Special Session 47: Singular limits in elliptic and parabolic PDEs

On nondegeneracy and stability in the one-phase singular perturbation problem

Nikola Kamburov
Pontificia Universidad Catolica de Chile
In this talk I will present some recent results on the singular perturbation problem that formally approximates the classical one-phase free boundary problem. We define a natural density condition on the transition layers of the solutions that guarantees the uniform nondegeneracy property is satisfied and preserved in the limit. We then apply our result to the problem of classifying global stable solutions of the underlying semilinear problem and we show that those have flat level sets in dimensions $n \leq 4$, provided the density condition is fulfilled.