Crandall–Lions uniqueness theorem

Remarks

The theorem is a consequence of a comparison principle: if $u$ is a bounded uniformly continuous subsolution and $v$ a bounded uniformly continuous supersolution with $u(\cdot, 0) \le v(\cdot, 0)$, then $u \le v$. The proof doubles the variables. One maximises $u(x, t) - v(y, s) - \abs{x - y}^2 / \varepsilon - \abs{t - s}^2 / \varepsilon$ over pairs of points, after adding small extra terms that force a maximum to exist on the unbounded domain. At the maximum point the two halves of the penalty serve as test functions for $u$ and for $v$, and the two inequalities of the definition are subtracted. The Lipschitz hypotheses on $H$ make the error terms tend to $0$ with $\varepsilon$, which gives a contradiction if $u - v$ had a positive supremum.

Example: for $H(x, p) = \abs{p}$ the hypotheses hold with $C = 1$. The unique bounded uniformly continuous viscosity solution is given by the Hopf–Lax formula $u(x, t) = \min_{\abs{y - x} \le t} g(y)$.

The viscosity sense is needed. For $H(x, p) = \abs{p}$ and $g = 0$, both $u = 0$ and $w(x, t) = \min(0, \abs{x} - t)$ (with $n = 1$) are Lipschitz and satisfy the equation almost everywhere. Only $u = 0$ is a viscosity solution: at a point $(0, t)$ the function $w - \varphi$ with $\varphi(x, s) = -s$ has a local minimum, while $\varphi_t + \abs{\varphi_x} = -1 < 0$.

Going deeper

For a student

Uniqueness is what makes viscosity solutions useful in control theory. To show that a value function equals a given formula, it is enough to check that both are viscosity solutions of the same Hamilton–Jacobi–Bellman equation with the same data. The same doubling-of-variables argument gives comparison for stationary equations such as $u + H(x, Du) = 0$, and, with more work, for second-order degenerate elliptic equations. Existence comes separately, for instance from the value function of a control problem or from the vanishing viscosity method.