In the Timoshenko shear model, we investigate the solvability of a geometrically nonlinear boundary value problem for arbitrary inhomogeneous isotropic shallow elastic shells with free edges. Our method is based on integral representations for generalized displacements containing arbitrary holomorphic functions. The holomorphic functions are found from some boundary conditions on the generalized displacements. We reduce the problem to a nonlinear operator equation for generalized displacements in the Sobolev space and, with the help of the principle of contraction mappings, establish its solvability.

中文翻译：

---

自由边缘的Timoshenko型各向同性浅壳非线性非线性边值问题的可解性问题

在季莫申科剪切模型中，我们研究了具有自由边缘的任意非均质各向同性浅弹性壳的几何非线性边界值问题的可解性。我们的方法基于包含任意全纯函数的广义位移的积分表示。从广义位移的某些边界条件中找到全纯函数。我们将问题简化为一个针对Sobolev空间中广义位移的非线性算子方程，并借助压缩映射原理建立其可解性。