Thursday, February 24, 2022 - 16:15 in H7 + Zoom
Asymptotic expansion of the fractional heat content
A talk in the Nonlocal Equations: Analysis and Numerics series by
Julia Lenczewska from Wrocław
| Abstract: |
For a Borel set $\Omega \subset \mathbb{R}^d$ and $\alpha \in (0,2)$, the fractional heat content of $\Omega$ is defined as
$$
H^{(\alpha)}_{\Omega}(t) = \int_{\Omega} \int_{\Omega^c} p^{(\alpha)}_t(x-y) \, \mathrm{d}y \mathrm{d}x,
$$
where $\mathcal{F} p^{(\alpha)}_t (\xi) = e^{-t|\xi|^{\alpha}}$. For $\alpha \in (0,1)$, the fractional perimeter of $\Omega$ is given by
$$
\operatorname{Per}_{(\alpha)} (\Omega) = \int_{\Omega} \int_{\Omega^c} \nu(x-y) \, \mathrm{d}y \mathrm{d}x,
$$
where $\nu(x) = \mathcal{A}_{d,-\alpha} |x|^{-d-\alpha}$. It is known that for $\alpha \in (0,1)$
the first two terms in the asymptotic expansion of the fractional heat content include the volume of the set and its fractional perimeter.
In this talk, we will present further terms of the asymptotic expansion. This is a joint work with Tomasz Grzywny (WUST).
Within the CRC this talk is associated to the project(s): A7 |
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