二维积分系统的一维对称性
One-dimensional symmetry for integral systems in two dimensions
摘要：
本文的目的是证明二维非局部积分方程组$$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$的稳定解的De Giorgi型结果，其中$u=(u_i)_{i=1}^m$对$u_i: \mathbb R^n\to \mathbb R$，$H=(H_i)_{i=1}^m$是一般非线性项.对于某些核$K$，算子$L$由$$L(u_i (x)):= \int_{\mathbb R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$给出。这个想法是应用线性刘维尔定理的商的偏导数，就像在证明经典的德乔治猜想在较低的维度。由于没有与上述算子相关的Caffarelli-Silvestre局部扩张问题，我们通过某些积分估计直接处理这个问题。
英文摘要：
The purpose of this brief paper is to prove De Giorgi type results for stable solutions of the following nonlocal system of integral equations in two dimensions $$ L(u_i) = H_i(u) \quad \text{in} \ \ \mathbb R^2 , $$ where $u=(u_i)_{i=1}^m$ for $u_i: \mathbb R^n\to \mathbb R$, $H=(H_i)_{i=1}^m$ is a general nonlinearity. The operator $L$ is given by $$L(u_i (x)):= \int_{\mathbb R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$ for some kernel $K$. The idea is to apply a linear Liouville theorem for the quotient of partial derivatives, just like in the proof of the classical De Giorgi's conjecture in lower dimensions. Since there is no Caffarelli-Silvestre local extension problem associated to the above operator, we deal with this problem directly via certain integral estimates.
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