Wave Fronts for Hamilton-Jacobi Equations:¶The General Theory for Riemann Solutions in |
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Authors: | J Glimm HC Kranzer D Tan FM Tangerman |
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Institution: | Department of Applied Mathematics and Statistics, The University at Stony Brook, Stony Brook,?NY 11794-3600, USA. E-mail: glimm@ams.sunysb.edu, US
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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.
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.
Received: 9 September 1996 / Accepted: 22 April 1997 |
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