A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities
Abstract
In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type F(x,D2u)+ℋ(x,Du)=ℱ(x,u+,u−) in B1 ⊂ ℝn, where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely ℋ(x,ξ):= ⟨ ℬ(x),ξ⟩+ϱ(x)|ξ|σ, 0< σ≤2, σ≠ 1, and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type ℱ(x,u+,u−):=g(x)[(u+)m−(u−)m] for 0<m<1. Under suitable structural assumptions on the coefficients and the operator F, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order (σ≠ 1), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.
Related
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BibTeX
@misc{bessa2026fully,
title = {{A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities}},
author = {Junior da Silva Bessa and João Vitor da Silva and Yuwei Hu and Mayra Soares},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23216},
eprint = {2609.23216},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23216},
}
RIS
TY - UNPB
TI - A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities
AU - Junior da Silva Bessa
AU - João Vitor da Silva
AU - Yuwei Hu
AU - Mayra Soares
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23216
UR - https://arxiv.org/abs/2609.23216
AB - In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type F(x,D^2u)+ℋ(x,Du)=ℱ(x,u^+,u^−) in B_1 ⊂ ℝ^n, where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely ℋ(x,ξ):= ⟨ ℬ(x),ξ⟩+ϱ(x)|ξ|^σ, 0< σ≤2, σ≠ 1, and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type ℱ(x,u^+,u^−):=g(x)[(u^+)^m−(u^−)^m] for 0<m<1. Under suitable structural assumptions on the coefficients and the operator F, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order (σ≠ 1), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.
ER -