Geometric properties of solutions of Hamilton-Jacobi equations
The paper studies some geometric properties of the level sets of the solution of the following Cauchy problem for a Hamilton-Jacobi equation: \(u_ t+H(x,t,Du)=0\) in \(R^ n\times(0,\infty)\); \(u(\cdot,0)=u_ 0(\cdot)\) in \(R^ n\), employing the method of moving parallel planes, previously used for elliptic and parabolic PDE's [see e.g. \textit{B. Gidas, W.-M. Ni} and \textit{L. Nirenberg}, Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)]. The adaptation of this method to Hamilton-Jacobi equations is possible in the framework of the theory of viscosity solutions [see e.g. \textit{M. G. Crandall, L. C. Evans} and \textit{P. L. Lions}, Trans Am. Math. Soc. 282, 487-502 (1984)], using the comparison theorem for such solutions as a kind of maximum principle. Under suitable assumptions, the main of which are the radial symmetry of the Hamiltonian and nonnegative and compact supported initial data, it is shown that at some fixed time to every point of the boundary of a level set it is possible to attach two open truncated cones, one lying inside the level set and the other outside. This implies that the boundary of the level set is a Lipschitz surface. Under the additional hypothesis that the level sets are expanding it is shown that they become asymptotically spherical as \(t\to \infty\), the convergence taking place in the Lipschitz norm. Finally conditions are given under which the level sets are expanding.
scientific article; zbMATH DE number 3849674
scientific article