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August, 2021 A Gradient Estimate Related Fractional Maximal Operators for a $p$-Laplace Problem in Morrey Spaces
Thanh-Nhan Nguyen, Minh-Phuong Tran, Cao-Kha Doan, Van-Nghia Vo
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Taiwanese J. Math. 25(4): 809-829 (August, 2021). DOI: 10.11650/tjm/210202


In the present paper, we deal with the global regularity estimates for the $p$-Laplace equations with data in divergence form \[ \operatorname{div}(|\nabla u|^{p-2} \nabla u) = \operatorname{div}(|F|^{p-2} F) \quad \textrm{in $\Omega$}, \] in Morrey spaces with natural data $F \in L^p(\Omega;\mathbb{R}^n)$ and nonhomogeneous boundary data belongs to $W^{1,p}(\Omega)$. Motivated by the work of [M.-P. Tran, T.-N. Nguyen, New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data, J. Differential Equations 268 (2020), no. 4, 1427–1462], this paper extends that of global Lorentz–Morrey gradient estimates in which the `good-$\lambda$' technique was undertaken for a class of more general equations, and further, global regularity of weak solutions will be given in terms of fractional maximal operators.


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Thanh-Nhan Nguyen. Minh-Phuong Tran. Cao-Kha Doan. Van-Nghia Vo. "A Gradient Estimate Related Fractional Maximal Operators for a $p$-Laplace Problem in Morrey Spaces." Taiwanese J. Math. 25 (4) 809 - 829, August, 2021.


Received: 10 April 2020; Revised: 9 February 2021; Accepted: 22 February 2021; Published: August, 2021
First available in Project Euclid: 8 March 2021

Digital Object Identifier: 10.11650/tjm/210202

Primary: 35J62 , 35J92 , 46E30

Keywords: $p$-Laplace , fractional maximal operators , Gradient estimates , Lorentz–Morrey spaces , Quasilinear elliptic equation

Rights: Copyright © 2021 The Mathematical Society of the Republic of China


Vol.25 • No. 4 • August, 2021
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