Researchers: Anders Lindquist in cooperation with C. I. Byrnes (Washington University, St Louis).
Sponsors: The Swedish Research Council (VR) and the Göran Gustafsson Foundation.
Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. We have studied ``well-connected'' pairs of such problems which are not necessarily related by critical point considerations. We have also studied constrained problems of the kind which arise in mathematical programming as well as constraints of a geometric nature where a solution is sought on a leaf of a foliation. In these cases we are interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. In [A4] have proved a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result has been illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers' equation, by a geometric analysis of the existence of interior points for linear programming problems, and by a derivation of the existence of positive definite solutions of matrix Riccati equations without first analyzing the nonlinear matrix Riccati differential equation.