Homogeneous solutions to fully nonlinear elliptic equations
Homogeneous solutions to fully nonlinear elliptic equations
复制标题
全非线性椭圆方程的齐次解
DOI：
10.1090/s0002-9939-06-08367-5
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Yu Yuan
中科院分区：
文献类型：
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作者：
N. Nadirashvili;Yu Yuan
我们将齐次度 d 6 2 解分类为完全非线性椭圆方程。在这篇文章中，我们证明除了完全非线性椭圆方程的 2 解之外，任何齐次解都必须是“调和”的。考虑完全非线性椭圆方程 F D 2 u � = 0，其中 µI ≤ (Fij) = FMij (M) � ≤ µ −1 I。Nirenberg (N) 在 1950 年代导出了上述方程在 2 维中的先验 C 2,� 估计。 Krylov (K) 和 Evans (E) 在假设 F 是凸的情况下，在一般维度上对上述方程显示了相同的先验估计。作为对无凸性条件的一般完全非线性椭圆方程的先验估计的适度研究，我们研究了齐次解。定理0.1。令u 为R n 中椭圆方程F D 2 u � = 0 的R n \ {0} 齐次度d 6 2 连续解，其中F ∈ C 1 。那么 u 在 R n 中一个可能的新坐标系中是调和的，即 n X i,j=1 Fij (0)Diju(x) = 0。
We classify homogeneous degree d 6 2 solutions to fully nonlinear elliptic equations. In this note, we show that any homogeneous degree other than 2 solution to fully nonlinear elliptic equations must be "harmonic". Consider the fully nonlinear elliptic equation F D 2 u � = 0 with µI ≤ (Fij) = FMij (M) � ≤ µ −1 I. Nirenberg (N) derived the a priori C 2,� estimates for the above equa- tion in dimension 2 in 1950s. Krylov (K) and Evans (E) showed the same a priori estimates for the above equations in general dimensions under the assumption that F is convex. As a modest investigation of a priori estimates for general fully nonlinear elliptic equations without convexity condition, we study the homogeneous solutions. Theorem 0.1. Let u be a continuous in R n \ {0} homogeneous degree d 6 2 solution to the elliptic equation F D 2 u � = 0 in R n with F ∈ C 1 . Then u is harmonic in a possible new coordinate system in R n , namely n X i,j=1 Fij (0)Diju(x) = 0.