Mathematical Sciences: Operator Theory and Differential Equations

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9403785 Goldstein This award supports mathematical research in the generalarea of operator-theoretic functional analysis, differential equations and mathematical physics. The work focuses on problems from three overlapping areas. The first area, quantum theory, involves Thomas-Fermi theory with the Fermi-Amaldi correction for spin polarized systems, scattering theory (including obstacle scattering for elastic waves), nonrelativisitic limits and mHartree-Fock theory. The second area, nonlinear partial differential equations, involves the anisotropic porous medium equation, degenerate parabolic equations (including the Kompaneets equations of plasma physics), the Wentzell boundary condition, reaction-diffusion systems, the Korteweg-deVreiss equation and higher dimensional dispersive equations, singular conservation laws and factored equations of Euler-Poisson-Darboux type. The final area, semigroups of operators, involves Favard classes and degenerate parabolic problems, extremal characterizations of eigenvectors, singular perturbations and hemion-Cosine functions. The problems are mostly well-posed, but also of concern are ill-posed problems, including inverse problems, nonconvex minimization and numerical solutions. 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. ***