Steklov-Poincaré operator for Helmholtz equation

The standard topological derivative methodology developed by the authors in many papers requires knowledge of point-wise values of solutions to partial differential equations or variational equalities. However, the contact or unilateral problems are studied in the energy space setting, wsznumer2005here point-wise values are not defined. The authors proposed the approach based on domain decomposition and expansion of the SteklovPoincare operator for dealing with such cases. This allows application of variational inequalities analysis on cones in in solution sets. To this end the point values of solutions are represented by regular enough perturbation of the bilinear form. The appropriate formulas were given for Laplace and elasticity operator in 2D and 3D problems. In this paper we extend the method on Helmholz equations and derive the formulas for the perturbation of the bilinear form in such a case.