2025/01/22 by João Vítor da Silva, Ariel Salort, da Silva, João Vítor +1 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2501.13063
openalex publication_date 2025/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of p-Laplace type (1 < p< ∞) with strong absorption condition: -div (Φ(x, u, ∇ u)) + λ0(x) u+q(x) = 0 in Ω⊂ ℝN, where Φ: Ω× ℝ+ × ℝN → ℝN is a vector field with an appropriate p-structure, λ0 is a non-negative and bounded function and 0≤ q0\ ∩ Ω, where the regularity exponent is given explicitly by γ= (p)/(p-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 -Δp u + λ0 uqχ_\u>0\ = 0 for any λ0>0. \newline \newline \noindent Keywords: Quasi-linear elliptic operators of p-Laplace type, improved regularity estimates, Free boundary problems of dead core type, Liouville type results, Hausdorff measure estimates.