Consider the following iterative procedure for solving the $p$-Laplace equation $\nabla \cdot (|\nabla u|^{p-2} \nabla u) = 0$ with fixed Dirichlet boundary data:
- $u_0$ is our initial guess, for example a harmonic function.
- $u_k$ solves the equation $\nabla \cdot (|\nabla u_{k-1}|^{p-2} \nabla u_k) = 0$.
One would hope that, at least under some conditions, $u_k$ would converge to the $p$-harmonic function with the correct boundary values.
If convenient, we may assume $p > 2$ or other similar condition. $p=2$ is trivial.
This type of iteration has certainly been used to solve other nonlinear equations, so it would be a surprise if nobody had thought to use it for the $p$-Laplace equation. My questions:
- Does this type of iterative scheme have a name or associated keywords?
- Are convergence results for this iteration and $p$-Laplace equation known? I am interested in the both the type of convergence and rate of convergence.
