Regularity for the fully nonlinear dead-core problem

Abstract

We establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms. The model is given by a fully nonlinear elliptic equation with measurable coefficients and a $\mu$-Hölder continuous convection term, $F(X, D^2u) = f(u)$. The lack of Lipschitz regularity of the map $u \mapsto f(u)$ allows the existence of plateaus, i.e., nonnegative solutions may vanish identically within an a priori unknown region—the dead-core of the solution. We prove that at any touching ground point $Z \in \partial \{u>0 \}$, solutions are ${\varkappa (\mu )}$-differentiable for a sharp value $\varkappa (\mu ) \ge 2$, and in fact $\varkappa (1^{-}) = +\infty$. The proof is based on a new flatness improvement method. We apply this new regularity estimate to establish a Liouville-type theorem for entire solutions to dead-core problems and also to obtain measure estimates on the touching ground boundary. The results obtained in this article are new even for dead-core problems ruled by linear equations.