Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heat equation on the Heisenberg group: observability and applications   We investigate observability and Lipschitz stability for the Heisenberg heat
equation on the rectangular domain $$\Omega =
(-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices
of the form $\omega=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $\omega
= (a,b) \times \omega_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that
observability fails for an arbitrary time $T>0$ but both observability and
Lipschitz stability hold true after a positive minimal time, which depends on
the distance between $\omega$ and the boundary of $\Omega$: $$T_{\min}
\geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our proof follows a mixed
strategy which combines the approach by Lebeau and Robbiano, which relies on
Fourier decomposition, with Carleman inequalities for the heat equations that
are solved by the Fourier modes. We extend the analysis to the unbounded domain
$(-1,1)\times\mathbb{T}\times\mathbb{R}$.

13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga