Mathematical Sciences: Nonlinear Partial Differential Equations"

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9306199 Phillips This project studies differential equations from two different areas, the calculus of variations related to nonlinear elasticity and the evolution of defects for nonlinear systems. Continuing work on the regularity of solutions to problems from nonlinear elasticity builds on recently established results showing that certain boundary value problems from two dimensional elasticity have solutions which are locally Lipschitz continuous homeomorphisms. Efforts will now be made to derive estimates for these solutions and to use them to address regularity questions for general boundary value problems. A particular class of elliptic variational problems called polyconvex will also be studied to determine gradient estimates for solutions to this type of problem and to investigate the possibility of isolated singularities in their solutions. The second line of work concerns smooth solutions to the parabolic Ginzburg-Landau system. For each fixed moment of time, the solution is viewed as a vector field over the spacial domain. A defect is defined as place where the field is null. The evolution of the defect pattern and how it depends on the nonlinear structure of the system is to be investigated. In particular the phenomena of creations, mutual annihilation and stable pattern formation of defects are to be studied. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***