Mathematical Sciences: Operator Theory and Differential Equations

数学科学：算子理论和微分方程

• 批准号: 9403785
• 负责人: Gisele Goldstein
• 金额: $ 5.14万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-01 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403785&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Operator; Theory; Differential

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. ***

9403785 Goldstein该奖项支持在算子理论泛函分析、微分方程和数学物理等一般领域的数学研究。这项工作主要关注三个重叠领域的问题。第一个领域是量子理论，涉及托马斯-费米理论和自旋极化系统的费米-阿马尔迪修正，散射理论（包括弹性波的障碍散射），非相对论极限和哈特里-福克理论。第二个领域是非线性偏微分方程，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、反应扩散系统、Korteweg-deVreiss方程和高维色散方程、奇异守恒定律和欧拉-泊松-达布克斯型因子方程。最后一个领域，半群算子，涉及法瓦德类和退化抛物问题，特征向量的极值刻画，奇异摄动和半余弦函数。这些问题大多是适定问题，但也涉及不适定问题，包括逆问题、非凸极小化问题和数值解。微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊