Controlling monotonicity of nonlinear operators

TL;DR

This paper systematically analyzes the monotonicity and growth control of nonlinear operators like the p-Laplacian, providing corrected estimates and extending results to Orlicz growth, with a focus on clear proofs and elementary methods.

Contribution

It offers a comprehensive, corrected collection of estimates for Leray--Lions' operators, including new results for Orlicz growth, emphasizing accessible proofs and elementary arguments.

Findings

Corrected and unified estimates for monotonicity and growth control.

Extension of results to operators with Orlicz growth.

Emphasis on elementary proof techniques.

Abstract

Controlling the monotonicity and growth of Leray--Lions' operators including the $p$-Laplacian plays a fundamental role in the theory of existence and regularity of solutions to second order nonlinear PDE. We collect, correct, and supply known estimates including the discussion on the constants. Moreover, we provide a comprehensive treatment of related results for operators with Orlicz growth. We pay special attention to exposition of the proofs and the use of elementary arguments.