Work will be done in four related areas of analysis. First, radial solutions of nonlinear partial differential equations will be sought. Existence and uniqueness of solutions for the ground state problem are of particular concern. Bifurcation problems for positive solutions of the Dirichlet problem will be investigated along with a study of critical dimensions for the polyharmonic operator. Asymptotic behavior and global structure for the generalized Emden's equation will be analyzed and multi-valued ground states for the mean curvature operator studied. Work on the foundations of thermodynamics and continuum mechanics focuses on the existence of entropy and internal energy, especially as a consequence of non-cyclic thermodynamical laws. A derivation of a generalized energy equation for weak first law thermodynamics is planned. Emphasis on shock conditions and shock structure in weak first law thermodynamics will characterize work on shock wave theory. The Joule-Thompson experiment will be viewed as a limiting case of shock wave theory. A continuation of a fundamental derivation of thermodynamics as deduced from mathematical principles will be pursued. This work relates to several important areas of mathematical analysis including nonlinear partial differential equations and global analysis. In addition applications to the study of surfaces of prescribed mean curvature, thermodynamics and chemical reactions lie close to this undertaking.
