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.


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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