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The concept of mapped coercivity for nonlinear operators in Banach spaces

Roland Becker, Malte Braack

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.

The concept of mapped coercivity for nonlinear operators in Banach spaces

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.
Paper Structure (17 sections, 20 theorems, 72 equations)

This paper contains 17 sections, 20 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Mapped coercive operators
  3. Coercive operators
  4. Linear mappings
  5. Nonlinear mappings
  6. Proof of existence of solutions
  7. Finite dimensional equations with coercive operators
  8. Finite dimensional equations with mapped coercive operators
  9. Proof of Theorems \ref{['lemma:1']}, \ref{['thm:2']} and Corollary \ref{['lemma:cond']}
  10. Duality maps
  11. Duality maps w.r.t. equivalent norms
  12. Projection properties of duality maps
  13. Sum of a linear and nonlinear operator
  14. Applications
  15. Semi-linear Poisson equation in $W^{1,p}(\Omega)$
  16. ...and 2 more sections

Key Result

Theorem 2.1

We assume that $A:\mathcal{X}\to\mathcal{X}'$ is weakly continuous (eq:H1) and coercive (eq:standardcoercivity). Then, equation (eq:1) has a solution.

Theorems & Definitions (46)

  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 36 more