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

Abstract

This paper studies connections among observable sets, the observability inequality, the Hölder-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.

References in corpus (2)

• Nazarov's uncertainty principles in higher dimension
• Quantitative unique continuation for the heat equation with Coulomb potentials