Weakly Elliptic Systems with Obstacle Constraints III. Complex Eigenvalues and Singular Systems

Using traditional methods, especially that of eigenfunction expansions combined with the penalty method, sharp conditions are given for the existence of solutions to general weakly elliptic linear N×N second order systems subjected to obstacle type constraints. The result applies to systems with defining coefficient matrices A and B having complex eigenvalues as well as to the case of singular systems—a situation not well treated in the literature. These conditions are placed on the matrix pencil Aλ−B. The effectiveness of the result is then tested on some examples, first in control theory and then for the systems that arise in Part II of this series (D. R. Adams, 2000, J. Geom. Anal.10, 375–412).