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

DA SILVA, JOÃO VITOR; ROSSI, JULIO D.; SALORT, ARIEL M.

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES

OXFORD UNIV PRESS

Lugar: oxford; Año: 2019 vol. 99 p. 69 - 96

0024-6107

We study regularity issues and the limiting behavior as p → ∞ of non‐negative solutions for elliptic equations of p − Laplacian type ( 2 ⩽ p < ∞ ) with a strong absorption:− Δ p u ( x ) + λ 0 ( x ) u + q ( x ) = 0 in Ω ⊂ R N ,−Δ𝑝𝑢(𝑥)+𝜆0(𝑥)𝑢𝑞+(𝑥)=0inΩ⊂ℝ𝑁, where λ 0 > 0 is a bounded function, Ω is a bounded domain and 0 ⩽ q < p − 1 . When p 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 p − dead core solutions. Afterwards, assuming that ℓ : = lim p → ∞ q ( p ) / p ∈ [ 0 , 1 ) exists, we establish existence for limit solutions as p → ∞ , as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp C γ regularity estimates for limit solutions along free boundary points, that is, points on ∂ { u > 0 } ∩ Ω where the sharp regularity exponent is given explicitly by γ = 1 / ( 1 − ℓ ) . Finally, some weak geometric and measure theoretical properties as non‐degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.