Mathematical Sciences: Operator Theory and Differential Equations

该奖项支持的工作将侧重于从强调泛函分析技术和非线性算子半群工具的观点出发的各种数学主题。将解决三个问题领域。第一个是量子理论，涉及基态电子密度的性质和托马斯-费米理论的几个变体，涉及自旋极化系统、交换修正、逆问题和核理论。非线性偏微分方程表示广义原子的自旋极化托马斯-费米理论。要做的工作是使化学势对之间的相互作用最小化。能量泛函的欧拉-拉格朗日方程得到一个耦合椭圆系统，有待进一步分析。算子半群的第二个研究方向是关于一类一阶和二阶半线性演化方程的Hille-Yosida型理论。工作将集中于寻找半线性方程的充分必要条件，以保证算子解的强连续半群，并获得给定泛函解的增长条件。相关工作将涉及继续研究能量均分、近似定理、遍历理论和两点边界问题。第三个领域是非线性偏微分方程，涉及流体动力学和生物学中的模型方程。方程为非线性退化抛物型边值问题；这项工作的目的是确定相应的抽象微分算子何时是耗散的。

Work supported by this award will focus on a variety of mathematical topics approached from a point of view which emphasizes techniques from functional analysis and the tools of nonlinear operator semigroups. Three problem areas will be addressed. The first, quantum theory, involves properties of ground state electron densities and several variants of Thomas- Fermi theory dealing with spin polarized systems, exchange corrections, inverse problems and nuclear theory. Nonlinear partial differential equations represent the spin polarized Thomas-Fermi theory of generalized atoms. Work will be done to minimize the interaction of pairs of chemical potentials. The Euler-Lagrange equations for the energy functional yield a coupled elliptic system which remains to be analyzed. A second line of research on semigroups of operators concerns a Hille-Yosida type theory for certain first and second order semilinear evolution equations. Work will concentrate on finding necessary and sufficient conditions on semilinear equations to guarantee not only a strongly continuous semigroup of operator solutions but also to obtain growth conditions on solutions in terms of prescribed functionals. Related work will be concerned with continuing investigations into equipartition of energy, approximation theorems, ergodic theory and two-point boundary problems. The third area, nonlinear partial differential equations, concerns model equations arising in fluid dynamics and biology. The equations are nonlinear degenerate parabolic boundary value problems; the object of the work is to determine when the corresponding abstract differential operator is dissipative.