Second Variation and its Application in a Domain Optimization Problem

Abstract

In this paper, the second order necessary conditions for optimality in domain optimization problems are studied. The second variation, corresponding to a boundary variation, of the solution to a boundary value problem is shown to exist and is given as the solution of a boundary value problem of the same type. The boundary data are shown to be given in terms of the solution and of the first variation of the solution. From these results, the second variation of the object function is calculated to derive the second order necessary conditions of Kuhn-Tucker type.