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.
