Regularity for the fully nonlinear dead-core problem

We establish newgeometric 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-Holder continuous convection term, F(X, D(2)u) = f (u). The lack of Lipschitz regularity of the map u bar right arrow 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 is an element of partial derivative{u > 0}, solutions are aleph (mu)-differentiable for a sharp value aleph (mu) >= 2, and in fact aleph (1(-)) = +infinity. The proof is based on a newflatness 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.