2018/07/22 by João Vitor da Silva, João Vítor da Silva, Julio D. Rossi +2 · 17 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Boundary (topology) #Bounded function #Degeneracy (biology) #Elliptic operator #Exponent #Laplace operator #Limit (mathematics) #Limiting #Mathematical analysis #Mathematics #Monotonic function #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Pure mathematics
paper · pdf · doi:10.1112/jlms.12161
published in Journal of the London Mathematical Society 99(1), 69-96 (Wiley)
openalex publication_date 2018/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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 , 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.