The concept of mapped coercivity for nonlinear operators in Banach spaces

TL;DR

This paper introduces the concept of mapped coercivity for nonlinear operators in Banach spaces, providing a new framework for proving existence of solutions without requiring monotonicity, and demonstrates its application to complex PDEs.

Contribution

It presents a generalized coercivity concept for nonlinear operators, extending classical conditions, and applies it to semi-linear elliptic and Navier-Stokes problems.

Findings

Established existence results under generalized coercivity

Unified framework for linear and nonlinear coercivity

Applied to complex PDEs like Navier-Stokes

Abstract

We provide a concise proof of existence for nonlinear operator equations in separable Banach spaces. Notably, the operator is not assumed to be monotone. Instead, our main hypotheses consist of a continuity assumption and a generalized coercivity property. Mapped coercivity is a generalization of the usual coercivity property for nonlinear operators. In the case of linear operators, we recover linear coercivity and the traditional inf-sup condition. To illustrate the applicability of this general concept, we apply it to semi-linear elliptic problems and the Navier-Stokes equations.