Rate of Convergence in Periodic Homogenization of Hamilton–Jacobi Equations: The Convex Setting

Abstract

We study the rate of convergence of ${u^\varepsilon}$, as ${\varepsilon \to 0+}$, to u in periodic homogenization of Hamilton–Jacobi equations. Here, ${u^\varepsilon}$ and u are viscosity solutions to the oscillatory Hamilton–Jacobi equation and its effective equation

$$\left.\begin{array}{ll}{\rm (C)_\varepsilon}\qquad\begin{cases}u_t^{\varepsilon}+H\left(\frac{x}{\varepsilon}, Du^{\varepsilon}\right) = 0 \qquad & {\rm in} \, \mathbb{R}^{n} \times (0, \infty),\\ u^{\varepsilon}(x, 0) = g(x) \qquad & {\rm on} \, \mathbb{R}^{n},\end{cases}\end{array}\right.$$

and

$$\left.\begin{array}{ll}{\rm (C)} \qquad \begin{cases}u_t+\overline{H} \left(Du\right)=0 \qquad & {\rm in} \, \mathbb{R}^{n} \times (0, \infty),\\ u(x, 0) = g(x) \qquad & {\rm on} \, \mathbb{R}^{n},\end{cases}\end{array}\right.$$

respectively. We assume that the Hamiltonian H = H(y, p) is coercive and convex in the p variable and is ${\mathbb{Z}^n}$-periodic in the y variable, and the initial data g is bounded and Lipschitz continuous. Here, ${\overline{H}}$ is the effective Hamiltonian.

We prove that

• (i)
  $$u^{\varepsilon}(x, t) \geqq u(x, t)- C\varepsilon \quad {{\rm for all} \, (x, t)\in \mathbb{R}^{n} \times [0,\infty)},$$

  where C depends only on H and ${\|Dg\|_{L^\infty(\mathbb{R}^n)}}$ ;

• (ii) For fixed ${(x, t) \in \mathbb{R}^{n} \times (0, \infty)}$, if u is differentiable at (x, t) and ${\overline{H}}$ is twice differentiable at ${p = Du(x,t)}$, then

  $$u^\varepsilon(x, t) \leqq u(x, t) + \widetilde{C}_{p} t{\varepsilon} + C\varepsilon,$$

  provided that ${g \in C^{2}(\mathbb{R}^n)}$ with ${\|g\|_{C^{2}(\mathbb{R}^n)} < \infty}$. The constant ${\widetilde{C}_p}$ depends only on ${H, \overline{H}, p}$ and g. If g is only Lipschitz continuous, then the above inequality in (ii) is changed into ${u^{\varepsilon}(x, t) \leqq u(x, t) + C_{p} \sqrt{t\varepsilon} + C\varepsilon}$.

When n = 2 and H is positively homogeneous in p of some fixed degree ${k \geqq 1}$, utilizing the Aubry–Mather theory, we obtain the optimal convergence rate ${O(\varepsilon)}$

$$|u^{\varepsilon}(x, t)-u(x, t) | \leqq C\varepsilon \quad {{\rm for all}\, (x, t)\in \mathbb{R}^2\times [0, \infty).}$$

Here C depends only on H and ${\|Dg\|_{L^{\infty}(\mathbb{R}^2)}}$.

When n = 1, the optimal convergence rate ${O(\varepsilon)}$ is established for any coercive and convex H.

The convergence rate turns out to have deep connections with the dynamics of the underlying Hamiltonian system and the shape of the effective Hamiltonian ${\overline{H}}$. Some related results and counter-examples are obtained as well.