Hadamard's variational formula for the Bergman kernel
This article (and a number of subsequent works of the author) is concerned with the dependence of the Bergman kernel function \(K_ D(z,w)\) on a domain \(D\subset \subset {\mathbb{C}}^ N\). Let \(D_ 0=\{z\in {\mathbb{C}}^ N,\quad r(z)<0\},\) \(N\geq 1\), be a domain with defining function \(r\in C^{\infty}({\mathbb{C}}^ N)\). For a smooth, small, real-valued function \(\rho\) on \(\partial D_ 0\) there exists a domain \(D_{\rho}\) such that \(\partial D_{\rho}=\{z+\rho(z)\nu(z);\quad z\in \partial D_ 0\},\) where \(\nu \in {\mathbb{C}}^ N\) is the outward normal to \(\partial D_ 0\) at \(z\in \partial D_ 0\). The so called (first) Hadamard variation is defined as \(\delta K_{D_ 0}(z,w)=\frac{d}{d\epsilon}K_{D_{\epsilon \rho}}(z,w)|_{\epsilon =0}.\) In the case \(n=1\) it was observed by \textit{M. Schiffer} [Duke Math. J. 13, 529-540 (1946; Zbl 0060.237); see also \textit{S. Bergmann} and \textit{M. Schiffer}, ibid. 14, 609-638 (1947; Zbl 0029.39901) and ``Kernel functions and elliptic differential equations in mathematical physics'' (1953; Zbl 0053.390)] that \[ -\delta K_{D_ 0}(z,w)=\int_{\partial D_ 0}K_{D_ 0}(z,\xi)K_{D_ 0}(\xi,w)\rho(\xi)dS^ 0(\xi) \] where \(dS^ 0(\xi)\) denotes the volume element of \(\partial D_ 0\) at \(\xi\). In the present paper the author proves that the above formula is valid for every strictly pseudoconvex domain.
scientific article; zbMATH DE number 3849607
scientific article