Existence of a solution for complete least squares optimal shape problems

We show the existence of a solution for the complete least squares formulation associated with an elliptic optimal shape problem which appears in semiconductor device theory. The traditional approach for optimal shape problems is to minimize a cost functional subject to a number of governing equations. In the complete least squares formulation, which is considered in this paper, the cost functional and terms associated with the governing equations are all part of a quadratic functional, to be minimized over a very little constrained set.