Lᵖ error estimates for numerical approximation of convex Hamilton–Jacobi equations

We establish Lᵖ L p error estimates for monotone numerical schemes approximating convex Hamilton–Jacobi equations on the d -dimensional torus. Using the adjoint method and semiconcavity estimates, we first prove an L¹ L 1 error bound of order one for classical monotone schemes of Crandall–Lions type and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian and semiconcavity assumptions on the initial datum. By interpolation with the classical L^∞ L ∞ estimate, we obtain Lᵖ L p estimates for every 1≤ p<+∞ 1 ≤ p < + ∞ .