Weighted Lorentz Gradient Estimates for a Class of Quasilinear Elliptic Equations with Measure Data

2020 ◽  
Author(s):  
Fengping Yao
Keyword(s):  
Elliptic Equations ◽  
Gradient Estimates ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Quasilinear Elliptic

scholarly journals Lorentz–Morrey global bounds for singular quasilinear elliptic equations with measure data

2019 ◽  
Vol 22 (05) ◽  
pp. 1950033 ◽  
Author(s):  
Minh-Phuong Tran ◽  
Thanh-Nhan Nguyen
Keyword(s):  
Elliptic Equations ◽  
Gradient Estimates ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Singular Case ◽  
Global Estimate ◽  
Quasilinear Elliptic Problem ◽  
Text Type ◽  
Local Gradient ◽  
Quasilinear Elliptic

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: [Formula: see text] in Lorentz–Morrey spaces, where [Formula: see text] ([Formula: see text]), [Formula: see text] is a finite Radon measure, [Formula: see text] is a monotone Carathéodory vector-valued function defined on [Formula: see text] and the [Formula: see text]-capacity uniform thickness condition is imposed on the complement of our domain [Formula: see text]. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann. 346 (2010) 571–627] at least for the case [Formula: see text], where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl. 102 (2014) 99–123] for regular case [Formula: see text]. Here in this study, we particularly restrict ourselves to the singular case [Formula: see text]. The results are central to generalize our technique of good-[Formula: see text] type bounds in the previous work [M.-P. Tran, Good-[Formula: see text] type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal. 178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results.

scholarly journals Global W1,p(·) estimate for renormalized solutions of quasilinear equations with measure data on Reifenberg domains

2018 ◽  
Vol 7 (4) ◽  
pp. 517-533 ◽  
Author(s):  
The Anh Bui
Keyword(s):  
Elliptic Equations ◽  
Variable Exponent ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Renormalized Solutions ◽  
Lebesgue Spaces ◽  
Quasilinear Equations ◽  
Bmo Coefficients ◽  
Reifenberg Flat ◽  
Quasilinear Elliptic

AbstractIn this paper, we prove the gradient estimate for renormalized solutions to quasilinear elliptic equations with measure data on variable exponent Lebesgue spaces with BMO coefficients in a Reifenberg flat domain.

scholarly journals Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: 1 < p ≤ 3n−2 2n−1

2020 ◽  
pp. 2050056
Author(s):  
Le Cong Nhan ◽  
Le Xuan Truong
Keyword(s):  
Global Regularity ◽  
Gradient Estimates ◽  
Quasilinear Elliptic Equations ◽  
Measure Data ◽  
Singular Case ◽  
Regularity Estimates ◽  
Monotonicity Conditions ◽  
Vector Valued Function ◽  
Quasilinear Elliptic ◽  
Vector Valued

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form [Formula: see text] where [Formula: see text] is a finite signed Radon measure in [Formula: see text], [Formula: see text] is a bounded domain such that its complement [Formula: see text] is uniformly [Formula: see text]-thick and [Formula: see text] is a Carathéodory vector-valued function satisfying growth and monotonicity conditions for the strongly singular case [Formula: see text]. Our result extends the earlier results [19,22] to the strongly singular case [Formula: see text] and a recent result [18] by considering rough conditions on the domain [Formula: see text] and the nonlinearity [Formula: see text].

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