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The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces

Panu Lahti

Abstract

In the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of BV functions with zero boundary values. In particular, we show that the class is the closure of compactly supported BV functions in the BV norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and BV analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

MSC 2010: 30L99; 31E05; 26B30

Communicated by Frank Duzaar


Funding statement: The research was funded by a grant from the Finnish Cultural Foundation.

Acknowledgements

Most of the research for this paper was conducted during the author’s visit to the University of Cincinnati, whose hospitality the author wishes to acknowledge. The author also wishes to thank Anders and Jana Björn for discussions on variational p-capacities, as well as the anonymous referee for suggestions on how to improve the paper.

References

[1] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal. 10 (2002), no. 2–3, 111–128. Search in Google Scholar

[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Clarendon Press, New York, 2000. Search in Google Scholar

[3] L. Ambrosio, M. Miranda, Jr. and D. Pallara, Special functions of bounded variation in doubling metric measure spaces, Calculus of Variations: Topics From the Mathematical Heritage of E. De Giorgi, Quad. Mat. 14, Seconda Università degli Studi di Napoli, Caserta (2004), 1–45. Search in Google Scholar

[4] L. Beck and T. Schmidt, Convex duality and uniqueness for BV-minimizers, J. Funct. Anal. 268 (2015), no. 10, 3061–3107. Search in Google Scholar

[5] A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts Math. 17, European Mathematical Society (EMS), Zürich, 2011. Search in Google Scholar

[6] A. Björn and J. Björn, The variational capacity with respect to nonopen sets in metric spaces, Potential Anal. 40 (2014), no. 1, 57–80. Search in Google Scholar

[7] A. Björn and J. Björn, Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology, Rev. Mat. Iberoam. 31 (2015), no. 1, 161–214. Search in Google Scholar

[8] A. Björn, J. Björn and N. Shanmugalingam, The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173–203. Search in Google Scholar

[9] A. Björn, J. Björn and N. Shanmugalingam, Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 (2008), no. 4, 1197–1211. Search in Google Scholar

[10] A. Björn, J. Björn and N. Shanmugalingam, The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities, J. Differential Equations 259 (2015), no. 7, 3078–3114. Search in Google Scholar

[11] E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. Search in Google Scholar

[12] M. Carriero, G. Dal Maso, A. Leaci and E. Pascali, Relaxation of the nonparametric plateau problem with an obstacle, J. Math. Pures Appl. (9) 67 (1988), no. 4, 359–396. Search in Google Scholar

[13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math., CRC Press, Boca Raton, 1992. Search in Google Scholar

[14] H. Federer, Geometric Measure Theory, Grundlehren Mathe. Wiss. 153, Springer, New York, 1969. Search in Google Scholar

[15] H. Federer and W. P. Ziemer, The Lebesgue set of a function whose distribution derivatives are p-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. Search in Google Scholar

[16] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, EMS Monogr. Math. 80, Birkhäuser, Basel, 1984. Search in Google Scholar

[17] H. Hakkarainen and J. Kinnunen, The BV-capacity in metric spaces, Manuscripta Math. 132 (2010), no. 1–2, 51–73. Search in Google Scholar

[18] 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. Search in Google Scholar

[19] H. Hakkarainen, J. Kinnunen, P. Lahti and P. Lehtelä, Relaxation and integral representation for functionals of linear growth on metric measure spaces, Anal. Geom. Metr. Spaces 4 (2016), 288–313. Search in Google Scholar

[20] H. Hakkarainen, R. Korte, P. Lahti and N. Shanmugalingam, Stability and continuity of functions of least gradient, Anal. Geom. Metr. Spaces 3 (2015), 123–139. Search in Google Scholar

[21] H. Hakkarainen and N. Shanmugalingam, Comparisons of relative BV-capacities and Sobolev capacity in metric spaces, Nonlinear Anal. 74 (2011), no. 16, 5525–5543. Search in Google Scholar

[22] J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001. Search in Google Scholar

[23] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover, Mineola, 2006. Search in Google Scholar

[24] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. Search in Google Scholar

[25] 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. Search in Google Scholar

[26] J. Kinnunen, R. Korte, N. Shanmugalingam and H. Tuominen, The De Giorgi measure and an obstacle problem related to minimal surfaces in metric spaces, J. Math. Pures Appl. (9) 93 (2010), no. 6, 599–622. Search in Google Scholar

[27] J. Kinnunen, R. Korte, N. Shanmugalingam and H. Tuominen, Pointwise properties of functions of bounded variation in metric spaces, Rev. Mat. Complut. 27 (2014), no. 1, 41–67. Search in Google Scholar

[28] R. Korte, P. Lahti, X. Li and N. Shanmugalingam, Notions of Dirichlet problem for functions of least gradient in metric measure spaces, preprint (2016), ; to appear in Rev. Mat. Iberoam. Search in Google Scholar

[29] P. Lahti, A Federer-style characterization of sets of finite perimeter on metric spaces, Calc. Var. Partial Differential Equations 56 (2017), no. 5, Article ID 150. Search in Google Scholar

[30] P. Lahti, Superminimizers and a weak Cartan property for p=1 in metric spaces, preprint (2017), ; to appear in J. Anal. Math. Search in Google Scholar

[31] P. Lahti, Approximation of BV by SBV functions in metric spaces, preprint (2018), . Search in Google Scholar

[32] P. Lahti, Strong approximation of sets of finite perimeter in metric spaces, Manuscripta Math. 155 (2018), no. 3–4, 503–522. Search in Google Scholar

[33] P. Lahti and N. Shanmugalingam, Fine properties and a notion of quasicontinuity for BV functions on metric spaces, J. Math. Pures Appl. (9) 107 (2017), no. 2, 150–182. Search in Google Scholar

[34] P. Lahti and N. Shanmugalingam, Trace theorems for functions of bounded variation in metric spaces, J. Funct. Anal. 274 (2018), no. 10, 2754–2791. Search in Google Scholar

[35] T. Mäkäläinen, Adams inequality on metric measure spaces, Rev. Mat. Iberoam. 25 (2009), no. 2, 533–558. Search in Google Scholar

[36] J. Malý and W. P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations, Math. Surveys Monogr. 51, American Mathematical Society, Providence, 1997. Search in Google Scholar

[37] V. G. Maz’ja, The Dirichlet problem for elliptic equations of arbitrary order in unbounded domains, Dokl. Akad. Nauk SSSR 150 (1963), 1221–1224. Search in Google Scholar

[38] V. G. Maz’ja, Sobolev Spaces, Springer Ser. Soviet Math., Springer, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. Search in Google Scholar

[39] J. M. Mazón, J. D. Rossi and S. Segura de León, Functions of least gradient and 1-harmonic functions, Indiana Univ. Math. J. 63 (2014), no. 4, 1067–1084. Search in Google Scholar

[40] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. Search in Google Scholar

[41] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam. 16 (2000), no. 2, 243–279. Search in Google Scholar

[42] N. Shanmugalingam, Harmonic functions on metric spaces, Illinois J. Math. 45 (2001), no. 3, 1021–1050. Search in Google Scholar

[43] 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. Search in Google Scholar

[44] W. P. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, New York, 1989. Search in Google Scholar

[45] W. P. Ziemer and K. Zumbrun, The obstacle problem for functions of least gradient, Math. Bohem. 124 (1999), no. 2–3, 193–219. Search in Google Scholar

Received: 2018-05-01
Revised: 2018-09-12
Accepted: 2018-09-25
Published Online: 2018-10-21
Published in Print: 2021-04-01

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