## The topological derivative of the Dirichlet integral under formation of a thin ligament.(Russian, English)Zbl 1071.35037

Sib. Mat. Zh. 45, No. 2, 410-426 (2004); translation in Sib. Math. J. 45, No. 2, 341-355 (2004).
The authors study a mixed problem for the Poisson equation $-\Delta_x u(h,x) = f(h,x)$ in a singularly-perturbed domain $$\Omega(h)$$ with smooth boundary and compact closure. The domain $$\Omega(h)$$ depends on a small parameter $$h \in (0,h_0]$$ and represents a joint of sets with different limit dimensions. The parameter $$h\in (0,h_0]$$ characterizes the geometrical shape of the domain $$\Omega(h) = \Omega\cup\Lambda_h$$, $$\Omega\subset\mathbb R^2$$, where $$\Lambda_h$$ is a curvilinear strip. The function $$f(h,x)$$ is assumed to be of the form $f(h,x) = \tilde f(h,x) + \begin{cases} f_\Omega(x) & \text{for } x\in\Omega, \\ f_{\Lambda}(\tau) & \text{for }x\in\Lambda(h), \end{cases}$ where $$f_{\Omega}$$ and $$f_{\Lambda}$$ are given functions on the domain $$\Omega$$ and on the axis $$\Gamma$$ of the ligament $$\Lambda$$, respectively. The function $$\tilde f$$ is a small residual which cannot be considered under asymptotic analysis.
The main aim is to construct and justify the asymptotics for a solution $$u(h,x)$$ as $$h\to + 0$$. Moreover, the authors determine the principal part of a decomposition of the so-called energy functional in the case when a thin ligament $$\Lambda$$ is created. The authors show that this result has an important application to the theory of shape optimization in the context of studying curved boundaries.

### MSC:

 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35J25 Boundary value problems for second-order elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs

### Keywords:

asymptotic; thin ligament; energy functional; shape optimization
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