Mathematical Sciences: Free Boundary Problems, Fully        Nonlinear Equations, Nonlinear Parabolic Equations, and the Navier-Stokes Equation
{'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}
The range of questions to be dealt with in this research                   covers broad fundamental investigations into the nature of                      solutions of partial differential equations.  The equations arise               from physical as well as geometrical sources and, for the most                  part, are nonlinear.  One class of equations to be treated are                  known as fully nonlinear equations.  They describe basic                        phenomena of geometry, elasticity and physics.  Typical of these                is the Monge-Ampere equation which effectively describes the                    curvature of a domain.  The more interesting, and difficult                     question is that of deciding whether a given function can be the                curvature of a domain.  Work is planned on characterizing                       interior smoothing properties and related stability properties of               solutions.                                                                           Another line of investigation will continue work on free                   boundary problems which represent physical quantities whose                     properties change discontinuously at some level of the unknown,                 such as in multi-phase systems.  Here the boundary between phases               is a dynamic part of the problem and must be found along with                   solutions of the attendant differential equations.  Work here                   will concentrate on geometric properties of the interphase and                  efforts will continue in seeking a unified theory coupled with                  minimal surface regularity theory.  A singular perturbation                     approach will be employed.                                                           Bridging the two main themes of this project will be a study               of generalized surfaces for which an elementary symmetric                       function of the curvature is prescribed.  Results of this nature                provide insight into various envelopes associated with arbitrary                sets such as the convex or minimal area hulls.                                       By-products of this work are expected to have extensive                    applications to the physical and mathematical sciences.
{'FirstName': 'Luis', 'LastName': 'Caffarelli', 'PI_MID_INIT': 'A', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Luis A Caffarelli', 'EmailAddress': 'caffarel@math.utexas.edu', 'NSF_ID': '000387147', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Institute For Advanced Study', 'CityName': 'PRINCETON', 'ZipCode': '085404952', 'PhoneNumber': '6097348000', 'StreetAddress': '1 EINSTEIN DR', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'New Jersey', 'StateCode': 'NJ', 'CONGRESSDISTRICT': '12', 'CONGRESS_DISTRICT_ORG': 'NJ12', 'ORG_UEI_NUM': 'MNMJWSC5Z4H1', 'ORG_LGL_BUS_NAME': 'INSTITUTE FOR ADVANCED STUDY - LOUIS BAMBERGER & MRS FELIX FULD FOUNDATION', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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