Iterative Methods for Globally Lipschitz Nonlinear Laplace Equations

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 $F:\mathbb{C}\to \mathbb{C}$ 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.

31 pages, title changed, extending the results from Euclidean case to compact manifolds with boundary cases