Sharp regularity estimates for quasi-linear elliptic dead core problems and applications

Abstract

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type (1 < p< ∞) with strong absorption condition: (Formula presented.). R+× RN→ RNis a vector field with an appropriate p-structure, λ0is a non-negative and bounded function and 0 ≤ q< p- 1. Such a model permits existence of solutions with dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. We establish sharp and improved Cγregularity estimates along free boundary points, namely F0(u, Ω) = ∂{ u> 0 } ∩ Ω , where the regularity exponent is given explicitly by γ=pp-1-q≫1. Some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density and porosity of free boundary are proved. As an application, a Liouville-type result for entire solutions is established provided that their growth at infinity can be controlled in an appropriate manner. Finally, we obtain finiteness of (N- 1) -Hausdorff measure of free boundary for a particular class of dead core problems. The approach employed in this article is novel even to dead core problems governed by the p-Laplace operator - Δ pu+ λ0uqχ{ u > 0 }= 0 for any λ0> 0.