Mathematical Sciences: Constrained Problems in the Calculus of Variations
{'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}
A three-year continuing grant is recommended in support of              mathematical models and methods related to problems in                          continuum physics.  The systems of differential equations which                 occur and the associated variational principles used to derive                  the equations are often unstable, incompletely posed or not                     well-posed.  Investigations of several aspects of these                         problems, requiring an assortment of methods, will be carried                   out.  Objectives, from the viewpoint of continuum theory,                       concern stability, phase transitions and defects which occur in                 ordered materials.  Primary examples appropriate to this work                   are liquid crystals and crystalline solids.  Mathematically,                    one wishes to obtain analytic interpretations of certain                        continuum phenomena, expressed in terms of constraints and                      invariance groups and the behavior of solutions for typical                     problems.  Of special interest is the change of constraints or                  groups, leading them to become the mechanisms of phase                          transitions.  Singular behavior may be exhibited as well in                     these circumstances.                                                                    Solutions of the static equilibrium configuration                       correspond to stationary points of the energy density.  They                    take the form of harmonic mappings into the sphere in three                     dimensions.  When boundary conditions are placed on the model,                  singularities can arise in solutions (called defects in the                     literature).  Defects play a central role in understanding the                  behavior of the material.  Major new theoretical work and                       initial numerical procedures have been developed by a number of                 researchers in the past two years.  The present work will                       continue analyzing the theory of defects and establishing                       validation of numerical procedures.  Additional work will focus                 on uniqueness of solutions and measurements of the size                         (Hausdorff dimension) of the singular set.                                              This research, part of a larger effort dedicated to the                 study of crystal structures, represents an impressive                           combination of mathematically sophisticated ideas and models of                 the physical world.  The steps taken by the principal                           investigator in clarifying some of the difficulties has been                    impressive.  Others are now being attracted to this area.                       Funding should be continued without question.
{'FirstName': 'David', 'LastName': 'Kinderlehrer', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'David Kinderlehrer', 'EmailAddress': 'davidk@andrew.cmu.edu', 'NSF_ID': '000204348', 'StartDate': '06/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Minnesota-Twin Cities', 'CityName': 'MINNEAPOLIS', 'ZipCode': '554552009', 'PhoneNumber': '6126245599', 'StreetAddress': '200 OAK ST SE', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Minnesota', 'StateCode': 'MN', 'CONGRESSDISTRICT': '05', 'CONGRESS_DISTRICT_ORG': 'MN05', 'ORG_UEI_NUM': 'KABJZBBJ4B54', 'ORG_LGL_BUS_NAME': 'REGENTS OF THE UNIVERSITY OF MINNESOTA', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1987&All=true', 'xml': '8706782.xml'}