Mathematical Sciences: Non-linear Partial Differential Equations

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Caffarelli 9401168 This project continues mathematical research on several broad areas involving nonlinear partial differential equations. Of the two central themes, one concerns Lagrangian coordinates and allocation problems. In its physical interpretation, the work is concerned with the deformation or evolution of a material body. In particular, efforts will be made to understand how compression and rotation control the deformation. For instance, if the infinitesimal compression is controlled, what other quantity of the rotation must be controlled to assure that the deformation be smooth? This question, a Lagrangian version of the Helmholtz divergence-curl decomposition for a velocity or instantaneous deformation field, has arisen naturally in several areas where one must compute large deviations (computational plasticity, models for front formation in meteorological sciences, discretization of incompressible flows) as well as problems of allocation, where one tries to allocate resources into a new distribution which minimizes the allocation costs. A second theme concerns free boundaries and their singular perturbations. These arise in continuum mechanics when a rapid transition takes place in which the model assumes the presence of a sharp interphase (the burning of a flame, the solidification of a liquid, etc.). Studies of how well the sharp transition approximates the continuous phenomena will be carried out. For instance, the geometric properties of the sharp transition versus a narrow band of continuous transition, or when the limit of the continuous phenomena is actually the expected sharp transition will be analyzed. 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 gr owth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***