On the method of potential for a fourth-order elliptic equation arising in anisotropic elasticity theory (Q2713926)

The author studies the following boundary value problem which comes from the modelling of anisotropic properties of materials in the framework of elasticity theory: NEWLINE\[NEWLINE \begin{aligned} & \left(k^2\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)\Delta^k = 0, \quad k = 1, 2, \\ & u^k|_{\partial Q} = f_1(s),\quad \left.\frac{u^k}{\partial n}\right|_ {\partial Q} = f_2(s),\\ &f_1(s) \in C^{1,\alpha}(\partial Q),\quad f_2(s)\in C^{0,\alpha}(Q) \end{aligned} NEWLINE\]NEWLINE in a bounded simply connected domain \(Q\subset \mathbb{R}^2\) with Lyapunov boundary.NEWLINENEWLINENEWLINEThe aim of the article is to construct a system of Fredholm-type equations of the second order. This allows to prove an existence and uniqueness result for a solution to the problem in the Hölder class of functions. The author observes that passing to the limit in the equation under consideration allows to obtain an analogous system of the integral equations for the biharmonic equation.