Mathematical Sciences: Partial Differential Equations and   Geometric Analysis
{'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}
Much of the research sponsored by this award will focus on                 variational problems in mathematical analysis and the partial                   differential equations which derive when extrema are reached.                   The variational problems are normally reflections of some                       physical principle such as the minimization of action or energy.                     The proposal focuses on four areas, most of which represent                continuations of research currently in progress.  Elliptic                      variational problems represents one area.  The form of the                      associated differential equation is one of inhomogeneous                        divergence type.  It is formulated to reflect a variational                     problem involving an obstacle.  The equation has a forcing term                 which is a measure, or distribution, rather than a function.  A                 goal of this work is to determine the regularity properties of                  the solution and the geometry of the contact set of the obstacle                (which may not be known) in terms of assumed smoothness of the                  boundary of the obstacle.                                                            Parabolic variational problems will also be treated.  The                  obstacle problem will again be considered.  This class of                       problems is more difficult to analyze than the elliptic because                 there is a lack of symmetry between the space and time components               of the differential operator.  Still, some results from the                     elliptic case are expected to be applicable here.                                    Another variational question to be addressed is that of                    minimizing (integrals of) gradients of functions on domains in                  which the function's boundary values are specified.  A question                 of concern here is that of smoothness - there are examples of                   minimizing functions which need not be smooth across the interior               of the domain.  Work will be done to characterize when one can                  expect smooth extremal functions in this context.                                    The fourth area to be treated is that of non-linear                        potential theory.  This concerns a number of questions about a                  differential operator, the p-Laplacian, which has been the                      subject of considerable interest recently.  Solutions of the                    corresponding differential equations enjoy many of the properties               of harmonic functions.  Emphasis will be placed on understanding                the critical set where the gradient of solutions equals zero.
{'FirstName': 'William', 'LastName': 'Ziemer', 'PI_MID_INIT': 'P', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'William P Ziemer', 'EmailAddress': 'ziemer@indiana.edu', 'NSF_ID': '000333369', 'StartDate': '06/01/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Indiana University', 'CityName': 'BLOOMINGTON', 'ZipCode': '474057000', 'PhoneNumber': '3172783473', 'StreetAddress': '107 S INDIANA AVE', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Indiana', 'StateCode': 'IN', 'CONGRESSDISTRICT': '09', 'CONGRESS_DISTRICT_ORG': 'IN09', 'ORG_UEI_NUM': 'YH86RTW2YVJ4', 'ORG_LGL_BUS_NAME': 'TRUSTEES OF INDIANA UNIVERSITY', 'ORG_PRNT_UEI_NUM': 'YH86RTW2YVJ4'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8821904.xml'}