Geometric Aspects of Ambrosetti-Prodi operators with Lipschitz nonlinearities

TL;DR

This paper investigates the properties of Ambrosetti-Prodi operators with Lipschitz nonlinearities, showing that key features like the Lyapunov-Schmidt decomposition remain valid despite non-differentiability.

Contribution

It demonstrates that many properties of the smooth Ambrosetti-Prodi operator extend to the Lipschitz case, including a useful global decomposition for numerical purposes.

Findings

Critical set properties are preserved with Lipschitz nonlinearities.

A global Lyapunov-Schmidt decomposition remains available.

The operator retains useful structure despite non-differentiability.

Abstract

For Dirichlet boundary conditions on a bounded domain, what happens to the critical set of the Ambrosetti-Prodi operator if the nonlinearity is only a Lipschitz map? It turns out that many properties which hold in the smooth case are preserved, despite of the fact that the operator is not even differentiable at some points. In particular, a global Lyapunov-Schmidt decomposition of great convenience for numerical inversion is still available.