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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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