Wave Fronts for Hamilton-Jacobi Equations:¶The General Theory for Riemann Solutions in $\calR^n$

Abstract

The Hamilton-Jacobi equation describes the dynamics of a hypersurface in \(\). 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 fist main result of this paper is a general framwork for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framwork ist to unterstand 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 \(\)), 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.