Iterative Methods for Globally Lipschitz Nonlinear Laplace Equations

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

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