Wave fronts for Hamilton-Jacobi equations: The general theory for Riemann solutions in ℝⁿ

The Hamilton-Jacobi equation describes the dynamics of a hypersurface in ℝⁿ. This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. These intersections define (n - 1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher-dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems. The first main result of this paper is a general framework for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framework is to understand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two-dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces embedded in ℝ³), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips. We also define elementary waves as Riemann solutions which possess a common group velocity. Our second main result, for elementary waves, is a complete characterization in terms of algebraic constraints on the data. When satisfied, these constraints allow a consistently defined closed form expression for the solution. We also give a computable characterization for the admissibility of an elementary wave which is inductive in the codimension of the wave, and which generalizes the classical Oleinik condition for scalar conservation laws in one dimension.