Mathematical Sciences: Applications of Harmonic and         Parabolic Measure to Problems in Function Theory and PartialDifferential Equations
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
This project involves mathematical research combining                      certain advances in measure theory with partial differential                    equations and classical function theory.  Problems in the                       latter area stem from several sources.  One is the Littlewood                   conjecture which predicts the mean value of the spherical                       derivature of polynomials and rational functions.  The                          conjecture has been shown to be sharp for rational functions                    but not for polynomials.  Work will be done in obtaining                        sharp constants in the rational case and working out the                        correct power of the polynomial's degree in the other.  These                   results are expected to derive from better estimates for                        harmonic measure.  These should follow from recent work on                      the dimension of sets which can support harmonic measure.                            Work on elliptic partial differential equations will be                    concerned with measuring the growth of solutions to the                         biharmonic equation in terms of the Riesz capacity of set                       where the gradient vanishes.  Sharp results for lower                           dimensions have been obtained.  In the area of parabolic and                    heat equations, the question of absolute continuity of the                      parabolic measure with respect to Lebesgue measure will be                      taken up. That this is not always the case has already been                     established for two-dimensional domains.  In this context,                      good necessary and sufficient conditions on the boundary for                    absolute continuity have been established.  The present work                    looks to extend the theory to higher dimensions.                                     Applications to the representation theory for solutions                    of parabolic equations will follow.
{'FirstName': 'John', 'LastName': 'Lewis', 'PI_MID_INIT': 'L', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'John L Lewis', 'EmailAddress': 'john@ms.uky.edu', 'NSF_ID': '000408967', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Kentucky Research Foundation', 'CityName': 'LEXINGTON', 'ZipCode': '405260001', 'PhoneNumber': '8592579420', 'StreetAddress': '500 S LIMESTONE', 'StreetAddress2': '109 KINKEAD HALL', 'CountryName': 'United States', 'StateName': 'Kentucky', 'StateCode': 'KY', 'CONGRESSDISTRICT': '06', 'CONGRESS_DISTRICT_ORG': 'KY06', 'ORG_UEI_NUM': 'H1HYA8Z1NTM5', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF KENTUCKY RESEARCH FOUNDATION, THE', 'ORG_PRNT_UEI_NUM': None}
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[{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}, {'Code': '1271', 'Text': 'COMPUTATIONAL MATHEMATICS'}]
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