MA40203: Theory of Partial Differential Equations

We now turn from linear to nonlinear elliptic equations. While there are many ways to apply our linear techniques to such problems, we will focus on the celebrated (and beautiful) method of moving planes. The idea is to compare the function $u$ to its reflections about a hyperplane, and then to slide this hyperplane to a ‘critical position’ where the full power of the maximum principle can be applied.