Year: 2026
Type: preprint
Abstract: Zugmeyer~\cite{Zug} established sharp trace Gagliardo--Nirenberg--Sobolev inequalities on convex cones for $a\geq n>p>1$. We classify all nonzero real-valued extremals of these inequalities. Let the cone be the epigraph of a finite convex function that is positively homogeneous of degree one, and let $H$ be any norm. We prove that equality holds precisely for functions of the form $A H(x+te+\ell)^{-(a-p)/(p-1)}$, where $A\in\mathbb{R}\backslash\{0\}$, $t>0$, $e=(0,\ldots,0,1)$, and $\ell$ belongs to the largest vector subspace contained in the cone. The result includes the critical Sobolev trace inequality at $a=n$, as well as nonsmooth norms and cones, half-spaces, and cones containing lines.
Source: arXiv (Cornell University)
Authors: Zhou, Yuxuan, Zhang, Yingfang, Zou, Wenming
Institution: Tsinghua University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8593688678
OpenAlex ID: W7223082588