New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data

2019/05/13 by Minh‐Phuong Tran, Tran, Minh-Phuong, Thanh‐Nhan Nguyen +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.1905.04891

Abstract

This paper studies a new gradient regularity in Lorentz spaces for solutions to a class of quasilinear divergence form elliptic equations with nonhomogeneous Dirichlet boundary conditions: \begincases div(A(x,∇ u)) amp;= div(|F|^p-2F) in Ω,
\hspace1.2cm u amp;= σ on ∂ Ω. \endcases where Ω⊂ ℝⁿ (n ≥ 2), the nonlinearity A is a monotone Carathéodory vector valued function defined on W^1,p₀(Ω) for p>1 and the p-capacity uniform thickness condition is imposed on the complement of our bounded domain Ω. Moreover, for given data F ∈ L^p(Ω;ℝⁿ), the problem is set up with general Dirichlet boundary data σ∈ W^1-1/p,p(∂Ω). In this paper, the optimal good-λ type bounds technique is applied to prove some results of fractional maximal estimates for gradient of solutions. And the main ingredients are the action of the cut-off fractional maximal functions and some local interior and boundary comparison estimates developed in previous works \cite55QH4, MPT2018, MPT2019 and references therein.

Citations

• The Wiener test and potential estimates for quasilinear elliptic equations
• Boundary regularity for quasilinear elliptic equations with general Dirichlet boundary data
• Solution of the Plateau problem for m-dimensional surfaces of varying topological type

Cited by

• Calderón-Zygmund gradient estimates for p-Laplace systems with BMO complex coefficients

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