Mathematical Sciences: Operator Theory and Differential Equations

9403785戈尔茨坦该奖项支持算子理论泛函分析、微分方程式和数学物理等一般领域的数学研究。这项工作的重点是三个重叠领域的问题。第一个领域，量子理论，涉及自旋极化系统的Thomas-Fermi理论和Fermi-AMaldi修正，散射理论(包括弹性波的障碍物散射)，非相对论极限和mHartree-Fock理论。第二个领域，非线性偏微分方程组，涉及各向异性多孔介质方程、退化抛物方程(包括等离子体物理的Kompaneets方程)、Wentzell边界条件、反应扩散方程、Korteweg-deVreiss方程和高维色散方程、奇异守恒律和Euler-Poisson-DarBoux型因式分解方程。最后一个领域，算子半群，涉及Favard类和退化抛物问题，特征向量的极值刻画，奇异摄动和Hemion-CoSine函数。这些问题大多是适定的，但也有一些不适定的问题，包括反问题、非凸极小化和数值解。微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

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. ***