Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4

Сибирский математический журнал

M. V. Kuznetsov¹ 

[1] Sobolev Institute of Mathematics

关键词: sub-Laplacian;    nilpotent Lie group;    extension method;   

DOI  :  10.1134/S0037446619010129

学科分类：数学（综合）

来源: Izdatel stvo Instituta Matematiki Rossiiskoi Akademii Nauk

【 摘 要 】

Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where \(\Delta=X_1^2+X_2^2\) is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation).

【 授权许可】

CC BY