On Minimax and Idempotent Generalized Weak Solutions to the Hamilton–Jacobi Equation

This paper provides a direct equivalence proof for minimax solutions of A.I. Subbotin and generalized weak solutions in the sense of idempotent analysis. It is shown that the Hamilton-Jacobi equation Vt+H(t, x,DxV ) = 0 (with the Hamiltonian H(t, x, s) concave in s), considered in the context of minimax generalized solutions, is linear w.r.t. ⊕ = min and ̄ = +. This leads to a representation formula for minimax solutions of the Cauchy problem. Using this representation, lower semicontinuous minimax solutions are proven to be equivalent to idempotent generalized weak solutions. Besides, it is shown that for the non-autonomous Hamilton-Jacobi equation Vt + H(t,DxV ) = 0 the formula presenting minimax solutions transforms to an explicit formula, which generalizes the Lax-Olĕınik formula. Thus for continuous Cauchy data, the notions of minimax, viscosity and generalized weak solution to the Cauchy problem coincide.