Comparison and uniqueness results are obtained for viscosity solutions of Hamilton-Jacobi equations. The main objective is the characterization of the value function associated with a variational problem of the Bolza type This is accomplished, in particular, in the presence of certain conditions reminiscent of the classical Tonelli conditions.

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition L(t, x, v) ≥ 1 2 m(|v| 2 −ω 2 |x| 2 )−C where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S t +H (t, x, ∇S) = 0 in Q T = (0, T )×R n , S| t=0 = S 0 , without any conditions at infinity on the solution.Here ωT < π/2. To the Hamilton-Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity-measure solution (S, ρ) ofThis system arises in the WKB method. The measure solution is defined by means of the Filippov flow of ∇S.

Let L(x, v) be a Lagrangian which is convex and superlinear in the velocity variable v, and let H (x, p) be the associated Hamiltonian. Conditions are obtained under which every viscosity solution u ∈ C((0, T ] × R n ) of the Hamilton-Jacobi equationis an action function in the large, i.e., u(t, x) = inf{u(0, X(0)) + t 0 L(X(τ ),Ẋ(τ )) dτ : X ∈ W 1,1 (0, t; R n ), X(t) = x} for all (t, x) ∈ (0, T ] × R n .

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This paper revisits the duality between differentiability and strict or strong convexity under the Legendre-Fenchel transform f → f * . Functions f defined on a Banach space X are considered. For a lower semicontinuous but not necessarily convex function f we relate essential Fréchet differentiability of the conjugate function f * to essential strong convexity of f .