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We present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including the eikonal equation in geometric optics. where n G C{Q) and n{x) > 0 on fl. Here the Hamiltonian H{x,u,p) = \p\ — n{x) is independent of u, and so the uniqueness of a viscosity solution of the Dirichlet problem for (0.2) is not a direct consequence of the uniqueness theory mentioned above. It is usually deduced from the uniqueness theory after converting equation (0.2) into equation (0.1) with a strictly increasing Hamiltonian H{x,u,p) in u via a transformation of the unknown (a device of S. N. Kruzkov (9)). Indeed, if u is a viscosity solution of (0.2), then the function v = —e~u is a viscosity solution of n{x)v + \Dv\ — 0. We refer the interested reader to Kruzkov (9), and P. L. Lions (10) and (2) for the details of this approach. The main purpose here is to present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including (0.2) as a special case in the framework of viscosity solutions. §1 contains comparison results which yield the desired uniqueness theorem. An example of application of our comparison theorem is presented in §2. Our techniques are also useful in proving the rate of convergence of the vanishing viscosity method. This subject will be discussed in (7).

A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type