P., Shanmugalingam, N.: Stability and continuity of functions of least gradient. Anal. Geom. Metr. Spaces 3 (2015). Art. 9

  18. 18.

    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations, p xii+404. Dover Publications Inc., Mineola, NY (2006)

    MATH  Google Scholar 

  19. 19.

    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev spaces on metric measure spaces: an approach based on upper gradients, New Mathematical Monographs, vol. 27, pp i–xi+448. Cambridge University Press (2015)

  21. 21.

    Kinnunen, J., Korte, R., Lorent, A., Shanmugalingam, N.: Regularity of sets with quasiminimal boundary surfaces in metric spaces. J. Geom. Anal. 23(4), 1607–1640 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: A characterization of Newtonian functions with zero boundary values. Calc. Var. 43(3-4), 507–528 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: Lebesgue points and capacities via the boxing inequality in metric spaces. Indiana Univ. Math. J. 57(1), 401–430 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kinnunen, J., Korte, R., Shanmugalingam, N., Tuominen, H.: Pointwise properties of functions of bounded variation on metric spaces. Rev. Mat. Complut. 27 (1), 41–67 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Kinnunen, J., Latvala, V.: Fine regularity of superharmonic functions on metric spaces, Future trends in geometric function theory, 157–167, Rep. Univ. Jyväskylä Dep. Math Stat. 92, Univ. Jyväskylä, Jyväskylä (2003)

  26. 26.

    Korte, R.: A Caccioppoli estimate and fine continuity for superminimizers on metric spaces. Ann. Acad. Sci. Fenn Math. 33(2), 597–604 (2008)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Lahti, P., Shanmugalingam, N.: Fine properties and a notion of quasicontinuity for BV functions on metric spaces, to appear in J. Math. Pures Appl.

  28. 28.

    Lewis, J.: Uniformly fat sets. Trans. Amer. Math. Soc. 308(1), 177–196 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Malý, J., Ziemer, W. P.: Fine regularity of solutions of elliptic partial differential equations, Mathematical Surveys and Monographs, 51, p xiv+291. American Mathematical Society, Providence, RI (1997)

    Book  MATH  Google Scholar 

  30. 30.

    Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 82(8), 975–1004 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Shanmugalingam, N.: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16(2), 243–279 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Ziemer, W. P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, 120, p xvi+308. Springer-Verlag, New York (1989)

    MATH  Google Scholar 

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Correspondence to Panu Lahti.

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Lahti, P. A Notion of Fine Continuity for BV Functions on Metric Spaces. Potential Anal 46, 279–294 (2017). https://doi.org/10.1007/s11118-016-9582-x

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Keywords

  • Bounded variation
  • Metric measure space
  • Quasicontinuity
  • Fine topology
  • Fine continuity
  • Upper hemicontinuity

Mathematics Subject Classification 2010

  • 30L99
  • 26B30
  • 43A85