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Differential equations on complex manifolds. (English) Zbl 1335.32029
Kielanowski, Piotr (ed.) et al., Geometric methods in physics. XXXIII workshop, Białowieża, Poland, June 29 – July 5, 2014. Cham: Birkhäuser/Springer (ISBN 978-3-319-18211-7/hbk; 978-3-319-18212-4/ebook). Trends in Mathematics, 273-292 (2015).
The article is an overview on the techniques used for solving high-order partial differential equations in \(\mathbb C^n\). More precisely, the authors consider the following Cauchy problem \[ \sum_{|\alpha|\leq m}a_\alpha D^\alpha u=f\quad\text{in}\quad\mathbb C^{n},(*) \] such that \(m\) is the multiplicity of zero of \(u(\cdot)\) on \(X\), a hypersurface in \(\mathbb C^n\), the \(a_\alpha\) are complex numbers, \(f\) is an analytic function, \(D\) is the classical Newton-Leibniz operator, and \(\alpha,\beta\) are the standard multi-indices. The authors provide two concrete examples for \(n\in\{1,2\}\). For solving \((*)\) one needs to use \(F\)-transforms (resp. \(R\)-transforms), appropriate integral transforms for homogeneous (resp. inhomogeneous) functions introduced by Sternin and Shatalov. Definitions and properties of these integral transforms are given the second section. For example, an \(F\)-transform acts on homogeneous analytic functions satisfying certain conditions, and an expression for it is given in terms of an integral over a particular homology class, where the integrand is expressed in terms of the topological Leray residue of a differential form on the set of vanishing phase functions. An example of an \(R\)-transform is supplied. The fourth section focuses on an application to the problem of balayage inwards for a domain \(\Omega\). Also, this last section focuses on explicit examples for \(\Omega\). The article is equipped with a few delightful pictures.
For the entire collection see [Zbl 1329.00214].
MSC:
32W50 Other partial differential equations of complex analysis in several variables
58J32 Boundary value problems on manifolds
35E99 Partial differential equations and systems of partial differential equations with constant coefficients
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