Mathematical Sciences: Operator Theory and Differential Equations

该奖项支持的工作将集中在各种 数学主题从一个角度来看， 强调功能分析的技术和工具， 非线性算子半群 三个问题领域将 处理。 第一个是量子理论，涉及到 基态电子密度和托马斯- 费米理论处理自旋极化系统，交换 修正、逆问题和核理论。 非线性 偏微分方程表示自旋极化 广义原子的费米子理论。 将开展工作， 使化学势对的相互作用最小化。 的 能量泛函的欧拉-拉格朗日方程 耦合椭圆方程组，有待进一步分析。 算子半群的第二类研究 关于某些第一和第二类的Hille-Yosida型理论 阶半线性发展方程 工作将集中在 半线性的充要条件 方程不仅保证强连续半群 的算子解，而且还获得增长条件 用规定的泛函来表示。 相关工作将 关注继续调查平均分配 能量，逼近定理，遍历理论和两点 边界问题。 第三个领域，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。 方程是非线性退化抛物型边值问题 问题;工作的目的是确定何时 相应的抽象微分算子是耗散的。

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.