a fully nonlinear PDE

At the far end of the nonlinearity ladder sits the hardest class: fully nonlinear PDEs, where even the highest-order derivatives enter nonlinearly. There is no linearity left to lean on anywhere — the top derivatives can be squared, multiplied together, or buried inside a function. If linear equations are gentle, semilinear ones tame at the top, and quasilinear ones linear-at-the-top-but-coupled, then fully nonlinear equations are the genuinely untamed frontier.

The defining test is what happens to the leading derivatives. In a fully nonlinear equation they appear through a nonlinear function — for example (u_x)^2 + (u_y)^2 = 1, the eikonal equation, where the first derivatives are squared, or the Monge-Ampere equation u_xx u_yy - (u_xy)^2 = f, where second derivatives are multiplied together. Contrast this with quasilinear, where the very same top derivatives would appear only to the first power. Because the principal part is no longer linear, you cannot read off a simple type or apply superposition or transform methods; the classical recipes mostly break down.

These equations are not exotic curiosities — they govern geometric optics and wavefronts (the eikonal equation), optimal transport and surfaces of prescribed curvature (Monge-Ampere), and optimal control and finance (the Hamilton-Jacobi-Bellman equation). Making sense of solutions often requires giving up on classical, everywhere-differentiable solutions altogether and adopting weaker notions such as viscosity solutions, a framework built precisely to handle fully nonlinear first- and second-order equations where derivatives may not exist in the ordinary sense but a robust, well-posed solution concept still survives.