Research

Since receiving the Ph.D. in Mathematics at the University of Illinois as an NSF Graduate Fellow, Ralph authored or co-authored over a hundred research articles and three research monographs; he supervised 22 Ph.D. dissertations. His research was supported by the National Science Foundation 1972-2000, the Office of Naval Research 1988-1991, and by the Department of Energy in 2005-2010, including the DOE Multiscale Pacific Northwest Consortium. His research interests include singular or degenerate nonlinear evolution equations and partial differential equations, related variational inequalities and free-boundary problems, and applications to initial-boundary-value problems of mechanics and diffusion. His current work is on the analysis of multi-scale models of coupled fluid-solid dynamics, flow in deformable porous media, and models of plasticity. 

Among his technical contributions are the development of existence-uniqueness-regularity theory for pseudo-parabolic and Sobolev-type partial differential equations, existence theory of degenerate evolution equations, particularly the doubly-nonlinear cases and nonlinear systems in mixed form. More applied contributions include the formulation and existence theory for Stefan free-boundary problems for a parabolic system, for the pseudo-parabolic equation, and for the hyperbolic telegraphers' equation, the phase-change problem of advection of methane in the hydrate zone, Biot systems of poroelasticity and related contact mechanics, and the coupled Biot-Stokes system. He introduced the fissured medium equation and the layered medium equation as models for diffusion in heterogeneous media and contributed to the development of the Darcy-Brinkman model of diffusion, and the development of distributed systems of partial differential equations with microstructure and those with hysteresis.