两相奇异摄动完全非线性椭圆方程的一致Lipschitz正则性
Uniform Lipschitz regularity for two-phase singularly perturbed fully nonlinear elliptic equations
浏览论文内容
中文总结 AI 辅助
该研究针对两相奇异摄动完全非线性椭圆方程的变号粘性解，证明了与摄动参数无关的一致Lipschitz正则性估计，消除了相关分析的长期紧性障碍，为后续锐界面分析提供了关键框架。
中文摘要 AI 辅助
我们研究奇异摄动完全非线性方程$F(D^2u_\varepsilon) = \frac{\alpha}{\varepsilon} \beta\left(\frac{u_\varepsilon}{\varepsilon}\right)$在$\mathbb{R}^n$中单位球$B_1$内的变号粘性解，其中$F$是一致椭圆算子，$\beta \in C_c(-1,1)$为非负函数。我们证明了尺度尖锐估计：$\\|\nabla u_\varepsilon\\|_{L^\infty(B_{1/2})} \leq C\left( \\|u_\varepsilon\\|_{L^\infty(B_1)}+\sqrt\alpha \right)$，其中常数$C$仅依赖于空间维数、椭圆性常数和$\beta$，且与$\varepsilon$和$\alpha$无关。这一结果消除了完全非线性两相奇异摄动分析中长期存在的紧性障碍。该问题的结构性难点在于：当$\varepsilon>0$时，既不存在自由边界也不存在规定的传输定律，而一般完全非线性框架下没有能够控制两相相互作用的单调性公式。证明过程发展了De Silva–Savin的衰减与Lipschitz二选一准则的扩散对应形式，精确平面过渡提供了局部比较几何，曲线测试紧性将该几何推广到坍缩反应层，本征过渡区域估计将问题简化为缓冲水平边界的线性增长，所得二进延拓通过大斜率停止论证闭合：累积斜率有界时直接得到所需增长，而斜率无界时归一化后有效反应强度消失，导致矛盾。该估计是定量最优的，为后续锐界面分析提供了尺度不变的紧性框架。
英文摘要
We study sign-changing viscosity solutions of the singularly perturbed fully nonlinear equation $$ F(D^2u_\varepsilon) = \fracα{\varepsilon} β\left(\frac{u_\varepsilon}{\varepsilon}\right) \qquad\text{in }B_1\subset\mathbb R^n, $$ where $F$ is uniformly elliptic and $β\in C_c (-1,1)$ is nonnegative. We prove the scale-sharp estimate $$ \|\nabla u_\varepsilon\|_{L^\infty(B_{1/2})} \leq C\left( \|u_\varepsilon\|_{L^\infty(B_1)}+\sqrtα\right), $$ with $C$ depending only on the dimension, the ellipticity constants, and $β$, and independent of $\varepsilon$ and $α$. This removes a longstanding compactness obstruction in the analysis of fully nonlinear two-phase singular perturbations. The difficulty is structural: at positive $\varepsilon$ there is neither a free boundary nor a prescribed transmission law, while the general fully nonlinear setting provides no monotonicity formula capable of controlling the interaction of the two phases. The proof develops a diffuse counterpart of the De Silva--Savin decay-versus-Lipschitz alternative. Exact planar transitions furnish the local comparison geometry, and curved-test compactness carries this geometry across collapsing reaction layers. An intrinsic transition-region estimate reduces the problem to linear growth from buffered level boundaries. The resulting dyadic continuation is closed by a large-slope stopping argument: bounded accumulated slopes yield the desired growth directly, whereas unbounded slopes force the effective reaction strength to vanish after normalization and lead to a contradiction. The estimate is quantitatively optimal and supplies the scale-invariant compactness framework required for the subsequent sharp-interface analysis.
