LIPSCHITZ DEPENDENCE OF VISCOSITY SOLUTIONS OF HAMILTON-JACOBI EQUATIONS WITH RESPECT TO THE PARAMETER

Let M be a closed and smooth manifold and H-epsilon : T* M -> R-1 be a family of Tonelli Hamiltonians for epsilon >= 0 small. For each phi is an element of C(M, R-1), T-t(epsilon)phi(x) is the unique viscosity solution of the Cauchy problem {d(t)w + H-epsilon(x, d(x)w) = 0, in M x (0, +infinity), w vertical bar(t=0) = phi, on M, where T-t(epsilon) is the Lax-Oleinik operator associated with H-epsilon. A result of Fathi asserts that the uniform limit, for t -> +infinity, of T-t(epsilon)phi + c(epsilon)t exists and the limit (phi) over bar (epsilon) is a viscosity solution of the stationary Hamilton-Jacobi equation H-epsilon (x, d(x)u) = c(epsilon), where c(epsilon) is the unique k for which the equation H-epsilon (x, d(x)u) = k admits viscosity solutions. In the present paper we discuss the continuous dependence of the viscosity solution (phi) over bar (epsilon) with respect to the parameter epsilon.