We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting the corresponding boundary value problems as boundary integral equations, we prove that their solutions depend holomorphically upon perturbations of the arcs' parametrizations. These results are key to prove the shape (domain) holomorphy of the domain-to-solution maps for the associated boundary integral equations with applications in uncertainty quantification, inverse problems and deep learning.

翻译：我们针对由二阶偏微分方程在含多个有限长开弧的无界二维区域中导出的一般弱奇异与超奇异边界积分算子，建立了形状全纯性结果。在将相应边值问题重新表述为边界积分方程后，我们证明了这些方程的解关于弧参数化扰动的全纯依赖性。这些结果对于证明相关边界积分方程中域到解映射的形状（区域）全纯性至关重要，并在不确定性量化、反问题与深度学习领域具有应用价值。