Geometric regularity estimates for quasi-linear elliptic models in non-divergence form with strong absorption
Claudemir Alcantara, João Vitor da Silva, Ginaldo Sá
TL;DR
The paper addresses sharp geometric regularity for quasi-linear elliptic models in non-divergence form with strong absorption, allowing degenerate or singular behavior when the gradient vanishes and admitting dead-core regions. The authors adopt a viscosity-solution framework and employ scaling, flatness, barrier constructions, and stability arguments to obtain an explicit growth rate near free boundary points, with the key exponent $\beta=\frac{\gamma+2}{\gamma+1-m}$, for $m\in[0,\gamma+1)$. They establish improved $C^{\kappa}_{\text{loc}}$ regularity along the free boundary $\mathscr{F}_0$, along with non-degeneracy, uniform positive density, porosity, and an $L^2$-average second-derivative bound; they also prove Liouville-type results for entire solutions and a sharp strong maximum principle in the critical case $m=\gamma+1$. The results extend to Henon-type and multi-phase models, offering a robust toolkit for understanding dead cores and free-boundary geometry in non-divergence form equations with strong absorption, with potential implications for game-theoretic diffusion models like Tug-of-War with noise.
Abstract
In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under strong absorption conditions of the form: \[ |\nabla u(x)|^γ Δ_p^{\mathrm{N}} u(x) = f(x, u) \quad \text{in} \quad B_1, \] where $γ> -1$, $p \in (1, \infty)$, and the mapping $u \mapsto f(x, u) \lesssim \mathfrak{a}(x) u_{+}^m$ (with $m \in [0, γ+ 1)$) does not decay sufficiently fast at the origin. This condition allows for the emergence of plateau regions, i.e., a priori unknown subsets where the non-negative solution vanishes identically. We establish improved geometric $\mathrm{C}^κ_{\text{loc}}$ regularity along the set $\mathscr{F}_0 = \partial \{u > 0\} \cap B_1$ (the free boundary of the model) for a sharp value of $κ\gg 1$, which is explicitly determined in terms of the structural parameters. Additionally, we derive non-degeneracy results and other measure-theoretic properties. Furthermore, we prove a sharp Liouville theorem for entire solutions exhibiting controlled growth at infinity.