论文标题
椭圆形伪差异操作员的光谱不平等现象
Spectral inequalities for elliptic pseudo-differential operators on closed manifolds
论文作者
论文摘要
令$（m，g）$为封闭的riemannian歧管。这项工作的目的是证明Lebeau-Robbiano频谱不平等，以实现椭圆形伪差异操作员$ e（x，d）$ e（x，d）$ on $ m，订单$ν> 0，$ nhörmander类$ class $ class $ class $ψ^nν_{ρ，δ}（m）。为了应用这一基本结果，我们确定了与$ e（x，d）相关的（分数）热方程的无数可控制性。歧管，控制理论的经典结果，是由于Lebeau和Robbiano引起的光谱不平等，及其对热方程的零控制性的结果，在封闭的歧管的设置中为受试者提供了完整的图像。为了证明频谱不平等的证明，我们引入了一种由Ruzhansky和Turunen引起的全球伪差分计算启发的定期方法。
Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $ν>0,$ in the Hörmander class $Ψ^ν_{ρ,δ}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $ω\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral inequality for an elliptic operator and the null-controllability for its diffusion model) extend in the setting of closed manifolds, classical results of the control theory, as the spectral inequality due to Lebeau and Robbiano and their result on the null-controllability of the heat equation giving a complete picture of the subject in the setting of closed manifolds. For the proof of the spectral inequality we introduce a periodization approach in time inspired by the global pseudo-differential calculus due to Ruzhansky and Turunen.
