On a variational problem with unknown boundaries and the determination of optimal shapes of elastic bodies : PMM vol. 39, n 6, 1975, pp. 1082–1092

The optimization problem is considered for a partial differential equation of elliptic type. The boundary of the domain in which the equation is given emerges as the control function and is to be determined from the condition of the extremum of the integral of the solution of the boundary value problem. Seeking the extremals is reduced to solving a va national problem without differential constraints. Necessary conditions for optimality are obtained, and shapes of elastic bars possessing the maximum stiffness under torsion are found with their aid.