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$ L^p$ regularity of least gradient functions


Author: Wojciech Górny
Journal: Proc. Amer. Math. Soc. 148 (2020), 3009-3019
MSC (2010): Primary 35J20, 35J25, 35J75, 35J92
DOI: https://doi.org/10.1090/proc/15031
Published electronically: April 9, 2020
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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.


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Wojciech Górny
Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
Email: w.gorny@mimuw.edu.pl

DOI: https://doi.org/10.1090/proc/15031
Keywords: Least gradient problem, anisotropy, $L^p$ regularity
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
Article copyright: © Copyright 2020 American Mathematical Society