AI 中文总结
研究有限常系数贝尔曼方程，通过构造特定矩阵，找到具有非利普希茨黑塞矩阵的齐次解，揭示凸完全非线性一致椭圆方程的解不在\(C^{2,1}\)及\(C^{2, 1-\varepsilon}\)中。
AI 中文摘要
我们为有限常系数贝尔曼方程构造具有非利普希茨黑塞矩阵的齐次解。首先，对于每个\(\sigma\in(0,1)\)，在\(\mathbb{R}^4\)中找到两个一致椭圆矩阵\(A_1,A_2\in\mathcal{S}^4\)以及\(\max\bigl\{{\rm tr}\,(A_1D^2u),{\rm tr}\,(A_2D^2u)\bigr\}=0\)的非零\((2+\sigma)\)-齐次解\(u\)。其次，在\(\mathbb{R}^2\)中构造三个满足\({\rm Id}_2\leq A_j\leq3{\rm Id}_2\)的矩阵，其对应的贝尔曼方程有具有非利普希茨黑塞矩阵的齐次解。特别地，凸完全非线性一致椭圆方程的解不在\(C^{2,1}\)中，对于小的\(\varepsilon>0\)甚至不在\(C^{2, 1-\varepsilon}\)中。
英文摘要
We construct homogeneous solutions with non-Lipschitz Hessian for finite, constant-coefficient Bellman equations. First, for every $σ\in(0,1)$, we find two uniformly elliptic matrices $A_1,A_2\in\mathcal{S}^4$ and a nonzero $(2+σ)$-homogeneous solution $u$ of \[\max\bigl\{{\rm tr}\,(A_1D^2u),{\rm tr}\,(A_2D^2u)\bigr\}=0 \qquad\text{in }\mathbb{R}^4.\] Second, in $\mathbb{R}^2$ we construct three matrices satisfying ${\rm Id}_2\leq A_j\leq3{\rm Id}_2$ for which the corresponding Bellman equation admits a homogeneous solution with a non-Lipschitz Hessian. In particular, solutions to convex fully nonlinear uniformly elliptic equations are not in $C^{2,1}$, and not even in $C^{2, 1-\varepsilon}$ for $\varepsilon > 0$ small.
