The investigator and his colleagues study mathematical problems with large surfaces and small volumes that arise in physics (e.g., highly fragmented electrical conductors, high voltage electric discharges, electrolytic deposition, diffusion-limited aggregation), chemistry (catalytic converters, surface chemistry), biology (cell membranes, vascular systems), engineering (hydraulic fracturing in oil wells, thin ramified fibers, elastic thin bodies, towers and bridges in open space). In all these problems a lower-dimensional physical body -- the "surface" -- intrudes and interacts with a full dimensional surrounding body -- the "volume." The volume is small, the surface is large, possibly fractal and with infinite area. In this project, the mathematics of fractal structures in space is focused on two fundamental issues: 1) second order transmission conditions for second order operators; 2) singular homogenization with fractal terms. New tools of analysis are developed, like Hoelder metrics and measure-valued Lagrangeans. Finite element numerical approximations provide the quantitative and flexible setting required by prospective applications such as those mentioned above. The principal investigator and his colleagues develop new mathematics aimed at meeting the challenges and taking advantage of the new technologies of the time.
The mathematics developed in this project is motivated by, and oriented to, the study of applied problems of unusual type, occurring in the materials sciences and manufacturing, nanotechnology, biotechnology, and the environmental sciences. The unusual feature of these problems consists in the presence and interaction, within the same system, of two physical phenomena, one confined in a "small volume," the second unfolding itself on a "large surface." Nanotechnology can produce very fine fractal surfaces that can be assembled to form composite materials with enhanced internal surface effects (catalytic devices, absorbers, surface chemistry, radiators and similar). Environment-driven fractal networks can support irrigation of large surfaces with scarce fluid volume (as in vascular and biological systems). Interlaced thin fibers and fractal thin structures can tune wave propagation in space. Despite their complexity, these applications hinge on an unusual volume vs. surface relation of the kind mentioned before. The project involves young researchers and students -- with attention to underrepresented minorities -- who receive a broad, flexible training that provides access to science and technology as well to professional activities outside academic institutions.
