AI 中文总结
本文研究完全非线性两相Alt-Phillips自由边界问题，通过精细爆破和稳定性分析，建立了高阶奇异结点处的增长估计、定量非退化结果及Liouville定理，统一并推广了已有结论。
AI 中文摘要
本文分析了一类由带低阶项的完全非线性椭圆方程驱动的Alt-Phillips型两相自由边界问题。更确切地说，我们考虑如下形式的方程：\\[ F(x,D^{2}u)+\mathscr{H}(x,Du)=\mathscr{F}(x,u^{+},u^{-}) \quad \text{在 } B_1 \subset \mathbb{R}^{n} \text{中}，\\] 其中Hamiltonian项表现出混合线性和非线性梯度依赖，即 \\[ \mathscr{H}(x,\xi):= \langle \mathcal{B}(x),\xi\rangle+\varrho(x)|\xi|^{\sigma}, \quad 0< \sigma\leq2, \\,\\, \sigma \neq 1，\\] 反应项由半线性型的两相幂型非线性控制：\\[ \mathscr{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)\big[(u^{+})^{m}-(u^{-})^{m}\big] \quad \text{对于} \quad 0<m<1。\\] 在系数和算子$F$的适当结构假设下，我们发展了一个稳健的分析框架来捕捉自由边界附近解的精细性质。特别地，我们在高阶奇异结点（即两相相遇且解表现出临界退化的点）处建立了改进的增长估计。结果基于精细的爆破论证、稳定性和Liouville型定理。我们进一步证明了定量非退化结果，确保解在每个相上以受控速率脱离零。作为我们分析的推论，我们推导出该类中全局解的Liouville型定理。我们的结果扩展并统一了若干先前已知的情形，即使在存在阶为（$\sigma \neq 1$）的梯度依赖非线性时也是新的，从而涵盖了相变和反应扩散现象中出现的具有非线性漂移和吸收效应的模型。
英文摘要
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 \[ F(x,D^{2}u)+\mathscr{H}(x,Du)=\mathscr{F}(x,u^{+},u^{-}) \quad \text{in } B_1 \subset \mathbb{R}^{n}, \] where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely \[ \mathscr{H}(x,ξ):= \langle \mathcal{B}(x),ξ\rangle+\varrho(x)|ξ|^σ, \quad 0< σ\leq2, \,\,\, σ\neq 1, \] and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type \[ \mathscr{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)\big[(u^{+})^{m}-(u^{-})^{m}\big] \quad \text{for} \quad 0<m<1. \] 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 ($σ\neq 1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.
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