Hopf–Lax-Type Formula forut+H(u, Du)=0

An explicit formula is derived for the Hamilton Jacobi equationut+H(u, Du)=0 on (0, ∞)×Rnwithu(0, x)=g(x). The hamiltonianH(γ, p) is assumed to be nondecreasing inγ R1and convex and positively homogeneous of degree 1 inp Rn. The unique solution is given by[formula]wherehis a quasiconvex function given as the quasiconvex dual ofH, that is,h(q)=inf{γ R1:H(γ, p) p•q, p.