Fully nonlinear dead-core systems

We study fully nonlinear dead-core systems coupled with strong absorption terms. We discover a chain reaction, exploiting properties of an equation along the system. The lack of both the classical Perron's method and comparison principle for the systems requires new tools for tackling the problem. By means of a fixed point argument, we prove existence of solutions, and obtain higher sharp regularity across the free boundary. Additionally, we prove a variant of a weak comparison principle and derive several geometric measure estimates for the free boundary, as well as a Liouville type theorems for entire solutions. These results are new even for linear dead-core systems.