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Regularity properties for p − dead core problems and their asymptotic limit as p → ∞
Author(s) -
da Silva João Vitor,
Rossi Julio D.,
Salort Ariel M.
Publication year - 2019
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12161
Subject(s) - bounded function , limit (mathematics) , mathematics , limiting , degeneracy (biology) , exponent , laplace operator , mathematical analysis , boundary (topology) , elliptic operator , monotonic function , pure mathematics , mechanical engineering , bioinformatics , linguistics , philosophy , engineering , biology
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 , whereλ 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.