Gradient rearrangement estimates for singular and degenerate anisotropic p-Laplace equations
Abstract
In this paper, we prove global pointwise estimates on the decreasing rearrangement u∗ and [H(∇ u)]∗ of SOLAs u of an elliptic Poisson equation driven by the anisotropic p-Laplace operator ΔpH for given L1-data and equipped with either homogeneous Dirichlet or Neumann boundary conditions. Here, we assume that the dimension n≥ 2, 1<p<n and H is norm on Rn. In the degenerate range 2 ≤ p <n, this gives an anisotropic extension and a full-interval completion of the rearrangement theory of Cianchi-Maz'ya [JEMS, 2014]. In particular, we obtain a new rearrangement estimate for gradients by decomposing the data for singular range 1<p<2, where the nonlinear interpolation method aren't available. Moreover, the endpoint weak-type gradient estimate is obtained for all 1<p<n. In the range p*(n)<p<n with p*(n):=(3n−2)/(2n−1), we prove a stability estimate for differences of gradients. These results extends the known ones in the literature to the singular range p*(n)<p<2.
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BibTeX
@misc{chen2026gradient,
title = {{Gradient rearrangement estimates for singular and degenerate anisotropic $p$-Laplace equations}},
author = {Lu Chen and Daniel Hauer and Yabo Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13226},
eprint = {2610.13226},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13226},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Gradient rearrangement estimates for singular and degenerate anisotropic p-Laplace equations
AU - Lu Chen
AU - Daniel Hauer
AU - Yabo Yang
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13226
UR - https://arxiv.org/abs/2610.13226
AB - In this paper, we prove global pointwise estimates on the decreasing rearrangement u^∗ and [H(∇ u)]^∗ of SOLAs u of an elliptic Poisson equation driven by the anisotropic p-Laplace operator Δ_p^H for given L^1-data and equipped with either homogeneous Dirichlet or Neumann boundary conditions. Here, we assume that the dimension n≥ 2, 1<p<n and H is norm on R^n. In the degenerate range 2 ≤ p <n, this gives an anisotropic extension and a full-interval completion of the rearrangement theory of Cianchi-Maz'ya [JEMS, 2014]. In particular, we obtain a new rearrangement estimate for gradients by decomposing the data for singular range 1<p<2, where the nonlinear interpolation method aren't available. Moreover, the endpoint weak-type gradient estimate is obtained for all 1<p<n. In the range p_*(n)<p<n with p_*(n):=(3n−2)/(2n−1), we prove a stability estimate for differences of gradients. These results extends the known ones in the literature to the singular range p_*(n)<p<2.
ER -