Wave fronts for Hamilton-Jacobi equations: The general theory for Riemann solutions in \(\mathbb{R}^n\)
The authors consider a class of 1-homogeneous Hamiltonian functions \(H(p)\) and weak solutions, that is, Crandall-Evans-Lions viscosity solutions, for the Hamilton-Jacobi equation \(S_t+H(S_x)=0\) on the space-time \(\mathbb{R}^n\times\mathbb{R}^+\). The authors point out that propagation and bifurcation of discontinuities of such solutions can be studied by means of associated Riemann problems. Furthermore, they investigate those Riemann solutions forming an elementary wave pattern, thus generalizing the notion of shock waves and centered rarefaction waves of one-dimensional conservations laws. Among other things, there is an interesting remark about an application of Hamilton-Jacobi equations (different from the usual ones, like optics) to the manufacture of semiconductor chips.
Crandall-Evans-Lions viscosity solutions
scientific article; zbMATH DE number 1121172