A regularity result via fractional maximal operators for p-Laplace equations in weighted Lorentz spaces

ABSTRACT

We study the global regularity in weighted Lorentz spaces for the gradient of weak solutions to p-Laplace equations in an open-bounded non-smooth domain which satisfies a p-capacity condition. Moreover, the gradient estimate is presented in terms of fractional maximal operators which are related to the Riesz potential and the fractional derivative. Our main concern is to extend the results of Tran and Nguyen [New gradient estimates for solutions to quasi-linear divergence form elliptic equations with general Dirichlet boundary data. J Differ Equ. 2020;268(4):1427–1462] but in the weighted Lorentz spaces associated to the Muckenhoupt weights. The key technique that we use in this paper is the good-λ inequality and the interesting properties of the cut-off fractional maximal operators.

Keywords:

• p-Laplace equations
• quasi-linear elliptic equations
• divergence form
• gradient estimates
• p-capacity
• weighted Lorentz spaces

AMS Subject Classifications:

• 35J62
• 35J92

Disclosure statement

No potential conflict of interest was reported by the author(s).