Geometric properties of free boundaries in degenerate quenching problems
Damião J. Araújo, Rafayel Teymurazyan, José Miguel Urbano · arXiv (Cornell University) · 2024
We study minimizers of non-differentiable functionals modeled on the degenerate quenching problem. Our main result establishes the finiteness of the $(n-1)-$dimensional Hausdorff measure of the free boundary. The proof is based on optimal gradient decay estimates obtained from an intrinsic Harnack-type inequality, along with a detailed analysis in a flatness regime, where minimizers enjoy improved regularity. Our arguments provide an alternative proof of classical results of Phillips and, although developed in the degenerate setting, also offer insights relevant to the singular case.