Hopf-Lax formula and generalized characteristics

We study some differential properties of viscosity solution for Hamilton - Jacobi equations defined by Hopf-Lax formula $u(t,x)=\min_{y\in \R^n} \big\{σ(y)+tH^*\big (\frac {x-y}{t}\big)\big \}.$ A generalized form of characteristics of the Cauchy problems is taken into account the context. Then we examine the strip of differentiability of the viscosity solution given by the function $u(t,x).$

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