Shapes,measures and elliptic equations

A measurable set - a shape - can be considered as a measure; the present thesis treats the inverse problem - to characterize those measures which can be considered as shapes, in a very generalized sense - by solving some optimal shape and optimal shape design problems which are governed by linear or nonlinear elliptic equations. A new method is introduced for solving the usual optimal shape problems, and also a new set of problems which are defined in terms of a pair of elements, a shape (defined by its boundary) and an optimal control associated with it. The problems are considered in polar and cartesian coordinates separately. The new method to attack these problems, which is applicable to both system of coordinates, consists in using the variational form of the elliptic equations and then applying the process of embedding into some appropriate spaces of measures; thus the problem is replaced by a measure-theoretical one in which one seeks to minimize a linear form over a subset of positive Radon measures defined by linear equalities. The optimal solution is approximated then by a finite combination of atomic measures so that the op-timal shape design problem is transformed into a finite linear programming problem. The solution of this problem is used to construct the optimal shape and its associated optimal control. The advantages of this new method with respect to other methods, and the existence of the optimal solution in each case, have been carefully considered.