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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$L^p$ regularity of least gradient functions
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by Wojciech G贸rny 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.
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Additional Information
  • Wojciech G贸rny
  • 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, 鈥淎nisotropic least gradient problem鈥, 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