scholarly journals Interior gradient estimates for quasilinear elliptic equations

2016 ◽  
Vol 55 (3) ◽  
Author(s):  
Truyen Nguyen ◽  
Tuoc Phan
Keyword(s):  
Elliptic Equations ◽  
Gradient Estimates ◽  
Quasilinear Elliptic Equations ◽  
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 Up-to-Boundary Pointwise Gradient Estimates for Very Singular Quasilinear Elliptic Equations with Mixed Data

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tan Duc Do ◽  
Le Xuan Truong ◽  
Nguyen Ngoc Trong
Keyword(s):  
Weak Solutions ◽  
Elliptic Equations ◽  
Mixed Data ◽  
Gradient Estimates ◽  
Quasilinear Elliptic Equations ◽  
Pointwise Estimates ◽  
Quasilinear Elliptic

Abstract This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data { - div ⁡ ( 𝐀 ⁢ ( x , D ⁢ u ) ) = g - div ⁡ f in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(\mathbf{A}(x,Du))&% \displaystyle=g-\operatorname{div}f&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} is sufficiently flat in the sense of Reifenberg.

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