| Abstract: |
| This talk gives 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 the 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 in $\mathbb{R}^n$ 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. |
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