Metric Entropy for Hamilton-Jacobi Equations with Uniformly Directionally Convex Hamiltonian

The present paper studies the bounded variation-type regularity for viscosity solutions of the Hamilton-Jacobi equation ut(t, x) + H ( Dxu(t, x) ) = 0, (t, x) (0,∞) × Rd, with a coercive and uniformly directionally convex Hamiltonian H. More precisely, we establish a BV bound on the slope of backward characteristics DH(Dxu(t, )) starting at a positive time t. Relying on the BV bound, we quantify the metric entropy in W1,1 loc ( Rd ) for the map St that associates, to every given initial data u0 Lip ( Rd ) , the corresponding solution Stu0. Finally, a counterexample is constructed to show that both Dxu(t, ) and DH(Dxu(t, )) fail to be in BVloc for a general strictly convex and coercive H ∈ C 2 ( Rd).