Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

Minh-Phuong Tran; Thanh-Nhan Nguyen

doi:10.3934/dcds.2021043

Article Contents

2021, Volume 41, Issue 9: 4461-4476. Doi: 10.3934/dcds.2021043

This issue Previous Article Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L_∞ tensor coefficient under relaxed ellipticity condition Next Article Centralizers of partially hyperbolic diffeomorphisms in dimension 3

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

Minh-Phuong Tran^1, and
Thanh-Nhan Nguyen^2, ,

1.
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2.
Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

^* Corresponding author: Thanh-Nhan Nguyen
^* Corresponding author: Thanh-Nhan Nguyen

Received: September 30, 2020

Early access: March 2021

Published: September 2021

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Abstract

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations $ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $ is established via Wolff type potentials. It is worthwhile to note that the model case of $ \mathbb{A} $ here is the non-degenerate $ p $-Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case $ 1<p \le \frac{3n-2}{2n-1} $, where the data $ \mu $ on right-hand side is assumed belonging to some classes that close to $ L^1 $. Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that $ \Omega $ is sufficiently flat in the Reifenberg sense.

Keywords:

Singular quasilinear elliptic equations,
measure data,
pointwise gradient estimates,
Wolff's potential.

Mathematics Subject Classification: Primary: 35J62; 35J75; 35J92.

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References

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