NSF Org:	DMS Division Of Mathematical Sciences
Recipient:	REGENTS OF THE UNIVERSITY OF MICHIGAN
Initial Amendment Date:	June 16, 1989
Latest Amendment Date:	June 16, 1989
Award Number:	8902749
Award Instrument:	Standard Grant
Program Manager:	John V. Ryff DMS  Division Of Mathematical Sciences MPS  Directorate for Mathematical and Physical Sciences
Start Date:	June 15, 1989
End Date:	May 31, 1992 (Estimated)
Total Intended Award Amount:	$34,156.00
Total Awarded Amount to Date:	$34,156.00
Funds Obligated to Date:	FY 1989 = $34,156.00
History of Investigator:	Frederick Gehring (Principal Investigator) fgehring@umich.edu
Recipient Sponsored Research Office:	Regents of the University of Michigan - Ann Arbor 1109 GEDDES AVE STE 3300 ANN ARBOR MI  US  48109-1015 (734)763-6438
Sponsor Congressional District:	06
Primary Place of Performance:	DATA NOT AVAILABLE
Primary Place of Performance Congressional District:
Unique Entity Identifier (UEI):	GNJ7BBP73WE9
Parent UEI:
NSF Program(s):	CLASSICAL ANALYSIS
Primary Program Source:
Program Reference Code(s):
Program Element Code(s):	126100
Award Agency Code:	4900
Fund Agency Code:	4900
Assistance Listing Number(s):	47.049

ABSTRACT

Quasiconformal mappings evolved from studies of plane mappings whose infinitesimal distortions remained bounded within fixed limits. Such transformations were considered as the natural mathematical generalization of a conformal map (which infinitesimally maps circles onto circles). The theory has grown in several directions and dimensions, making important contact with nonlinear potential theory and Teichmuller theory of Riemann surfaces. This project is concerned with problems relating quasiconformal maps and the geometric behavior of solutions to degenerate elliptic partial differential equations. The work derives from a recent discovery that quasiconformal homeomorphisms mapping onto a ball have a stability property: in certain subsets of the domain, their distortion is globally controlled without regard to the geometry of the domain. Little is understood concerning domains which may be mapped onto a ball; work will be done to characterize such domains. This phenomenon was first observed in conformal maps where several deep results has subsequently been produced. A second line of investigation concerns the properties of supersolutions of the p-Laplace equation. These functions form the basis for a nonlinear potential theory. Two particular goals the study of possible fine topologies available which yield the best continuity results for such functions and to seek a boundary Harnack principle for domains other than a ball.

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