Free Boundary Problems for Systems: Toward a Comprehensive Theory of Computation

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This proposal investigates a new class of systems, formulated in terms of time dependent and stationary partial differential equations. The systemshave inherent switching properties, leading to their interpretation as freeboundary problems. What defines them as new is their characterization assystems rather than scalar equations, underscoring interlocking, complexdependencies. This work, part of which is joint with Siegfried Carl,extends the work of many mathematicians who have worked on free boundaryproblems for scalar equations during the last four decades. Systems of free boundary problems are associated with such diverse models aschange of phase, semi-classical band gap theories, threshold switching, and combustion. Using significant models for illustration, we employa multi-tiered analysis, permitting the development of an infrastructurefor analysis and computation. The principal investigator and Michael Roseintroduced the initial comprehensive finite element study of the two-phase Stefan problem in 1982. In this proposal, iteration and discretization are studied as companion entities. A particularly significant aspect is the identification of the system map and its efficient approximation via smoothing and linearization. The cooperation of Monica Torres has beensecured to study the interface of these systems with homogenizationtheory and the theory of fully nonlinear partial differential equations.Mathematical models of free boundary problems build upon a rich tradition.Free boundary problems were studied intensively during the 1970s and1980s. A major impetus in the U.S. was the construction of the trans-Alaskapipeline, which required knowledge of the complex process of meltingof permafrost. The free boundary refers to the so-called mushy regionbetween solid and liquid phases. A mathematical model of temperature,called the two-phase Stefan problem, was employed for this purpose. One ofthe principal contributors to this effort was John Wheeler of the Exxon Corporation. Progress was slow in coming in the modeling of systems,where several effects may be present. Associated free boundary problems mayarise in many application areas, due to discontinuous switching induced bysources or sinks within the system. An environmental example is thedischarge of a contaminant species into a river, the regulation of thedischarge, and the ensuing response of control mechanisms to ameliorate thedischarge. Sophisticated systems of partial differential equations,incorporating physical principles describing the contaminant density andthe river temperature, are utilized, and include the switching phenomena. The incorporation of quantum principles into models of nanostructures alsoinduces switching and free boundary problems, as does the study of certainprocesses of combustion. Another application of these ideas, which is not pursued in this proposal, is the pricing of certain financialinstruments. An area of broader impact of the proposal has been identified in relation to the international community sponsoring the Workshops onComputational Electronics. The principal investigator has been an advisorypanel member since 1990, and details of some of this work, part of the effort toward making quantum computing possible, will be disseminated tothe community at a regularly scheduled workshop, and in the journal,Computational Electronics. Publication will also occur in regularmathematics journals.