AI 中文总结
该论文构造特殊拉格朗日曲率方程的反例，二维中用平行曲面焦后分支构造非\(C^1\)解，二维矩形上构造\(C^1\)范数有界但曲率无界解，三维亚临界阶段构造梯度有跳跃间断的解，揭示相关先验估计中结构假设的必要性。
AI 中文摘要
我们为特殊拉格朗日曲率方程（SLCE）构造了三个反例。首先，在二维中，利用具有恒定正高斯曲率的平行曲面的焦后分支构造了一个非\(C^1\)的显式利普希茨粘性解。其次，同样在二维，在一个固定矩形上构造了一列光滑可允许解，其\(C^1\)范数一致有界但在一点处曲率无界，且对于\(\beta > 1/3\)任何一致的\(C^{1,\beta}\)估计都不成立。第三，在三维和亚临界阶段，构造了一个穆尼 - 萨文型利普希茨粘性解，其梯度在一个解析曲面上有跳跃间断。这些例子证明了邱和周最近的先验估计的精确性，表明其结构假设——凸性和临界阶段——是严格必要的。
英文摘要
We construct three counterexamples for the special Lagrangian curvature equation (SLCE). First, in dimension two, we use a post-focal branch of a parallel surface with constant positive Gauss curvature to construct an explicit Lipschitz viscosity solution which is not $C^1$. Second, still in dimension two, we construct a sequence of smooth admissible solutions on a fixed rectangle with uniformly bounded $C^1$-norm but unbounded curvature at one point; furthermore, we show that any uniform $C^{1,β}$ estimate fails for $β> 1/3$. Third, in dimension three and in the subcritical phase, we construct a Mooney-Savin type Lipschitz viscosity solution whose gradient has a jump discontinuity across an analytic surface. These examples demonstrate the sharpness of the recent a priori estimates by Qiu and Zhou, revealing that their structural assumptions-convexity and the critical phase-are strictly necessary.
