## $L^p$ regularity of least gradient functions

HTML articles powered by AMS MathViewer

- by Wojciech Górny PDF
- Proc. Amer. Math. Soc.
**148**(2020), 3009-3019 Request permission

## Abstract:

It is shown that in the anisotropic least gradient problem on an open bounded set $\Omega \subset \mathbb {R}^N$ with Lipschitz boundary, given boundary data $f \in L^p(\partial \Omega )$ the solutions lie in $L^{\frac {Np}{N-1}}(\Omega )$; the exponent is shown to be optimal. Moreover, the solutions are shown to be locally bounded with explicit bounds on the rate of blow-up of the solution near the boundary in two settings: in the anisotropic case on the plane and in the isotropic case in any dimension.## References

- M. Amar and G. Bellettini,
*A notion of total variation depending on a metric with discontinuous coefficients*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**11**(1994), no. 1, 91–133 (English, with English and French summaries). MR**1259102**, DOI 10.1016/S0294-1449(16)30197-4 - Luigi Ambrosio, Nicola Fusco, and Diego Pallara,
*Functions of bounded variation and free discontinuity problems*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR**1857292** - E. Bombieri, E. De Giorgi, and E. Giusti,
*Minimal cones and the Bernstein problem*, Invent. Math.**7**(1969), 243–268. MR**250205**, DOI 10.1007/BF01404309 - Lawrence C. Evans and Ronald F. Gariepy,
*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660** - Herbert Federer,
*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR**0257325** - Wojciech Górny,
*Planar least gradient problem: existence, regularity and anisotropic case*, Calc. Var. Partial Differential Equations**57**(2018), no. 4, Paper No. 98, 27. MR**3813249**, DOI 10.1007/s00526-018-1378-y - Wojciech Górny,
*Existence of minimisers in the least gradient problem for general boundary data*, Indiana Univ. Math. J., to appear. - Wojciech Górny, Piotr Rybka, and Ahmad Sabra,
*Special cases of the planar least gradient problem*, Nonlinear Anal.**151**(2017), 66–95. MR**3596671**, DOI 10.1016/j.na.2016.11.020 - H. Hakkarainen, R. Korte, P. Lahti, and N. Shanmugalingam,
*Stability and continuity of functions of least gradient*, Anal. Geom. Metr. Spaces**3**(2015), no. 1, 123–139. MR**3357766**, DOI 10.1515/agms-2015-0009 - Robert L. Jerrard, Amir Moradifam, and Adrian I. Nachman,
*Existence and uniqueness of minimizers of general least gradient problems*, J. Reine Angew. Math.**734**(2018), 71–97. MR**3739314**, DOI 10.1515/crelle-2014-0151 - José M. Mazón,
*The Euler-Lagrange equation for the anisotropic least gradient problem*, Nonlinear Anal. Real World Appl.**31**(2016), 452–472. MR**3490852**, DOI 10.1016/j.nonrwa.2016.02.009 - Leon Simon,
*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417** - Peter Sternberg, Graham Williams, and William P. Ziemer,
*Existence, uniqueness, and regularity for functions of least gradient*, J. Reine Angew. Math.**430**(1992), 35–60. MR**1172906** - Peter Sternberg and William P. Ziemer,
*The Dirichlet problem for functions of least gradient*, Degenerate diffusions (Minneapolis, MN, 1991) IMA Vol. Math. Appl., vol. 47, Springer, New York, 1993, pp. 197–214. MR**1246349**, DOI 10.1007/978-1-4612-0885-3_{1}4

## Additional Information

**Wojciech Górny**- Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
- Email: w.gorny@mimuw.edu.pl
- Received by editor(s): April 23, 2019
- Received by editor(s) in revised form: April 24, 2019, and November 25, 2019
- Published electronically: April 9, 2020
- Additional Notes: This work was supported in part by the research project no. 2017/27/N/ST1/02418, “Anisotropic least gradient problem”, funded by the National Science Centre, Poland.
- Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 3009-3019 - MSC (2010): Primary 35J20, 35J25, 35J75, 35J92
- DOI: https://doi.org/10.1090/proc/15031
- MathSciNet review: 4099787