a viscosity solution
Some important nonlinear PDEs simply have no smooth solution — the natural solution develops kinks where it is not differentiable, so 'plug it in and check' fails. The Hamilton-Jacobi equation and the equations of optimal control are like this. We need a notion of 'solution' that allows corners yet still picks out a single, correct answer. The viscosity solution is that notion: a clever weak definition, tailored to fully nonlinear first- and second-order equations, that delivers existence AND uniqueness even when classical derivatives do not exist.
The idea is disarmingly slick. To define a solution of F(x, u, grad u) = 0 without ever differentiating u, you test it against smooth functions that touch it. Say a smooth phi touches u from above at a point x_0 (phi >= u nearby, equal at x_0); then where the graph of u peeps below a smooth cap, demand F(x_0, u, grad phi) <= 0. Symmetrically, whenever a smooth phi touches u from below, demand F(x_0, u, grad phi) >= 0. A function satisfying both one-sided tests everywhere is a viscosity solution. The genius is that at a corner of u, smooth functions can only touch from one side, so only one inequality applies there — and that asymmetry is exactly what selects the physically right corner (the 'downward kinks' allowed, 'upward kinks' forbidden, say). The name comes from one way to justify it: add a tiny viscosity term, solve the smooth equation F = epsilon Laplacian u, and let epsilon go to 0 — the viscosity solution is the limit, the 'vanishing-viscosity' answer.
Why is it the right notion? Because it comes with a COMPARISON PRINCIPLE: if one viscosity subsolution sits below a viscosity supersolution on the boundary, it stays below everywhere, which immediately gives UNIQUENESS. This is the decisive advantage — many weak formulations admit infinitely many solutions, but the viscosity framework pins down exactly one, matching the value function of the control problem behind the equation. It is now the standard solution concept for Hamilton-Jacobi, Hamilton-Jacobi-Bellman, the eikonal equation, and many fully nonlinear elliptic equations. The honest caveat: it is the right notion for first-order and 'degenerate-elliptic' equations, not a universal cure-all — for divergence-form problems the weak (Sobolev) formulation is the natural one instead.
The eikonal equation |u_x| = 1 on the interval [-1, 1] with u(-1) = u(1) = 0 has two 'almost-everywhere' solutions: the tent u = 1 - |x| (a downward corner at 0) and its flip u = |x| - 1 (an upward corner). Only the tent is the viscosity solution — it is the distance-to-the-boundary function, the physically meaningful one — and the comparison principle rules the other out.
The weak notion that gives uniqueness for kinky Hamilton-Jacobi solutions.
A common confusion: a function that satisfies the equation almost everywhere is NOT automatically a viscosity solution, and there can be many such 'a.e. solutions'. The viscosity test (touching from above/below) is strictly stronger and is what restores uniqueness via the comparison principle.