The project aims at further extending the scope and applicability of boundary value problems in domains with fractal boundaries, along three main lines of investigation: boundary value and transmission problems, singular homogenization, and variational fractals. The investigators extend the range of applicability of their theory of singular boundary value and transmission problems, by considering wider classes of 3-dimensional domains, with possibly irregularly or randomly scaled fractal boundaries and layers. They give fractal models more plausible physical ground, by approximating fractals with thin fibers and fractal equations with partial differential operators, bringing dynamical fractal theory into the more familiar realm of singular homogenization. Finally, the investigators enhance connections with classical effective properties of elliptic and sub-elliptic operators, by developing a theory of variational fractals that replaces similarity with quasi-metric scaling and provides a rigorous foundation to the power-law formalism, which is widely used in physics and applications in the study of phenomena with no characteristic length. The project, focused on constructive and metric methods, opens new perspectives in applied boundary value problems in domains with complicated and interacting boundaries and in homogenization theory with fractal components.
Problems with large surfaces and interfaces confined in small volumes occur in many applications, including material sciences (e.g., porous films and dendritic structures grown by diffusion limited aggregation), physics (e.g., highly fragmented electrical conductors, high voltage electric discharges, electrolytic deposition and diffusion-limited-aggregation), chemistry (e.g., catalytic converters), biology (e.g., cell membranes, biological tissues, bones), and engineering (e.g., hydraulic fracturing in oil wells). The project is situated at a crossroad of applied mathematics where analysis, numerical analysis, and probability meet. The project develops new theoretical methods and approximation tools specifically suited for these types of problems. The new constructive and metric approach provides the flexibility and the parameter variety required by prospective applications in physics, biology and engineering.
