##
**Lower-bound gradient estimates for first-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts.**
*(English)*
Zbl 1015.35031

The paper is concerned with viscosity solutions of first-order Hamilton-Jacobi equations
\[
u_t(x,t) + H(x,t,D_x u(x,t)) = 0
\]
in \(\mathbb{R}^n \times (0,T)\), with initial data \(u(\cdot,0) = u_0\). The two main questions discussed in the paper are: (i) upper and lower uniform estimates for \(D_xu\) and properties of \(D_x^2 u\) (in the sense of measures), and (ii) consequences of these estimates for the regularity of the level sets \(\Gamma_t^\alpha = \{ x \mid u(x,t) = \alpha\}\). Regarding the regularity of solutions, the author proves upper bounds for \(|D_x u|\) and finite speed of propagation under the natural assumptions \(|D_x H(x,t,p)|\leq C(1+|p|)\), \(|D_p H(x,t,p)|\leq C(1+|x|)\). If \(H\) is additional convex in \(p\), there is a also a lower estimate, \(|D_xu(x,t)|\geq \epsilon e^-\lambda t\), provided such an estimate holds at \(t = 0\). Similarly, semiconcavity of \(u\) (i.e. \(u\) plus a smooth function is concave) follows from corresponding properties for \(H\).

Regarding the regularity of level sets, the author notes that level sets of semiconcave functions have one-sided \(C^1\) supporting hypersurfaces and proves that under the assumptions outlined above, for a.e. \(\alpha\), the approximate level sets \(\Gamma^\alpha_t\) are locally Lipschitz except at residual sets of \(n-1\) dimensional Hausdorff measure zero. Examples are given that show that little more can be said about specific level sets \(\Gamma^\alpha_t\) and that non-Lipschitz singularities of level sets may develop due to the evolution.

The paper is written very clearly and contains simplified proofs of many well-known results on this problem.

Regarding the regularity of level sets, the author notes that level sets of semiconcave functions have one-sided \(C^1\) supporting hypersurfaces and proves that under the assumptions outlined above, for a.e. \(\alpha\), the approximate level sets \(\Gamma^\alpha_t\) are locally Lipschitz except at residual sets of \(n-1\) dimensional Hausdorff measure zero. Examples are given that show that little more can be said about specific level sets \(\Gamma^\alpha_t\) and that non-Lipschitz singularities of level sets may develop due to the evolution.

The paper is written very clearly and contains simplified proofs of many well-known results on this problem.

Reviewer: Hans Engler (Bonn)