ABSTRACT
In this chapter we choose three model problems to explore extensions of the theory of elliptic PDE on Lipschitz domains to some nonlinear operators. We will handle: the Navier-Stokes problem with small data (or large viscosity) using the Banach fixed point theorem; a nonlinear diffusion problem which can be rewritten in terms of a Lipschitz strongly monotone operator; and a reaction-diffusion problem with cubic reaction as an application of the Browder-Minty theorem. Two of these problems will motivate us to take a stroll through the dense forest of theorems involving the embedding of Sobolev spaces in Lp spaces.
