The Topological Derivative of the Dirichlet Integral Under Formation of a Thin Ligament

We construct and justify the asymptotic expansion of a solution and the corresponding energy functional of the mixed boundary-value problem for the Poisson equation in a domain with a ligament, i.e., thin curvilinear strip connecting two small parts of the boundary outside the domain. Asymptotic analysis is required in the theory of shape optimization; therefore, in contrast to other publications, we use no simplifying assumptions of the flattening of the boundary near the junction zones.