Work on the project will continue research on the qualitative properties of solutions and solution operators of linear and nonlinear partial differential equations. Emphases will be on linear elliptic and on nonlinear evolution equations with minimal smoothness conditions. By minimal smoothness conditions one means either conditions on the equation, on the domain or both. From a practical point of view, information about solutions on nonsmooth domains is most often sought. Only recently have certain boundary value problems of elliptic type defined on domains with boundaries, with edges and corners been completely resolved. That is, one can predict the smoothness of the solutions at the boundary. Several important problems suggest themselves and will be the focus of work done on this project. They include questions of extentions to the general elliptic equation, elliptic systems and higher order elliptic equations. Some success has alresdy been achieved with fourth order equations. While the elliptic equations on nonsmooth domains are beginning to be understood, the corresponding evolution equations (involving time derivatives) remains a mystery. The ingredient missing is an inequality relating gradients with normal derivatives, usually classified as inequalities of Rellich type. The best results to date depend on positivity of solutions of classical evolution equations. Objectives in analyzing this class of solutions focus more on questions of long term behavior, boundedness and uniqueness. Some computational approximations may be employed in these studies.
