Stability and Continuity of Functions of Least Gradient

• Autorzy: H. Hakkarainen , R. Korte , P. Lahti , N. Shanmugalingam
• Języki publikacji: 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.

Słowa kluczowe

EN

least gradient; BV; metric measure spac; approximate continuity; continuity; stability; jump set; Dirichlet problem; minimal surface

Kategorie tematyczne

• 26B15: Integration: length, area, volume
• 31E99: None of the above, but in this section
• 26B30: Absolutely continuous functions, functions of bounded variation
• 31C45: Other generalizations (nonlinear potential theory, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Analysis and Geometry in Metric Spaces
• Rocznik: 2015
• Tom: 3
• Numer: 1

Daty

otrzymano

2014-10-13

zaakceptowano

2015-05-24

online

2015-06-15

Twórcy

autor

H. Hakkarainen

• Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

autor

R. Korte

• Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland

autor

P. Lahti

• Aalto University, School of Science and Technology, Department of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland

autor

N. Shanmugalingam

• Department of Mathematical Sciences, P.O.Box 210025, University of Cincinnati, Cincinnati, OH 452210025, U.S.A.

Bibliografia

• [1] F. J. Almgren, Jr., Almgren’s big regularity paper, Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2. With a preface by Jean E. Taylor and Vladimir Scheffer. World Scientific Monograph Series in Mathematics, 1. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. xvi+955 pp.
• [2] L. Ambrosio, Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces, Adv. Math. 159 (2001), no. 1, 51–67.
• [3] L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Calculus of variations, nonsmooth analysis and related topics. Set-Valued Anal. 10 (2002), no. 2-3, 111–128. [Crossref]
• [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]
• [5] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems,OxfordMathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
• [6] 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, 1–45, Quad. Mat., 14, Dept. Math., Seconda Univ. Napoli, Caserta, 2004.
• [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]
• [8] A. Björn and J. Björn, Nonlinear potential theory on metric spaces, EMS Tracts in Mathematics, 17. European Mathematical Society (EMS), Zürich, 2011. xii+403 pp. [WoS]
• [9] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268.
• [10] C. Camfield, Comparison of BV norms in weighted Euclidean spaces and metric measure spaces, Thesis (Ph.D.)–University of Cincinnati (2008), 141 pp.
• [11] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. [Crossref]
• [12] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions Studies in AdvancedMathematics series, CRC Press, Boca Raton, 1992.
• [13] E. Giusti,Minimal surfaces and functions of bounded variation, Monographs inMathematics, 80. Birkhäuser Verlag, Basel, 1984. xii+240 pp.
• [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.
• [18] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear potential theory of degenerate elliptic equations, Dover Publications, Inc., Mineola, NY, 2006. xii+404 pp.
• [19] J. Kinnunen and N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces, Manuscripta Math. 105 (2001), 401–423. [Crossref]
• [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]
• [24] F. Maggi, Sets of finite perimeter and geometric variational problems, An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, 135. Cambridge University Press, Cambridge, 2012. xx+454 pp. [WoS]
• [25] U. Massari and M. Miranda, Sr., Minimal surfaces of codimension one, North-Holland Mathematics Studies, 91. Notas de Matemática [Mathematical Notes], 95, North-Holland Publishing Co., Amsterdam, 1984. xiii+243 pp.
• [26] M. Miranda, Sr., Comportamento delle successioni convergenti di frontiere minimali, Rend. Sem. Mat. Univ. Padova, 38 (1967), 238–257.
• [27] M. Miranda, Jr., Functions of bounded variation on “good” metric spaces, J.Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. [Crossref]
• [28] H. Parks, Explicit determination of area minimizing hypersurfaces, Duke Math. J. 44 (1977), no. 3, 519–534. [Crossref]
• [29] H. Parks, Explicit determination of area minimizing hypersurfaces. II, Mem. Amer. Math. Soc. 60 (1986), no. 342, iv+90 pp.
• [30] E. R. Reifenberg, Solution of the Plateau Problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. [Crossref]
• [31] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces, Rev.Mat. Iberoamericana 16 (2000), no. 2, 243–279.
• [32] J. Simons, Minimal cones, Plateau’s problem, and the Bernstein conjecture, Proc. Nat. Acad. Sci. U.S.A. 58 (1967), 410–411. [Crossref]
• [33] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105.
• [34] E. Soultanis, Homotopy classes of Newtonian spaces, preprint http://lanl.arxiv.org/pdf/1309.6472.pdf.
• [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.

Identyfikatory

DOI

10.1515/agms-2015-0009