Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations

TL;DR

This paper establishes weighted norm inequalities for solutions to quasilinear equations on Reifenberg flat domains, enabling the resolution of existence problems for Riccati-type equations with gradient sources of arbitrary growth.

Contribution

It introduces Muckenhoupt-Wheeden type bounds for gradients of solutions on Reifenberg flat domains, addressing a key open problem in quasilinear PDEs.

Findings

Weighted norm inequality for gradients established

Existence of solutions for Riccati equations with arbitrary power growth proved

Extension of bounds to Reifenberg flat domains

Abstract

A weighted norm inequality of Muckenhoupt-Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.