The Hopf-Lax formula gives the unique viscosity solution
It is known that the Hopf-Lax formula \[ V(t,x) = \inf\left\{\varphi(\xi) +tL\left(\frac{x-\xi}{t}\right), \xi\in\mathbb{R}^n\right\}, \] which is sometimes called Lax-Olejnik formula (see, e.g., [\textit{D. B. Silin}, Monatsh. Math. 124, 343-364 (1997; Zbl 0895.49016)]), defines a viscosity solution to the initial-value problem for the Hamilton-Jacobi equation \[ u_t+H\left(u_x\right) =0, \quad \lim_{t\downarrow 0}u(t,x)=\varphi(x). \] Here \(L\) is the Fenchel conjugate function of the Hamiltonian \(H\). The main result of the paper is that under standard growth and smoothness conditions, a viscosity solution must satisfy the Hopf-Lax formula. Therefore, in particular, it is obtained that the viscosity solution is unique.
scientific article; zbMATH DE number 1836162