|Consider a first-order Hamilton-Jacobi equation
ut(t,x)+H(∇u(t,x))=0, x∈RN, t>0, (0.1)
with a strictly convex and coercive Hamiltonian H : RN → R. For every u ∈ W 1,1(RN , R), let Stu =. u(t,·) denote the unique viscosity solution of (0.1) with initial data u(0,·) = u. Having in mind the analysis recently developed for solutions to conservation laws [2-4], inspired by a question posed by Lax, we are interested in studying the compactifying effect of the operator St, at any fixed time t > 0, w.r.t the W1,1-topology. Namely, we wish to estimate the Kolmogorov ε-entropy in W1,1 of the image of bounded sets of initial data through the map St. We recall that, given a metric space (X, d), and a totally bounded subset K of X, we let Nε(K | X) denote the minimal number of sets in a cover of K by subsets of X having diameter ≤ 2ε, and define the Kolmogorov ε-entropy of K as Hε(K | X) =. log2 Nε(K | X). Entropy numbers play a central roles in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, as suggested by Lax, this concept could provide a measure of the order of “resolution” of a numerical method for (0.1).
Our main result in  shows that, for every fixed L,M > 0, letting C[L,M] denote the set of Lipschitz functions u : RN → R with Lipschitz constant L and with support contained in [−M,M]N, there holds
Hε ST (C[L,M ] ) | W 1,1 (RN , R) ≈ (1/εN ). (0.2)
Relying on fine properties of monotone operators we derive upper estimates on the ε-entropy of classes of semiconcave functions, which in turn yield upper estimates on Hε(ST(C[L,M])). Instead, lower bounds on Hε(ST (C[L,M])) are established in two steps. We first introduce a class of semiconcave functions SF defined as combinations of suitable bump functions, and with a combinatorial argument we provide an optimal lower estimate on the ε-entropy of such a class. Next, we prove a controllability result showing that any element of SF can be obtained, at any given time T > 0, as the value u(t,·) of a viscosity solution of (0.1), with initial data in C[L,M]. (Joint work with Fabio Ancona and Piermarco Cannarsa).
1. Ancona F., Cannarsa P., Nguyen K.T., Quantitative compactness estimates for Hamilton- Jacobi Equations, preprint (2013).
2. Ancona F., Glass O., Nguyen K.T., Lower compactness estimates for scalar balance laws, Comm. Pure Appl. Math. 65 (2012), no. 9, 1303-1329.
3. Ancona F., Glass O., Nguyen K.T., On compactness estimates for hyperbolic systems of conservation laws, preprint (2013).
4. De Lellis C., Golse F., A Quantitative Compactness Estimate for Scalar Conservation Laws, Comm. Pure Appl. Math. 58 (2005), no. 7, 989–998.|