For optimization problems of domains in which elliptic boundary value problems are defined a solution is proposed. The treated problems are those to determine the domain that minimizes an objective functional of the state functions under the conditions that the coefficient functions of the partial differential equations and the boundary value functions in the elliptic boundary value problems have smoothness and a one-to-one correspondence with domain variation and that the volumes of the domains are limited. Domain variation is formulated with a speed field. The derivative of the objective functional is obtained as a linear form of a shape gradient function. The solution is formulated by using the gradient method in the functional space of the speed field with the linear form of the shape gradient function. The solution is implemented to analyze the speed field with regard to the deformation field of the linear elastic continuum formed in the objective domain applying the force in proprtion to the shape gradient function.