Rademachers Theorem

Definition

Consider the first order, scalar PDE \[ F(x, u, \nabla u) = 0,\quad x \in \Omega \subseteq \mathbb{R}^n, \] and the boundary condition \[ u(x) = \bar{u}(x),\quad x \in \partial \Omega. \] Then, a function $u$ is a generallized solution to the PDE if $u$ is Lipschitz continuous on the closure $\bar{\Omega}$, takes the given boundary condition and satisfies the first order equation at almost every point $x \in \Omega$.