1 results
THE DISTRIBUTIONAL
$k$-HESSIAN IN FRACTIONAL SOBOLEV SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 23 October 2019, pp. 496-507
- Print publication:
- June 2020
-
- Article
- Export citation
-
We introduce the notion of a distributional
$k$-Hessian (
$k=2,\ldots ,n$) associated with fractional Sobolev functions on 
$\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in 
$\mathbb{R}^{n}$. We show that the distributional 
$k$-Hessian is weakly continuous on the fractional Sobolev space 
$W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional 
$k$-Hessian is well defined in 
$W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if 
$W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.
