Regularity properties for p-dead core problems and their asymptotic limit as p to infinity

Abstract

We study regularity issues and the limiting behavior as (Formula presented.) of non-negative solutions for elliptic equations of (Formula presented.) Laplacian type ((Formula presented.)) with a strong absorption: (Formula presented.) where (Formula presented.) is a bounded function, (Formula presented.) is a bounded domain and (Formula presented.). When (Formula presented.) is fixed, such a model is mathematically interesting since it permits the formation of dead core zones, that is, a priori unknown regions where non-negative solutions vanish identically. First, we turn our attention to establishing sharp quantitative regularity properties for (Formula presented.) dead core solutions. Afterwards, assuming that (Formula presented.) exists, we establish existence for limit solutions as (Formula presented.), as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp (Formula presented.) regularity estimates for limit solutions along free boundary points, that is, points on (Formula presented.) where the sharp regularity exponent is given explicitly by (Formula presented.). Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.