Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^n$

TL;DR

This paper characterizes observable sets for the heat equation in ^n, establishing their equivalence with observability, interpolation, and spectral inequalities, and explores weak forms of these inequalities with localized observations.

Contribution

It provides a complete characterization of observable sets for the heat equation and proves their equivalence with key inequalities, advancing understanding of control and observation in PDEs.

Findings

Observable sets are characterized by -thickness at a certain scale.

Equivalence among observability, interpolation, and spectral inequalities is established.

Weak inequalities are derived for observations over a ball.

Abstract

This paper studies connections among observable sets, the observability inequality, the H\"{o}lder-type interpolation inequality and the spectral inequality for the heat equation in $\mathbb R^n$. We present a characteristic of observable sets for the heat equation. In more detail, we show that a measurable set in $\mathbb{R}^n$ satisfies the observability inequality if and only if it is $\gamma$-thick at scale $L$ for some $\gamma>0$ and $L>0$.We also build up the equivalence among the above-mentioned three inequalities. More precisely, we obtain that if a measurable set $E\subset\mathbb{R}^n$ satisfies one of these inequalities, then it satisfies others. Finally, we get some weak observability inequalities and weak interpolation inequalities where observations are made over a ball.