Domain Derivatives for Energy Functionals with Boundary Integrals
This paper deals with domain derivatives of energy functionals related to elliptic boundary value problems. Emphasis is put on boundary conditions of mixed type which give rise to a boundary integral in the energy. A formal computation for rather general functionals is given. It turns out that in the radial case the first derivative vanishes provided the perturbations are volume preserving. In the simplest case of a torsion problem with Robin boundary conditions, the sign of the first variation shows that the energy is monotone with respect to domain inclusion for nearly circular domains. In this case also the second variation is derived.
KeywordsDomain derivative Optimality
Mathematics Subject Classification49Q10
The authors would like to thank the referee for having pointed out many misprints and a computational error in the second variation for the torsion problem in balls.