The Hopf-Lax formula gives the unique viscosity solution

Abstract

It is proved that the Hopf--Lax formula provides the unique viscosity solution of the Cauchy problem \begin{align*} u'_t(t,x)+H(u'_x(t,x)) & =0, \qquad(t,x)\in(0,T]\times {\bf R}^n,\\ \lim_{t\downarrow0} u(t,x) & =\varphi(x)\qquad\text{for all $x\in {\bf R}^n$.} \end{align*}