Local boundedness for $ p $-Laplacian with degenerate coefficients

Peter Bella; Mathias Schäffner; Peter Bella; Mathias Schäffner

doi:10.3934/mine.2023081

Mathematics in Engineering

2023, Volume 5, Issue 5: 1-20. doi: 10.3934/mine.2023081

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Local boundedness for $ p $-Laplacian with degenerate coefficients

Peter Bella ¹,
Mathias Schäffner ^{2
,
,}

1.
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, 44227 Dortmund, Germany
2.
MLU Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Straße 5, 06120 Halle (Saale), Germany

Received: 09 September 2022 Revised: 15 March 2023 Accepted: 15 March 2023 Published: 03 April 2023

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $ \nabla \cdot (\lambda |\nabla u|^{p-2}\nabla u) = 0 $, where the variable coefficient $ 0\leq\lambda $ and its inverse $ \lambda^{-1} $ are allowed to be unbounded. Assuming certain integrability conditions on $ \lambda $ and $ \lambda^{-1} $ depending on $ p $ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $ p > 1 $.
- elliptic regularity,
- nonuniform ellipticity,
- local boundedness,
- unbounded solutions,
- weak solutions
Citation: Peter Bella, Mathias Schäffner. Local boundedness for $ p $-Laplacian with degenerate coefficients[J]. Mathematics in Engineering, 2023, 5(5): 1-20. doi: 10.3934/mine.2023081

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Abstract

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $ \nabla \cdot (\lambda |\nabla u|^{p-2}\nabla u) = 0 $, where the variable coefficient $ 0\leq\lambda $ and its inverse $ \lambda^{-1} $ are allowed to be unbounded. Assuming certain integrability conditions on $ \lambda $ and $ \lambda^{-1} $ depending on $ p $ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $ p > 1 $.

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