Mathematical Sciences: Harmonic Analysis and Partial        Differential 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}
An analysis of qualitative properties of solutions of linear               and nonlinear partial differential equations will be made.   A                  portion of this effort will be aimed at the partial differential                operators themselves and the harmonic analysis which occurs                     naturally in this context.  The main emphasis will be on linear                 elliptic and parabolic equations with minimal smoothness                        conditions and on nonlinear evolution equations.                                     The genesis of this research lies in relatively recent work                of the principal investigator on basic boundary value problems                  for the Laplace operator on general domains whose boundaries may                have corners, edges and other features which appear in                          applications but present obstacles to mathematical analysis.                    Sharp maximal function estimates  were obtained for the Dirichlet               problem over a certain parameter range.  Similar results for the                Neumann problem were also obtained over a conjugate parameter                   range.  Work will proceed in an effort to find the appropriate                  analogous results for the general elliptic operator.  Equally                   important will be extensions of present theory in the direction                 of systems.  Although the two-dimensional case will work, the                   current theory fails for higher dimensional systems.                                 Work will also be done on the inhomogeneous Laplace equation               on nonsmooth domains.  In addition to the forcing function, zero                boundary values are prescribed.  It is known that averages of                   second derivatives cannot be estimated in this setting, but it                  may be sufficient to estimate gradients.                                             Nonlinear evolution equations in this project will be those                related to fundamental work of the principal investigator on                    parabolic equations involving only powers of the Laplacian.  The                object was to show that nonnegative solutions must always exist.                That being accomplished, work will now proceed to focus on the                  Laplace operator applied to powers of the unknown including                     changes of sign, the objective being that of showing existence of               variable-sign solutions and understanding their properties.                          Applications are expected to the general theory of                         differential equations in particular to potential theory.
{'FirstName': 'Bjorn E.', 'LastName': 'Dahlberg', 'PI_MID_INIT': 'J', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Bjorn E. J Dahlberg', 'EmailAddress': None, 'NSF_ID': '000383252', 'StartDate': '11/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Washington University', 'CityName': 'SAINT LOUIS', 'ZipCode': '63110', 'PhoneNumber': '3147474134', 'StreetAddress': 'ONE BROOKINGS DR', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Missouri', 'StateCode': 'MO', 'CONGRESSDISTRICT': '01', 'CONGRESS_DISTRICT_ORG': 'MO01', 'ORG_UEI_NUM': 'L6NFUM28LQM5', 'ORG_LGL_BUS_NAME': 'WASHINGTON UNIVERSITY, THE', 'ORG_PRNT_UEI_NUM': None}
{'Name': None, 'CityName': None, 'StateCode': None, 'ZipCode': None, 'StreetAddress': None, 'CountryCode': None, 'CountryName': None, 'StateName': None, 'CountryFlag': None, 'CONGRESSDISTRICT': None, 'CONGRESS_DISTRICT_PERF': '""'}
{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1987&All=true', 'xml': '8718011.xml'}