On Fredholm integral equations in two-dimensional anisotropic elasticity theory (Q2714025)

The author discusses the reduction of Fredholm integral equations which arise in two-dimensional anisotropic elasticity theory to the so-called Sherman equations for the second boundary value problem. The approach is based on the fact that the system of equations of anisotropic elasticity theory has simple complex characteristics. The boundary value problem has the form \((l_1^2\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) (l_2^2\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2}) = 0\), \(\frac{\partial w}{\partial x}|_{\partial Q} = \phi_1(s)\), \(\frac{\partial w}{\partial y}|_{\partial Q} = \phi_2(s)\), \(s\in\partial Q\), and is studied in a plane simply connected domain \(Q\) with Lyapunov boundary \(\partial Q\). Here \(w\) is the stress function, and \(l_k\) \((k=1,2)\) are positive constants. The following result is proven (which provides a key ingredient for studying the two-dimensional anisotropic elasticity theory):NEWLINENEWLINENEWLINELet \(\partial Q\in C^{1,\alpha}\), \(\psi_k(s)\in C^{0,\alpha} (\partial Q)\), \(k=1, 2\). In addition, assume that the solvability condition for the boundary value problem under consideration, \(\int_{\partial Q}(\phi_1(s)x'(s) + \phi_2(s)y'(s)) ds = 0\), is satisfied. Then the boundary value problem has a unique solution \(w\) (to within an additive constant) in the space \(H^2(Q)\cap C^{1,\alpha}(\partial Q)\), which can be represented as NEWLINE\[NEWLINE\begin{multlined} w = \operatorname{Re}\left\{\frac{1}{\pi i(\mu_1 - \mu_2)}\int_{\partial Q} \bigl(-\mu_1f_1(s) + f_2(s)\bigr)\ln(z_2 - t_2) dt_2\right. - \\ \left.\frac{1}{\pi i(\mu_1 - \mu_2)}\int_{\partial Q} \bigl(-\mu_2f_1(s) + f_2(s)\bigr)\ln(z_1 - t_1) dt_1\right\} + C, \end{multlined} NEWLINE\]NEWLINE where \(C\) is an arbitrary real constant.