Iterative Methods for Globally Lipschitz Nonlinear Laplace Equations

TL;DR

This paper presents an iterative approach to establish existence and uniqueness of complex nonlinear elliptic PDEs with Lipschitz nonlinearities on domains and manifolds, extending to integral formulations.

Contribution

The paper introduces a novel iterative method for solving complex nonlinear elliptic PDEs with Lipschitz nonlinearities on domains and Riemannian manifolds, including integral equation approaches.

Findings

Proved existence and uniqueness of solutions for specified PDEs.

Extended methods to Riemannian manifolds with boundary.

Discussed integral formulations using parametrix methods.

Abstract

We introduce an iterative method to prove the existence and uniqueness of the complex-valued nonlinear elliptic PDE of the form $ -\Delta u + F(u) = f $ with Dirichlet or Neumann boundary conditions on a precompact domain $ \Omega \subset \mathbb{R}^{n}$, where $ F : \mathbb{C} \rightarrow \mathbb{C} $ is Lipschitz. The same method gives a solution to $ - \Delta_{g} u + F(u) = f $ for these boundary conditions on a smooth, compact Riemannian manifold $ (M, g) $ with $ \mathcal{C}^{1} $ boundary, where $ - \Delta_{g} $ is the Laplace-Beltrami operator. We also apply parametrix methods to discuss an integral version of these PDEs.