Abstract or Additional Information
We consider boundary value problems which are Euler-Lagrange equations of certain energy-functionals. Important questions in this context are: How do they depend on the geometry of the domain on which they are defined? For instance, does the energy assume a minimum among all domains of given volume? How does the optimal region, if it exists, look like?
The technique of domain variations studies the changes of functionals under infinitesimal deformations. It is a differential calculus that allows to derive necessary conditions for the geometry of an optimal domain. Its beginnings go back to Hadamard in 1908, who calculated the first variation of Green's functions with Dirichlet boundary conditions. In this talk, the first and second variations of the energy of torsion problem with Robin boundary conditions will be discussed.