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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
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