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Green‘s Function and Poisson Integral in a Circle Disk for Strongly Elliptic Systems with Constant Coefficients
- Publisher
- Springer Journals
- Copyright
- Copyright © 2019 by Springer Nature Switzerland AG
- Subject
- Mathematics; Analysis; Mathematical Methods in Physics
- ISSN
- 1664-2368
- eISSN
- 1664-235X
- DOI
- 10.1007/s13324-019-00310-0
- Publisher site
- See Article on Publisher Site
Abstract
We extend a classical result about weighted averages of harmonic functions to solutions of second-order strongly elliptic systems of PDE with constant coefficients in disks in the complex plane. It is well known that a non-tangential cluster set of the (harmonic) Poisson integral with a given piecewise continuous boundary function f at every point $$\zeta $$ ζ in the unit circle is the segment joining the left- and right-hand side limits of f at $$\zeta $$ ζ being taken along the unit circle. Using the recently obtained Poisson-type integral representation formula for solutions of aforementioned systems, we establish an analogous result about weighted averages for solutions of such systems. Furthermore, we illustrate the nature of the obtained results by presenting some special mappings of the unit disk by solutions with piecewise constant boundary data.
Journal
Analysis and Mathematical Physics – Springer Journals
Published: May 13, 2019