On a transmission problem for two systems of elastostatics with polygonal interface
Let \(D\) be a simply connected domain in \(\mathbb{R}^2\) whose boundary \(\partial D\) is a simple closed curvilinear polygon. The following interface problem \[ Lu= 0\quad\text{in } D,\quad \widetilde L u= 0\quad\text{in }\mathbb{R}^2- \overline D,\tag{1} \] \[ u^+- u^-= h\in H^1(\partial D),\;\Biggl({\partial u\over \partial\nu_L}\Biggr)^+- \Biggl({\partial u\over \partial \nu_{\widetilde L}}\Biggr)^-= \ell\in L^2(\partial D)\tag{2} \] is analyzed, where \(L\) is the Lamé operator with the Lamé's constants \(\lambda\), \(\mu\) that is \[ Lu= \mu \Delta u+ (\lambda+ \mu)\nabla\text{ div } u,\tag{3} \] and \(\widetilde L\) denotes an another Lamé operator whose Lamé's constants are \(\widetilde \lambda\), \(\widetilde \mu\). \({\partial u\over \partial\nu_L}\) is so-called traction its definition is \[ {\partial u\over \partial\nu_L}= \mu(\nabla u+ u\nabla) n+ \lambda(\text{div } u)n,\tag{4} \] the definition of \({\partial u\over \partial\nu_{\widetilde L}}\) is the same, but the \(\widetilde \lambda\), \(\widetilde\mu\) re used instead of \(\lambda\), \(\mu\). \((\cdots)^+\) and \((\cdots)^-\) denote respectively the values of \((\cdots)\) when approaching the interface \(\partial D\) from the interior and exterior of \(D\). The solution of (1), (2) is sought in the form \[ u(x, y)= S(f)(x, y),\;(x, y)\in D,\;u(x, y)= \widetilde S(g)(x, y),\;(x, y)\in \mathbb{R}^2- \overline D,\tag{5} \] where \[ S(f)= \int_{\partial D} \Gamma(x- \xi, y- \eta) f(\xi, \eta)d\sigma\tag{6} \] and \(\Gamma(x, y)\) is the fundamental solution of \(Lu= 0\). \(\widetilde S(g)\) is defined analogously to (6) using the fundamental solution of \(\widetilde L u= 0\). The authors prove that the following tracesystem on \(\partial D\) \[ (7)\quad(Sf)^+- (\widetilde S g)^-= h\text{ in } H^1(\partial D),\quad (8)\quad \Biggl({\partial S(f)\over \partial\nu_L}\Biggr)^+- \Biggl({\partial \widetilde S(g)\over \partial\nu_{\widetilde L}}\Biggr)^-= \ell\text{ in } L^2(\partial D) \] has a unique solution \((f, g)\) in \(L^2(\partial D)\times L^2(\partial D)\). The organization of the paper is as follows: In Section 1 the parametrization of curvilinear polygon \(\partial D\) and the formulation a system of boundary integral equations are given. Here, the boundary operator \(L^2(\partial D)\times L^2(\partial D)\to H^1(\partial D)\times L^2(\partial D)\), \[ (f, g)\to \Biggl((Sf)^+- (\widetilde S g)^-, \Biggl({\partial S(f)\over \partial\nu_L}\Biggr)^+- \Biggl({\partial\widetilde S(g)\over \partial\nu_L}\Biggr)^-\Biggr)\tag{9} \] is introduced. Section 2 gives the explicit form of (9) in a plane sector. In Section 3 the invertibility of (9) is proven.
scientific article; zbMATH DE number 838568
scientific article