# On boundary behavior of solutions to elliptic systems in disks

On boundary behavior of solutions to elliptic systems in disks 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# On boundary behavior of solutions to elliptic systems in disks

, Volume 9 (3) – May 13, 2019
15 pages

/lp/springer-journals/on-boundary-behavior-of-solutions-to-elliptic-systems-in-disks-gVrBCraHjk
Publisher
Springer Journals
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 PhysicsSpringer Journals

Published: May 13, 2019