We establish the holomorphic dependence of the boundary integral operators (BIOs) comprising the Calderón projector for the Laplacian in two dimensions on the boundary shape. More precisely, we show that the Calderón projector, as an element of the Banach space of bounded linear operators satisfying suitable mapping properties, depends holomorphically on a set of boundaries given by a collection of \({\mathscr {C}}^2\)–regular Jordan curves in \({\mathbb {R}}^2\). In turn, this result implies that the solution of a well-posed first or second kind boundary integral equation (BIE) arising from the boundary reduction of the Laplace problem set on a domain of class \({\mathscr {C}}^2\) in two spatial dimensions depends holomorphically on the shape of the boundary, provided that the corresponding right-hand side does so as well. This property of shape holomorphy is of crucial significance to mathematically justify the construction of sparse parametric shape surrogates of polynomial chaos type, and to prove dimension-independent convergence rates for the approximation of parametric solution families of BIEs in forward and inverse computational shape uncertainty quantification.

我们建立了边界积分算子 (BIO) 的全纯依赖关系，这些算子构成了边界形状上二维拉普拉斯算子的 Calderón 投影仪。更准确地说，我们展示了 Calderón 投影仪，作为满足合适映射属性的有界线性算子的 Banach 空间的一个元素，全纯地依赖于一组由\({\mathscr {C}}^2\ ) - \({\mathbb {R}}^2\) 中的正则 Jordan 曲线。反过来，这个结果意味着一个适定的第一类或第二类边界积分方程 (BIE) 的解是由类\({\mathscr {C}}^2 \)在两个空间维度中，全纯取决于边界的形状，前提是相应的右侧也这样做。形状全纯的这种特性对于在数学上证明多项式混沌类型的稀疏参数形状替代品的构造的合理性，以及证明在正向和逆向计算形状不确定性量化中 BIE 参数解系列近似的维数无关收敛率具有至关重要的意义。